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Uniform stability of fractional neutral systems: a Lyapunov-Krasovskii functional approach

KeWei Liu12* and Wei Jiang1

Author Affiliations

1 School of Mathematical Sciences, Anhui University, Hefei, 230039, China

2 School of Mathematics, Hefei University of Technology, Hefei, 230009, China

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Advances in Difference Equations 2013, 2013:379  doi:10.1186/1687-1847-2013-379

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/379


Received:6 September 2013
Accepted:2 December 2013
Published:27 December 2013

© 2013 Liu and Jiang; 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

In this paper, we study the stability of nonlinear fractional neutral systems equipped with the Caputo derivative. We extend the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems. Conditions of uniform stability are obtained for the nonlinear fractional neutral systems.

MSC: 34K20, 34K37, 34K40.

Keywords:
fractional neutral systems; uniform stability; Lyapunov-Krasovskii approach

1 Introduction

In recent decades, fractional calculus and fractional differential equations have attracted great attention. It has been proved that fractional calculus and fractional differential equations are valuable tools in the modeling of many phenomena in various fields of engineering, physics and economics. For details and examples, see [1-5] and the references therein.

Stability analysis is always one of the most important issues in the theory of differential equations and their applications for both deterministic and stochastic cases. The analysis on stability of fractional differential equations is more complex than that of classical differential equations, since fractional derivatives are nonlocal and have weakly singular kernels. Recently, stability of fractional differential equations has attracted increasing interest. The earliest study on stability of fractional differential equations started in [6], the author studied the case of linear fractional differential equations with Caputo derivative and the same fractional order α, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M1">View MathML</a>. The stability problem comes down to the eigenvalue problem of system matrix. Since then, many researchers have done further studies on the stability of linear fractional differential systems [7-11]. For the nonlinear fractional differential systems, the stability analysis is much more difficult and a few results are available in [12-18]. For more details about the stability results and the methods available to analyze the stability of fractional differential equations, the reader may refer to the recent survey papers [19,20] and the references therein.

As we all know, Lyapunov’s second method provides a way to analyze the stability of a system without explicitly solving the differential equations. It is necessary to extend Lyapunov’s second method to fractional systems. In [12,13], the fractional Lyapunov’s second method was proposed, and the authors extended the exponential stability of integer order differential system to the Mittag-Leffler stability of fractional differential system. In [14], by using Bihari’s and Bellman-Gronwall’s inequality, an extension of Lyapunov’s second method for fractional-order systems was proposed. In [15-17], Baleanu et al. extended Lyapunov’s method to fractional functional differential systems and developed the Lyapunov-Krasovskii stability theorem, Lyapunov-Razumikhin stability theorem and Mittag-Leffler stability theorem for fractional functional differential systems. As far as we know, there are few papers with respect to the stability of fractional neutral systems. In this paper, we consider the stability of a class of nonlinear fractional neutral functional differential equations with the Caputo derivative. Motivated by Li et al.[12,13], Baleanu et al.[15] and Cruz and Hale [21], we aim in this paper to extend the Lyapunov-Krasovskii method for the nonlinear fractional neutral systems.

The rest of the paper is organized as follows. In Section 2, we give some notations and recall some concepts and preparation results. In Section 3, we extend the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems, results of uniform stability for the nonlinear fractional neutral systems are presented. Conclusions are presented in Section 4.

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts needed here. Throughout this paper, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M2">View MathML</a> be a real n-dimensional linear vector space with the norm <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M3">View MathML</a>, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M4">View MathML</a> be the space of continuous functions taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M5">View MathML</a> into <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M2">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M7">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M8">View MathML</a> defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M10">View MathML</a> be a real constant. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M12">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M13">View MathML</a>, then for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M14">View MathML</a>, we let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M15">View MathML</a> be defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M16">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M17">View MathML</a>.

Let us recall the following known definitions. For more details, we refer the reader to [1,2,4,5].

Definition 2.1 The fractional order integral of a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M18">View MathML</a> of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M19">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M20">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M21">View MathML</a> is the gamma function.

