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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Stability of planar nonautonomous dynamic systems

Gro Hovhannisyan

Author Affiliations

Kent State University at Stark, 6000 Frank Ave. NW, Canton, OH, 44720-7599, USA

Advances in Difference Equations 2013, 2013:144  doi:10.1186/1687-1847-2013-144

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/144


Received:28 February 2013
Accepted:6 May 2013
Published:23 May 2013

© 2013 Hovhannisyan; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are describing the stable nonautonomous planar dynamic systems with complex coefficients by using the asymptotic solutions (phase functions) of the characteristic (Riccati) equation. In the case of nonautonomous dynamic systems, this approach is more accurate than the eigenvalue method. We are giving a new construction of the energy (Lyapunov) function via phase functions. Using this energy, we are proving new stability and instability theorems in terms of the characteristic function that depends on unknown phase functions. By different choices of the phase functions, we deduce stability theorems in terms of the auxiliary function of coefficients <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M1">View MathML</a>, which is invariant with respect to the lower triangular transformations. We discuss some examples and compare our theorems with the previous results.

MSC: 34D20.

Keywords:
nonautonomous dynamic system; stability; attractivity to the origin; asymptotic stability; asymptotic solutions; characteristic function; Lyapunov function; energy function

1 Introduction

We are interested in the behavior of a given solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of the nonlinear planar dynamic system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M3">View MathML</a>

(1.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M4">View MathML</a> are complex-valued functions from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M5">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M6">View MathML</a>. Since we are assuming that the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of (1.1) is given (fixed), system (1.1) may be considered as a linear nonautonomous system with coefficients <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M8">View MathML</a> depending only on a time variable.

Here and further, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M9">View MathML</a> is the set of k times differentiable functions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M11">View MathML</a> is the set of Lebesgue absolutely integrable functions on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M13">View MathML</a> is the set of functions of bounded variation on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10">View MathML</a>.

Dynamic system (1.1) is said to be stable if for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M15">View MathML</a> and for any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of (1.1) there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M20">View MathML</a>. Dynamic system (1.1) is said to be attractive (to the origin) if for every solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of (1.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M22">View MathML</a>

(1.2)

Dynamic system (1.1) is asymptotically stable if it is stable and attractive.

A solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of (1.1) is stable if for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M15">View MathML</a> there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, whenever <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M20">View MathML</a>.

A solution of (1.1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> is asymptotically stable (attractive to the origin) if (1.2) is true.

It is well-known that for a nonautonomous system with the complex eigenvalues <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M30">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M31">View MathML</a>, the classical Routh-Hurvitz condition of stability <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M32">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M31">View MathML</a>, fails. Indeed, nonautonomous system (1.1) with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M34">View MathML</a>

(1.3)

is unstable if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M35">View MathML</a>, although the Routh-Hurvitz condition is satisfied. Necessary and sufficient conditions of asymptotic stability of this system,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M36">View MathML</a>

(1.4)

could be found from the explicit solutions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M37">View MathML</a>

(1.5)

This example shows that the description of stability of nonautonomous dynamic systems in terms of the eigenvalues is not accurate.

The usual method of investigation of asymptotic stability of differential equations is the Lyapunov direct method that uses energy functions and Lyapunov stability theorems [1-3].

The asymptotic representation of solutions and error estimates in terms of the characteristic function was used in [4-6] to prove asymptotic stability. In this paper we describe the stable dynamic systems by using energy approach with the use of the characteristic function (see (1.7) below), which is a more accurate tool than the eigenvalues.

The main idea of this paper is to construct the energy function in such a way that the time derivative of this energy is the linear combination of the characteristic functions (see (2.15) below). Using this energy, we prove main stability theorems for two-dimensional systems in terms of unknown phase functions (see Theorems 3.1-3.3).

Theorems 3.1-3.3 are similar to Lyapunov stability theorems with additional construction of an energy function in terms of the phase functions. Theorems 3.1-3.3 are applicable to a wide range of nonlinear systems with complex-valued coefficients (see Example 5.2 below or the linear Dirac equation with complex coefficients in [7]) since they have the flexibility in the choice of an energy function.

To show that our theorems are useful, we deduce different versions of stability theorems (old well-known and some new ones) by using different phase functions as asymptotic solutions of the characteristic equation (see (2.8) below). Moreover, we formulate some of the conditions of stability in terms of the auxiliary function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M1">View MathML</a> (see (2.10) below), which is invariant with respect to the lower triangular transformations (see Theorem A.1). Note that there is no universal stability theorem in terms of coefficients for nonautonomous system (1.1) since there is no universal formula for an asymptotic solution of the characteristic equation.

As an application (see Example 5.5), we prove the asymptotic stability of the nonlinear Matukuma equation from astrophysics [8,9].

Consider the second-order linear equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M39">View MathML</a>

(1.6)

Define the characteristic (Riccati) equation of (1.6)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M40">View MathML</a>

(1.7)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M41">View MathML</a> is said to be the characteristic function, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M42">View MathML</a> are the phase functions. In Section 6 (see Lemma 6.1) the following lemma is proved.

Lemma 1.1Assume that every solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M43">View MathML</a>of (1.6) approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M45">View MathML</a>

(1.8)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M46">View MathML</a>are solutions of characteristic equation (1.7).

In the proof of Lemma 1.1, it is shown that (1.8) is also a sufficient condition of attractivity of solutions of (1.6) to the origin under additional condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M47">View MathML</a>

(1.9)

If the asymptotic behavior of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M48">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a> is known, then the condition of attractivity (1.8) could be clarified. Unfortunately, there is no a simple formula for asymptotic behavior of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M48">View MathML</a> depending on the behavior of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M51">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M52">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>. Anyway, under some restrictions, one can obtain stability theorems for (1.6) by considering different asymptotic expansions of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M48">View MathML</a>.