Definition 2.2 For a function f given on the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M22">View MathML</a>, the α order Riemann-Liouville fractional derivative of f is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M23">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M24">View MathML</a>.

Definition 2.3 For a function f given on the interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M22">View MathML</a>, the α order Caputo fractional derivative of f is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M26">View MathML</a>

Some properties of the aforementioned operators are recalled below [1].

Property 2.1The following results are especially interesting:

(i) For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M27">View MathML</a>, we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M28">View MathML</a>.

(ii) When<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M29">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M30">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M31">View MathML</a>

(iii) For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M32">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M33">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M34">View MathML</a>, we have<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M35">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M36">View MathML</a>.

Remark 2.1 From Property 2.1, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M37">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M32">View MathML</a>, then for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39">View MathML</a>, we have

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M40">View MathML</a>.

(ii) In general, it is not true that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M41">View MathML</a> is nondecreasing in t.

In [21], Cruz and Hale studied a class of functional difference operators which are very useful in stability theory and the asymptotic behavior of solutions of functional differential equations of neutral type. In monograph [22], Hale et al. presented the following definitions and results of the difference operators.

For Banach spaces X and Y, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M42">View MathML</a> is the Banach space of bounded linear mappings from X to Y with the operator topology. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M43">View MathML</a>, then the Riesz representation theorem implies that there is an <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M44">View MathML</a> matrix function μ on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M5">View MathML</a> of bounded variation such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M46">View MathML</a>

Definition 2.4 Let Ω be an open subset of a metric space. We say <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M47">View MathML</a> has smoothness on the measure if, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M48">View MathML</a>, there is a scalar function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M49">View MathML</a> continuous for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M50">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M51">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M52">View MathML</a>, such that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M53">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M50">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M55">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M56">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M48">View MathML</a> and the matrix <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M58">View MathML</a> is nonsingular at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M59">View MathML</a>, we say <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M60">View MathML</a> is atomic at B at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M61">View MathML</a>. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M62">View MathML</a> is nonsingular on a set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M63">View MathML</a>, we say <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M60">View MathML</a> is atomic at B on K.

Definition 2.5 Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M65">View MathML</a> is open with elements <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M66">View MathML</a>. A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M67">View MathML</a> (not necessarily linear) is said to be atomic at B on Ω if D is continuous together with its first and second Fréchet derivatives with respect to ϕ; and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M68">View MathML</a>, the derivative with respect to ϕ, is atomic at B on Ω.

Remark 2.2 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a> is linear in ϕ and continuous in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M70">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M71">View MathML</a>

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M72">View MathML</a> is independent of ϕ and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M73">View MathML</a>

Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a> is atomic at B on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M75">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M51">View MathML</a>. In particular, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M77">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M78">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M80">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a> is atomic at B on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M75">View MathML</a> if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M83">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M51">View MathML</a>.

In the sequel, we consider the following nonlinear fractional neutral system:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M85">View MathML</a>

(2.1)

with the initial condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M86">View MathML</a>

(2.2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M87">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88">View MathML</a> is a constant, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M89">View MathML</a> are given continuous functions, nonlinear difference operator atomic at zero. For more details about the operator , the reader may refer to [21,22,24] and the references therein. In the sequel, we always assume that, for any given <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88">View MathML</a> and a given function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M93">View MathML</a>, there exists a unique continuous solution of (2.1), denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M94">View MathML</a>, such that it satisfies (2.1) for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M95">View MathML</a> and (2.2). To deal with stability, as usual, we assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M96">View MathML</a> so that (2.1) has the zero solution.