Assume that for some positive constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M55">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M57">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M58">View MathML</a>

(1.10)

Theorem 1.2 (Ignatyev [10])

Suppose that the functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M59">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M60">View MathML</a>are real, and they satisfy conditions (1.10) and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M61">View MathML</a>

(1.11)

Then linear equation (1.6) is asymptotically stable.

Condition that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M62">View MathML</a> is bounded above in (1.10) was removed in [11].

Note that if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M63">View MathML</a>

(1.12)

then condition (1.8) turns to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M64">View MathML</a>

(1.13)

and is an integral version of (1.11).

In [12] Ballieu and Peiffer introduced a more general condition than Ignatyev’s one (1.11) for the attractivity (see (1.15), (1.16) below) of a nonlinear second-order equation.

Theorem 1.3 (Pucci-Serrin [9], Theorem B)

Suppose that functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M59">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M60">View MathML</a>are real, and there exists a non-negative continuous function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M67">View MathML</a>of bounded variation on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M10">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M69">View MathML</a>

(1.14)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M70">View MathML</a>

(1.15)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M71">View MathML</a>

(1.16)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M72">View MathML</a>

(1.17)

then every bounded solution of the nonlinear equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M73">View MathML</a>

(1.18)

tends to zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

In this paper we prove general stability Theorems 3.1-3.3 in terms of unknown phase functions. Using these theorems we derive the versions of stability theorem of Pucci-Serrin [9], Smith [13], and some new ones.

2 Energy and some other auxiliary functions

Assuming <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M75">View MathML</a>, consider the following second-order nonlinear equation associated with system (1.1):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M76">View MathML</a>

(2.1)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M77">View MathML</a>

(2.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M78">View MathML</a>

(2.3)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M79">View MathML</a>

(2.4)

Remark 2.1 Note that using equation (1.1), one can eliminate dependence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M80">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M81">View MathML</a>. Indeed <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M82">View MathML</a>. Similar calculations show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M83">View MathML</a> depends only on t, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>, coefficients <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M4">View MathML</a>, and their derivatives.

Here and further, often we suppress the dependence on t and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> for simplicity.

Introduce the characteristic function of (2.1) that depends on an unknown phase function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M88">View MathML</a>

(2.5)

and the auxiliary function:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M89">View MathML</a>

(2.6)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M90">View MathML</a>

(2.7)

Define the characteristic (Riccati) function of system (1.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M91">View MathML</a>

(2.8)

Equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M92">View MathML</a> is the characteristic equation of system (1.1). For diagonal system (1.1), formulas (2.8) fail (for this case, see (A.23)).

Introduce the auxiliary functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M93">View MathML</a>

(2.9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M94">View MathML</a>

(2.10)

To explain the motivation for the choice of an energy function for system (1.1) (assuming <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M95">View MathML</a>), consider a representation of solutions of (1.1) in Euler form (see [6]):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M96">View MathML</a>

(2.11)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M97">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>, are exact solutions of the characteristic equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M99">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M100">View MathML</a> are defined as in (2.7), and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M101">View MathML</a>

(2.12)

For the case of linear system (1.1), representation (2.11) gives the general solution of (1.1), where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M102">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M103">View MathML</a> are constants. For a nonlinear system, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M102">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M103">View MathML</a> depend on a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>. Solving equations (2.11) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M107">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M108">View MathML</a>

(2.13)

Replacing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M97">View MathML</a> by arbitrary differentiable functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M110">View MathML</a>, we define auxiliary energy functions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M111">View MathML</a>

(2.14)

Remark 2.2 Although (2.14) are not constants for a nonlinear or nonautonomous system, they are useful for the study of stability. One can expect that for an appropriate choice of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87">View MathML</a> these energy functions are approximately conservative expressions for some nonlinear systems that are close to linear.

The derivative of the energy functions (2.14) may be written (see (6.23) below) as a linear combination of the characteristic functions:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M113">View MathML</a>

(2.15)

From (2.15) it follows that if for any given solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of (1.1) the phase functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87">View MathML</a> satisfy characteristic equation, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M116">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>, then energy conservation laws <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M118">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a> are satisfied.

Otherwise, (2.15) means that the error of asymptotic solutions is measured by the characteristic function.

Define (total) energy function as a non-negative quadratic form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M120">View MathML</a>

(2.16)

Remark 2.3 If the phase functions are chosen as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M121">View MathML</a>

(2.17)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M122">View MathML</a> is an arbitrary differentiable function, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M123">View MathML</a>

(2.18)

3 Stability theorems in terms of unknown phase functions

In this section we formulate the main Theorems 3.1-3.3 of the paper.

Theorem 3.1Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125">View MathML</a>, and there exist the complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M126">View MathML</a>and the real numbers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127">View MathML</a>, αsuch that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M130">View MathML</a>

(3.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M131">View MathML</a>

(3.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M132">View MathML</a>

(3.3)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M133">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M134">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M135">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M136">View MathML</a>

(3.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M137">View MathML</a>

(3.5)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is stable.

Remark 3.1 Since stability conditions (3.1)-(3.3) of Theorem 3.1 are given in terms of estimates with constants that depend on solutions of (1.1), system (1.1) is stable if these estimates are satisfied uniformly for all solutions (with constants that do not depend on solutions).

Remark 3.2 Note that for a linear nonautonomus system (1.1) with the choice <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M139">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M141">View MathML</a>, the error function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M142">View MathML</a> and conditions (3.1), (3.3) are close to the necessary and sufficient condition of the stability.

Theorem 3.2Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125">View MathML</a>, there exist the complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M126">View MathML</a>, and the real numbers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127">View MathML</a>, αsuch that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>and conditions (3.1), (3.2),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M149">View MathML</a>

(3.6)

are satisfied with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M134">View MathML</a>as in (3.4), (3.5).

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is asymptotically stable.

Theorem 3.3Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125">View MathML</a>, and there exist the complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M126">View MathML</a>such that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M157">View MathML</a>

(3.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M158">View MathML</a>

(3.8)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159">View MathML</a>is defined in (3.5), and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M160">View MathML</a>

(3.9)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is unstable.

Example 3.1 From Theorem 3.3 it follows that the linear canonical equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M162">View MathML</a>

(3.10)

is unstable.

Remark 3.3 If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M163">View MathML</a>

(3.11)

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M164">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165">View MathML</a> and condition (3.2) is satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M166">View MathML</a>.

Otherwise (3.2) is satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M167">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M168">View MathML</a>.

Under condition (3.11), condition (3.1) turns to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M169">View MathML</a>

which is satisfied if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M170">View MathML</a>

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M171">View MathML</a>

(3.12)

Sometimes it is convenient to use other than (3.4) formula for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M172">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M173">View MathML</a>

(3.13)

Remark 3.4 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M174">View MathML</a>, and there exists a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M175">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M176">View MathML</a>

(3.14)

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M177">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M178">View MathML</a>. In this case formula (3.5) is simplified

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M179">View MathML</a>

(3.15)

and we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M180">View MathML</a> . From Theorem 3.1 it follows that in this case the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a> of system (1.1) is asymptotically stable if for some real numbers α, l

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M182">View MathML</a>

(3.16)

are satisfied (see (3.13), (3.6)).

Note that (3.16) is a nonautonomous analogue of the classical asymptotic stability criterion of Routh-Hurvitz.

If the phase functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M183">View MathML</a> are chosen by formula (2.17), then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M184">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M185">View MathML</a>

(3.17)

From Theorems 3.1-3.3 one can deduce stability theorems for second-order equation (2.1). The attractivity to the origin for the solution of equation (2.1) is valid even by removing condition (3.1) (compare Theorem 3.2 with the following theorem).

Theorem 3.4Suppose that for a given solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), there exist the complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M187">View MathML</a>such that conditions (3.2), (3.6) are satisfied with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M188">View MathML</a>defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M189">View MathML</a>

(3.18)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M190">View MathML</a>

(3.19)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1) approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

Choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M193">View MathML</a>

(3.20)

from Theorem 3.1 (in view of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M194">View MathML</a>), we obtain the following theorem.

Theorem 3.5Suppose that for a given solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125">View MathML</a>, and there exist complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M197">View MathML</a>such that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M200">View MathML</a>

(3.21)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M201">View MathML</a>

(3.22)

and (3.6) are satisfied, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M202">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M203">View MathML</a>

(3.23)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M204">View MathML</a>

(3.24)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is asymptotically stable.

By choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M206">View MathML</a>

(3.25)

we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M207">View MathML</a>, and assuming (3.11) we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165">View MathML</a>. From Theorem 3.2 we deduce the following theorem.

Theorem 3.6Suppose that for a given solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125">View MathML</a>, and there exist complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M197">View MathML</a>such that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M214">View MathML</a>

(3.26)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M215">View MathML</a>

(3.27)

and (3.6) are satisfied with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159">View MathML</a>is as in (3.5), and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M217">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M218">View MathML</a>

(3.28)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is asymptotically stable.

Theorem 3.7Suppose that for a given solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M125">View MathML</a>, there exist complex-valued function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M222">View MathML</a>and the real numbers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127">View MathML</a>, αsuch that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>and the conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M226">View MathML</a>

(3.29)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M227">View MathML</a>

(3.30)

equation (3.3) (or (3.6)) are satisfied, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M228">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M229">View MathML</a>

(3.31)

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M230">View MathML</a>

(3.32)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is stable (or asymptotically stable).

Theorem 3.8Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M233">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M234">View MathML</a>, there exist the real numbers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127">View MathML</a>, αand the complex-valued function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M236">View MathML</a>such that for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, conditions (3.29) and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M238">View MathML</a>

(3.33)

are satisfied, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M122">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159">View MathML</a>are given by (3.29), (3.32).

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of equation (2.1) approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242">View MathML</a>.

4 Stability of the planar dynamic systems

From Theorems 3.1-3.3 one can deduce more useful asymptotic stability theorems in terms of coefficients of (1.1) by choosing the phase functions as asymptotic solutions of the characteristic equation.

Theorem 4.1Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M244">View MathML</a>, and for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>the conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M246">View MathML</a>

(4.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M247">View MathML</a>

(4.2)

and (3.3) (or (3.6)) are satisfied, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M248">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M249">View MathML</a>

(4.3)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M250">View MathML</a>

(4.4)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is stable (or asymptotically stable).

Theorem 4.2Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M253">View MathML</a>, and for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M256">View MathML</a>

(4.5)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M257">View MathML</a>

(4.6)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is asymptotically stable.

Theorem 4.3Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M253">View MathML</a>, for some numbers<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127">View MathML</a>, α, and for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M264">View MathML</a>

(4.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M265">View MathML</a>

(4.8)

and (3.3) (or (3.6)) are satisfied with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M266">View MathML</a>, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M267">View MathML</a>

(4.9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M268">View MathML</a>

(4.10)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is stable (or asymptotically stable).

Example 4.1 From Theorem 4.3 it follows that system (1.1) with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M270">View MathML</a>

(small damping) is asymptotically stable.

By using Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation, we will prove the following theorem.

Theorem 4.4Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of (1.1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M272">View MathML</a>, for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, the conditions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>, (4.1),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M275">View MathML</a>

(4.11)

and (3.3) (or (3.6)) are satisfied, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M276">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M277">View MathML</a>

(4.12)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M278">View MathML</a>

(4.13)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is stable (or asymptotically stable).

The following theorem is proved by using the Hartman-Wintner approximation [14].

Theorem 4.5Suppose for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M244">View MathML</a>, there exist the constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M127">View MathML</a>, αsuch that and for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M129">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M285">View MathML</a>

(4.14)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M286">View MathML</a>

(4.15)

and (3.3) (or (3.6)) are satisfied, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M288">View MathML</a>

(4.16)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M289">View MathML</a>

(4.17)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is stable (or asymptotically stable).

Remark 4.1 Note that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M291">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M292">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M293">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M294">View MathML</a>

In this case, asymptotic stability condition (3.6) is simplified:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M295">View MathML</a>

(4.18)

Remark 4.2 For the Euler equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M296">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M297">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M298">View MathML</a>, and the Hartman-Wintner approximation fails. To consider this case, one may consider the choice <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M299">View MathML</a> with the other phase function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M300">View MathML</a> that could be found by solving the equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M301">View MathML</a> (see (6.56)).

The following theorem is deduced from Theorem 4.1 by taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M302">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M303">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M304">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M194">View MathML</a>.

Theorem 4.6Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M244">View MathML</a>and for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M309">View MathML</a>and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M310">View MathML</a>

(4.19)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M311">View MathML</a>

(4.20)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M312">View MathML</a>

(4.21)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M313">View MathML</a>

(4.22)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>of system (1.1) is asymptotically stable.

5 Stability theorems for the equations with real coefficients

Theorem 5.1Assume that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M316">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M317">View MathML</a>are real-valued, for some positive constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M318">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>, the conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M320">View MathML</a>

(5.1)

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M321">View MathML</a>

(5.2)

are satisfied.

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of equation (2.1) is asymptotically stable.

Example 5.1 By Theorem 5.1 the canonical linear equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M323">View MathML</a>

(5.3)

is asymptotically stable if one of the following conditions is satisfied:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M324">View MathML</a>,

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M325">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M326">View MathML</a>,

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M328">View MathML</a>,

(iiii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M330">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M331">View MathML</a>.

A region of asymptotic stability of equation (5.3) described in Example 5.1 may be extended to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M332">View MathML</a>

(5.4)

by using another asymptotic solution of (5.3) (see Example 5.4 or [15,16]).

Theorem 5.2Assume that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M334">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M317">View MathML</a>are real-valued, and for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M337">View MathML</a>

(5.5)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of equation (2.1) approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242">View MathML</a>.

Theorem 5.3Assume that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M341">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342">View MathML</a>are real-valued, and for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M344">View MathML</a>

(5.6)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M345">View MathML</a>

(5.7)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of equation (2.1) is asymptotically stable.

Theorem 5.4Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342">View MathML</a>are real functions, and condition (5.7) is satisfied. Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

Example 5.2 By Theorem 5.3 the equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M352">View MathML</a>

(5.8)

(where β, σ, μ are real numbers and b, k, γ are positive numbers) is asymptotically stable.

Theorem 5.5Assume that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342">View MathML</a>are real functions and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M356">View MathML</a>

(5.9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M357">View MathML</a>

(5.10)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>is asymptotically stable.

Theorem 5.6Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M341">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361">View MathML</a>are real and condition (5.10) is satisfied. Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

Example 5.3 By Theorem 5.5 the linear equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M364">View MathML</a>

(5.11)

is asymptotically stable.

Theorem 5.7Assume that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M366">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361">View MathML</a>are real functions, and for all<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M369">View MathML</a>

(5.12)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M370">View MathML</a>

(5.13)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M371">View MathML</a>

(5.14)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1) is asymptotically stable.

Example 5.4 From Theorem 5.7 the asymptotic stability of the equation (see also [9,15,16]) follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M373">View MathML</a>

(5.15)

Example 5.5 By Theorem 5.7, the nonlinear Matukuma equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M374">View MathML</a>

(5.16)

is asymptotically stable.

Theorem 5.8Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361">View MathML</a>are real functions, and the conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M378">View MathML</a>

(5.17)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M379">View MathML</a>

(5.18)

are satisfied, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M380">View MathML</a>

(5.19)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M381">View MathML</a>

(5.20)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1) approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

Remark 5.1 By taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M384">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M385">View MathML</a>, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M386">View MathML</a>, and Theorem 5.8 becomes a version of Pucci-Serrin Theorem 1.3. In this case, (5.18) is simplified to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M387">View MathML</a>

(5.21)

Example 5.6 Due to Theorem 5.8, every solution of (1.6) with

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M388">View MathML</a>

approaches zero as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>, since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M390">View MathML</a>

Theorem 5.9Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the coefficients<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M348">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M361">View MathML</a>are real functions, and for some constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M394">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M395">View MathML</a>

(5.22)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M396">View MathML</a>

(5.23)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M397">View MathML</a>

(5.24)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

Theorem 5.10Suppose that for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1), the functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M401">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M342">View MathML</a>are real and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M403">View MathML</a>

(5.25)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M404">View MathML</a>

(5.26)

Then the solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M186">View MathML</a>of (2.1) approaches zero as<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>.

If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M407">View MathML</a>

(5.27)

then the attractivity condition (5.25) is simplified

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M408">View MathML</a>

(5.28)

Note that (5.28) is Smith’s [13] necessary and sufficient condition of asymptotic stability of (2.1) in the case of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M409">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M410">View MathML</a> .

Theorems 5.1-5.10 are new versions of the stability theorem proved in [1-5,9-13,17-21] by a different technique of construction of the energy function.

6 Proofs

Lemma 6.1Assume that all the solutions of linear system (1.1) are attractive to the origin, and functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M411">View MathML</a>are solutions of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M412">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M414">View MathML</a>

(6.1)

Proof of Lemma 6.1 and Lemma 1.1 First, we derive formula (2.8) for the characteristic function. Solving for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M415">View MathML</a> the first equation of (1.1), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M416">View MathML</a>

(6.2)

To eliminate <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M415">View MathML</a>, we substitute it in the second equation of (1.1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M418">View MathML</a>, so we get (2.1): <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M419">View MathML</a>, where P, Q are as in (2.2). From definition (2.5), we get (2.8). Formula (A.22) (see the Appendix) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M420">View MathML</a> is proved similarly by elimination of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M421">View MathML</a>.

The first component of a solution of linear system (1.1) may be represented in the Euler form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M422">View MathML</a>

(6.3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M423">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a> are solutions of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M425">View MathML</a>. From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M184">View MathML</a> we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M427">View MathML</a>

(6.4)

Since we are assuming that the solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M100">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a> of linear system (1.1) are attractive to the origin, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M430">View MathML</a>

(6.5)

as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>, that is, (6.1) is satisfied. Note that if additional condition (1.9) is satisfied, then (6.1) is also a sufficient condition of attractivity of solutions of (1.6), since in view of (6.5) as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M433">View MathML</a>

To prove Lemma 1.1, rewrite equation (1.6) in the form of system (1.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M434">View MathML</a>

(6.6)

which means that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M435">View MathML</a>

(6.7)

Then (1.8) follows from (6.1). □

Lemma 6.2If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436">View MathML</a>is a Hermitian<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M437">View MathML</a>matrix with the entries<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M438">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M439">View MathML</a>

(6.8)

then the matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436">View MathML</a>is non-negative (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M441">View MathML</a>), and for any 2-vectoru

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M442">View MathML</a>

(6.9)

Remark 6.1 If

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M443">View MathML</a>

(6.10)

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M444">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M445">View MathML</a>

(6.11)

Proof of Lemma 6.2 From the quadratic equation for the real eigenvalues of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M447">View MathML</a>

(6.12)

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M448">View MathML</a>

(6.13)

From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M449">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M450">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M451">View MathML</a>

(6.14)

Further from

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M452">View MathML</a>

(6.15)

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M453">View MathML</a>

(6.16)

 □

Lemma 6.3If there exist the complex-valued functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M454">View MathML</a>, and a real-valued function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M455">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M456">View MathML</a>

(6.17)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M457">View MathML</a>

(6.18)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159">View MathML</a>is defined in (3.5), then the energy inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M459">View MathML</a>

(6.19)

is satisfied, where the energy functions are defined in a more general form than in (2.14):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M460">View MathML</a>

(6.20)

Proof of Lemma 6.3 Denoting

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M461">View MathML</a>

(6.21)

we can rewrite energy formula (6.20) in the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M462">View MathML</a>

(6.22)

By differentiation, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M463">View MathML</a>

(6.23)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M464">View MathML</a>

(6.24)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M465">View MathML</a>

(6.25)

By direct calculations

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M466">View MathML</a>

(6.26)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M467">View MathML</a>

(6.27)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M468">View MathML</a>

(6.28)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M469">View MathML</a>

(6.29)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M470">View MathML</a>

(6.30)

Further

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M471">View MathML</a>

(6.31)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M472">View MathML</a>

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M473">View MathML</a>

(6.32)

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M474">View MathML</a>

or using notation (3.5), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M475">View MathML</a>

(6.33)

By Lemma 6.2 to have the non-negativity of the matrix N (with the entries <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M476">View MathML</a>), it is sufficient to show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M477">View MathML</a>

The first condition is condition (6.17), and the second condition follows from (6.18) and (6.31):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M478">View MathML</a>

So, from conditions (6.17), (6.18) it follows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M479">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M480">View MathML</a>

(6.34)

or (6.19) by integration. □

Lemma 6.4If the phase functions<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M481">View MathML</a>are such that (3.1) is satisfied, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M482">View MathML</a>

(6.35)

Proof of Lemma 6.4 Introducing the Hermitian matrix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M436">View MathML</a> with the entries <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M438">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M485">View MathML</a>

(6.36)

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M486">View MathML</a>

(6.37)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M487">View MathML</a>

(6.38)

From condition (3.1) we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M488">View MathML</a>

(6.39)

Further, by using Lemma 6.2, we obtain (6.35)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M489">View MathML</a>

(6.40)

 □

Proof of Theorem 3.1 First let us check that under the conditions of Theorem 3.1, Lemma 6.3 is applicable. Condition (6.18) is satisfied by choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M490">View MathML</a>

(6.41)

Condition (6.17) is satisfied as well in view of condition (3.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M491">View MathML</a>

From Lemma 6.3 and Lemma 6.4, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M492">View MathML</a>

(6.42)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M493">View MathML</a>

(6.43)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M172">View MathML</a> is defined as in (3.13):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M495">View MathML</a>

(6.44)

Substituting here formula (2.9) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M496">View MathML</a>, we get (3.4). Further from (3.3) and (6.43) the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M497">View MathML</a> and the stability follow. □

Proof of Remark 3.2 Note that if for linear system (1.1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M498">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M500">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M501">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M502">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M503">View MathML</a>, and solutions of (1.1) could be represented in the form (see (6.2))

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M504">View MathML</a>

Solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M505">View MathML</a> of (1.1) is bounded and stable if and only if for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M98">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M508">View MathML</a>

These exact conditions are close to conditions (3.1), (3.3) of Theorem 3.1 which, under assumption <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M116">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M500">View MathML</a>, turn to (see also (3.13))

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M511">View MathML</a>

 □

Proof of Theorem 3.2 From (3.1), (3.2) we get estimate (6.43) as in the proof of Theorem 3.1. Further from (3.6) and (6.43) the boundedness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M497">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M513">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>, that is, the asymptotic stability, follow. □

Proof of Theorem 3.3 Choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M515">View MathML</a>

(6.45)

we have again <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M516">View MathML</a>. In view of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M517">View MathML</a>

from assumption (3.7), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M518">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M519">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M520">View MathML</a>

which implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M521">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M516">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M523">View MathML</a>, and from (6.25) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M524">View MathML</a>.

So,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M525">View MathML</a>

(6.46)

or by integration

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M526">View MathML</a>

(6.47)

where μ is the largest eigenvalue of the non-negative matrix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M527">View MathML</a>.

Since both eigenvalues of the matrix K are non-negative, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M528">View MathML</a>

(6.48)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M529">View MathML</a>

(6.49)

From this estimate and (3.8), it follows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M530">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M242">View MathML</a>. □

Proof of Example 3.1 We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M532">View MathML</a>

(6.50)

for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M19">View MathML</a>, and T sufficiently big positive. Choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M534">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M535">View MathML</a>

So, conditions (3.7), (3.8) are satisfied, and from Theorem 3.3 it follows that equation (3.10) is unstable. □

Proof of Theorem 3.4 Consider equation (2.1) written in the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M536">View MathML</a>

(6.51)

Let us choose

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M537">View MathML</a>

(6.52)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M159">View MathML</a> is defined in (3.5) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M539">View MathML</a>. Then the conditions of Lemma 6.3 are satisfied, and we get from Lemma 6.3

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M540">View MathML</a>

(6.53)

where the matrix K is defined in (6.36). Since from (3.11) it follows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165">View MathML</a>, by applying Lemma 6.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M542">View MathML</a>

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M543">View MathML</a>

It means that for equation (2.1) we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M544">View MathML</a>

From (3.11) it follows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M545">View MathML</a>, and we have also

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M546">View MathML</a>

(6.54)

Further, using notation (3.18), (2.6) from (6.52), (6.53), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M547">View MathML</a>

and from (3.6) it follows <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M548">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M44">View MathML</a>. □

Proof of Theorem 3.7 By substitution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M550">View MathML</a>

(6.55)

functions (2.8), (2.9) may be simplified

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M551">View MathML</a>

(6.56)

Theorem 3.7 follows from Theorem 3.1, Theorem 3.2 by taking a given function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M552">View MathML</a> and choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M553">View MathML</a>, and phase function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M554">View MathML</a> as follows (see (6.55)):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M555">View MathML</a>

(6.57)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M556">View MathML</a>

(6.58)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M557">View MathML</a>

(6.59)

Further from (6.56), (3.15)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M558">View MathML</a>

(6.60)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M559">View MathML</a>

(6.61)

So, conditions (3.1), (3.2) turn to (3.29), (3.30). From (3.13) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M560">View MathML</a>

(6.62)

or (3.31). □

Proof of Theorem 3.8 Theorem 3.8 follows from Theorem 3.4 applied to the system (6.51). By choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561">View MathML</a> and θ as in (3.31), in view of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M562">View MathML</a>, we get (3.33) from (3.6) and (3.18). □

Proof of Theorem 4.1 Theorem 4.1 follows from Theorems 3.1 and 3.2 by choosing, as the approximate solutions of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M92">View MathML</a> (see (6.56)), the eigenvalue approximation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M564">View MathML</a>

(6.63)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M565">View MathML</a>

(6.64)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M566">View MathML</a>

(6.65)

Condition (3.1) turns to (3.12) (see Remark 3.3), or to (4.2).

In view of (6.64) and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M567">View MathML</a>

(6.66)

we get from (3.4), (3.5) formulas (4.3), (4.4):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M568">View MathML</a>

(6.67)

From (4.1) we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M165">View MathML</a>, and condition (3.2) is satisfied since from (4.4) we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M570">View MathML</a>. □

Proof of Theorem 4.2 Theorem 4.2 follows from Theorems 3.1 and 3.2 by choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M571">View MathML</a>, and the special Riccati equation approximation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M572">View MathML</a>

(6.68)

By direct calculations,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M573">View MathML</a>

(6.69)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M574">View MathML</a>

(6.70)

Condition (3.2) is true, since (3.11) is satisfied (see Remark 3.3). Condition (3.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M575">View MathML</a> turns to (3.12): <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M576">View MathML</a> or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M577">View MathML</a>

which follows from (4.5). From (3.15), since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M578">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M579">View MathML</a>

(6.71)

Further from (3.13) we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M580">View MathML</a>

(6.72)

and condition (3.6) turns to (4.6). □

Proof of Theorem 4.3 Theorem 4.3 follows from Theorem 3.7 by choosing the linear equation approximation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M581">View MathML</a>

(6.73)

 □

Proof of Example 4.1 Example 4.1 follows from Theorem 4.3. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M582">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M583">View MathML</a>

Choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M584">View MathML</a>, by using l’Hospital’s rule, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M585">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M586">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M587">View MathML</a>

and conditions (4.7), (4.8) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M575">View MathML</a> are satisfied.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M589">View MathML</a>

(6.74)

Asymptotic stability condition (3.6) is satisfied as well:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M590">View MathML</a>

 □

Proof of Theorem 4.4 Theorem 4.4 follows from Theorem 3.1, Theorem 3.2 by choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561">View MathML</a>, and JWKB approximation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M592">View MathML</a>

(6.75)

We have from (6.56), (3.15)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M593">View MathML</a>

(6.76)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M594">View MathML</a>

(6.77)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M595">View MathML</a>

(6.78)

Conditions (3.11) and (3.2) are satisfied. Condition (3.1) turns to (3.12) or (4.11), and from (3.13) we get (4.12)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M596">View MathML</a>

 □

Proof of Theorem 4.5 We deduce Theorem 4.5 from Theorems 3.1 and 3.2 assuming <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561">View MathML</a>, and by choosing the Hartman-Wintner approximation [14]

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M598">View MathML</a>

(6.79)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M599">View MathML</a> are solutions of the quadratic equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M600">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M601">View MathML</a>

(6.80)

By calculations,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M602">View MathML</a>

(6.81)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M603">View MathML</a>

(6.82)

Denoting

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M604">View MathML</a>

(6.83)

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M605">View MathML</a>

(6.84)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M606">View MathML</a>

(6.85)

From (3.15)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M607">View MathML</a>

or (4.17)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M608">View MathML</a>

since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M609">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M610">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M611">View MathML</a>

From (3.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M612">View MathML</a>

From (4.14) we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M613">View MathML</a>, and in view of (3.12), condition (3.1) turns to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M614">View MathML</a>

and it follows from (4.15).

From (4.17) we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M615">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M616">View MathML</a>, and condition (3.2) is satisfied.

To prove Remark 4.2, note that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M617">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M618">View MathML</a>, and from the quadratic equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M619">View MathML</a>, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M620">View MathML</a>, or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M621">View MathML</a>. Further, from the equation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M622">View MathML</a>, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M623">View MathML</a> and the other phase function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M624">View MathML</a>. □

Proof of Theorems 5.1, 5.2 Theorem 5.1 follows from Theorem 4.1 applied to system (6.51). Indeed, by substitution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M625">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M626">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M627">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M628">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M561">View MathML</a>, condition (4.2) of Theorem 4.1 turns to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M630">View MathML</a>. Further, from condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M631">View MathML</a>, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M632">View MathML</a> and (4.1) is satisfied. From (4.4) we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M633">View MathML</a>.

By choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M634">View MathML</a>, the conditions of Theorem 4.1 turn to (5.1) (big damping case).

By choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M635">View MathML</a>, the conditions of Theorem 4.1 turn to (5.2) (small damping case).

Theorem 5.2 follows from Theorem 3.4 by choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M636">View MathML</a>

(6.86)

 □

Proof of Example 5.1 Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M637">View MathML</a>

(6.87)

from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M638">View MathML</a> we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M639">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M640">View MathML</a>

(6.88)

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M641">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M642">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M643">View MathML</a>, and condition (5.1) of Theorem 5.1 is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M644">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M645">View MathML</a> (small damping), then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M646">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M647">View MathML</a>, and condition (5.2) of Theorem 5.1 is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M648">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M649">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M650">View MathML</a>, then condition (5.1) is satisfied again:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M651">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M652">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M650">View MathML</a>, then condition (5.2) is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M654">View MathML</a>

Further, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M641">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M657">View MathML</a>, then in view of (6.87) condition (5.1) is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M658">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M660">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M661">View MathML</a>, then in view of (6.86) condition (5.2) is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M662">View MathML</a>

Finally, when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M330">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M665">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M666">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M667">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M668">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M669">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M670">View MathML</a>, and condition (5.1) is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M671">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M672">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M647">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M646">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M675">View MathML</a>, condition (5.2) is satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M676">View MathML</a>

 □

Proof of Theorem 5.3 We deduce Theorem 5.3 from Theorem 3.1 applied to system (6.51), and by substitution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M677">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M575">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M628">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M680">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M681">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M682">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M632">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M684">View MathML</a>

From (3.5), (3.13)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M685">View MathML</a>

Conditions (3.1), (3.6) turn to (5.6), (5.7). If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M686">View MathML</a>, then (3.2) is satisfied. The case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M687">View MathML</a> is trivial, since in this case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M688">View MathML</a> and the functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M689">View MathML</a> are exact solutions of (2.1). □

Proof of Theorem 5.4 We deduce Theorem 5.4 from Theorem 3.4 by choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M690">View MathML</a>

From

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M691">View MathML</a>

we get from (3.19)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M692">View MathML</a>

and from (3.18)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M693">View MathML</a>

so (3.2) is satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M694">View MathML</a> and condition (3.6) turns to (5.7). Case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M695">View MathML</a> is trivial. □

Proof of Example 5.2 This example follows from Theorem 5.3.

From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M696">View MathML</a> we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M697">View MathML</a>.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M698">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M699">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M700">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M701">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M702">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M700">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M704">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M641">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M707">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M708">View MathML</a>

In all these cases, (5.7) is satisfied since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M709">View MathML</a>. □

Proof of Theorem 5.5 Theorem 5.5 follows from Theorem 4.2 applied to (6.51). □

Proof of Theorem 5.6 We deduce Theorem 5.6 from Theorem 3.4 by choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M710">View MathML</a>

From <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M711">View MathML</a> we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M712">View MathML</a>

and from (3.19) we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M713">View MathML</a>

From (3.18) and (3.6) we get (5.10)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M714">View MathML</a>

Condition (3.2) is satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M715">View MathML</a>. The case <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M716">View MathML</a> is obvious since in that case the exact solutions of (2.1) are <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M717">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M718">View MathML</a>. □

Proof of Example 5.3 This example follows from Theorem 5.5:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M719">View MathML</a>

 □

Proof of Theorem 5.7 Theorem 5.7 follows from Theorem 3.2 applied to (6.51), and by choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M720">View MathML</a>

(6.89)

For this case (3.2) is true, (3.1) turns to (5.13), and (3.6) turns to (5.14). □

Proof of Example 5.4 Example 5.4 follows from Theorem 5.7.

In view of (6.89), we have conditions (5.12), (5.13) of Theorem 5.7 are satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M721">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M722','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M722">View MathML</a>

Further, in view of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M723">View MathML</a>

condition (5.14) or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M724">View MathML</a>

(6.90)

is satisfied if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M725">View MathML</a> since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M726">View MathML</a>

 □

Proof of Example 5.5 Example 5.5 follows from Theorem 5.7. Indeed

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M727','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M727">View MathML</a>

Choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M728','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M728">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M729">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M730">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M731">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M732">View MathML</a>

or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M733">View MathML</a>

and conditions (5.12), (5.13) are satisfied:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M734">View MathML</a>

From

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M735','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M735">View MathML</a>

condition (5.14) is satisfied since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M736','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M736">View MathML</a>

 □

Proof of Theorem 5.8 We deduce Theorem 5.8 from Theorem 3.4 by choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M737">View MathML</a>, the phase <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M552">View MathML</a> from the Hartman-Wintner approximation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M739">View MathML</a>

(6.91)

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M554">View MathML</a> from (6.57)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M741">View MathML</a>

or (5.20)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M742">View MathML</a>

(6.92)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M743">View MathML</a>, we get, from (3.18), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M744">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M745">View MathML</a>

(6.93)

Condition (3.6) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M746','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M746">View MathML</a> turns to (5.18). From (3.32) in view of (6.90), we get (5.19):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M747">View MathML</a>

Condition (3.2) is satisfied in view of Remark 3.3 and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M748">View MathML</a>. □

Proof of Theorem 5.9 We deduce Theorem 5.9 from Theorem 3.8 by choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M749','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M749">View MathML</a>

By calculations

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M750','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M750">View MathML</a>

we get (5.22) from (3.29). Further, from (3.33) we get (5.23) since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M751','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M751">View MathML</a>

 □

Proof of Theorem 5.10 We deduce Theorem 5.10 from Theorem 3.4 by taking

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M752">View MathML</a>

and assuming <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M753','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M753">View MathML</a> or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M754">View MathML</a>.

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M755','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M755">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M756','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M756">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M757','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M757">View MathML</a>

From (3.33) and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M758">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M759">View MathML</a>

where in view of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M754">View MathML</a> we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M761">View MathML</a>

Further from (3.15)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M762">View MathML</a>

and (3.6) turns to (5.28). □

Appendix:  Some invariants of the planar dynamic systems

By a linear time-dependent non-singular lower triangular transformation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M763">View MathML</a>

(A.1)

from linear system (1.1) (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M764">View MathML</a> does not depend on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M2">View MathML</a>), we get another linear system

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M766','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M766">View MathML</a>

(A.2)

Define auxiliary functions associated with system (A.2) that depend on phase functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M767">View MathML</a> as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M768">View MathML</a>

(A.3)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M769">View MathML</a>

(A.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M770">View MathML</a>

(A.5)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M767">View MathML</a> are the phase functions of system (A.2).

Theorem A.1Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M772','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M772">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M773','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M773">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M774','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M774">View MathML</a>is a non-singular lower triangular transformation, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M767">View MathML</a>are solutions of the characteristic equations of linear systems (1.1), (A.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M777','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M777">View MathML</a>

(A.6)

with the initial values

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M778">View MathML</a>

(A.7)

Then we have the invariance

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M779','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M779">View MathML</a>

(A.8)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M780">View MathML</a>

(A.9)

Remark A.1 From Theorem A.1 it follows the well-known result that the function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M781','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M781">View MathML</a>

(A.10)

is invariant of (1.6) with respect to the transformation <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M782">View MathML</a>.

Proof of Theorem A.1 By substitution

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M783','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M783">View MathML</a>

(A.11)

we get from (2.8)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M784','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M784">View MathML</a>

(A.12)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M1">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M786','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M786">View MathML</a> are defined in (2.10), (A.5).

By direct calculations, from (A.2) we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M787','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M787">View MathML</a>

(A.13)

and (A.8). Further, we get <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M788','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M788">View MathML</a>, or

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M789">View MathML</a>

(A.14)

In view of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M790">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M791','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M791">View MathML</a>

(A.15)

assuming initial conditions (A.7), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M792">View MathML</a>

(A.16)

So, the solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M793">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M794">View MathML</a> of characteristic equations <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M795','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M795">View MathML</a> are connected:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M796','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M796">View MathML</a>

(A.17)

From these expressions, we get (A.9). □

Proof of Remark A.1 Rewrite equation (2.1) in form (6.6). Choosing

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M797">View MathML</a>

(A.18)

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M798">View MathML</a>

(A.19)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M799">View MathML</a>

(A.20)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M800','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M800">View MathML</a>

(A.21)

and (A.8) becomes <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M801">View MathML</a>. □

Remark A.2 There are several characteristic functions of (1.1) depending on the structure of the matrix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M764">View MathML</a>. Indeed, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M75">View MathML</a>, then the characteristic function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M804">View MathML</a> of (1.1) is given by (2.8). If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M805','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M805">View MathML</a>, but <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M806','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M806">View MathML</a>, the characteristic function may be defined by the similar formula

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M807','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M807">View MathML</a>

(A.22)

If system (1.1) is diagonal, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M808','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M808">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M809','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/144/mathml/M809">View MathML</a>

(A.23)

Competing interests

The author declares that he has no competing interests.

Acknowledgements

This paper is dedicated to my mother Paytsar Hovhannisyan.

The author would like to thank anonymous reviewers for very useful and constructive comments that helped to improve the original manuscript.

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