Definition 2.6[23]

The zero solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M97">View MathML</a> of (2.1) is stable if for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88">View MathML</a> and any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M99">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M100">View MathML</a> such that any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M94">View MathML</a> of (2.1) with initial value φ at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103">View MathML</a> satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M104">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39">View MathML</a>. It is asymptotically stable if it is stable and for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88">View MathML</a> and any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M99">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M108">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M109">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M111">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M112">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M113">View MathML</a>. It is uniformly stable if it is stable and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M114">View MathML</a> can be chosen independently of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102">View MathML</a>. It is uniformly asymptotically stable if it is uniformly stable and there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M116">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M117">View MathML</a>, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M118">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M120">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M121">View MathML</a>. It is globally (uniformly) asymptotically stable if it is (uniformly) asymptotically stable and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M122">View MathML</a> can be an arbitrary large, finite number.

For a nonlinear operator , in [24], Zhang gave the following definition.

Definition 2.7<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a> is said to be uniformly stable if there exist positive constants a, b, c such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M125">View MathML</a> and any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M127">View MathML</a>, with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M128">View MathML</a>, the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M129">View MathML</a> of

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M130">View MathML</a>

(2.3)

satisfies the following estimate:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M131">View MathML</a>

(2.4)

The following lemma plays a major role in our analysis.

Lemma 2.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a>be uniformly stable, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M133">View MathML</a>be the solution of equation (2.3). Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M134">View MathML</a>is any continuous and nondecreasing function with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M135">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M136">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137">View MathML</a>. Then, for small<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M55">View MathML</a>, there is a continuous and strictly increasing function<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M139">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M140">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M141">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137">View MathML</a>such that

(i) for each small<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M143">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M145">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M147">View MathML</a>

(2.5)

(ii) for each small<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M143">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M149">View MathML</a>and a nonnegative constantL, there exists<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M150">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M151">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M152">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146">View MathML</a>imply

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M154">View MathML</a>

(2.6)

Proof (i) From Definition 2.5, for sufficiently small <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M143">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M157">View MathML</a>

Trivially, we can choose a continuous and increasing function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M140">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M141">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137">View MathML</a>, so that (2.5) holds.

(ii) For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M162">View MathML</a> and sufficiently small <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M149">View MathML</a>, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M152">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M146">View MathML</a>, then by Definition 2.5 it suffices to show that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M167">View MathML</a>

which implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M168">View MathML</a>

Therefore, if we take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M169">View MathML</a>, then (2.6) holds. □

3 Main results

In this section, we consider the stability of nonlinear fractional neutral system (2.1). Here, we always assume that fractional neutral system (2.1) with initial condition (2.2) has a unique continuous solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M170">View MathML</a> which depends continuously upon <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102">View MathML</a>, φ. By Property 2.1(iii), we can obtain that initial value problem (2.1)-(2.2) is equivalent to the integral equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M172">View MathML</a>

(3.1)

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M173">View MathML</a> is continuously differentiable, we define the Caputo fractional derivative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M174">View MathML</a> along the solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M175">View MathML</a> of (2.1)-(2.2) as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M176">View MathML</a>

Now, we give the following Lyapunov-Krasovskii methods for nonlinear fractional neutral systems as counterpart to Lyapunov-Krasovskii methods for classical neutral systems proposed in [21].

Theorem 3.1Suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a>is uniformly stable, ftakes closed bounded sets into bounded sets, and suppose that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M178">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M179">View MathML</a>are strictly increasing functions with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M180">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M181">View MathML</a>is a continuous, nonnegative, nondecreasing function. If there exists a continuously differentiable functional<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M182">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M183">View MathML</a>

(3.2)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M184">View MathML</a>. Then the zero solution of (2.1) is uniformly stable. If, in addition, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M185">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M55">View MathML</a>, then it is uniformly asymptotically stable.

Proof It is possible to choose a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M134">View MathML</a> so that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M135">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M189">View MathML</a> for small <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M191">View MathML</a> for small <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M137">View MathML</a>. For the above-chosen <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M134">View MathML</a>, by Lemma 2.1, we can find a corresponding <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M194">View MathML</a> with the desired properties. Now, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M99">View MathML</a>, we can find a sufficiently small δ such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M196">View MathML</a>. Hence, for any initial time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M102">View MathML</a> and any initial condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M198">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M103">View MathML</a>, (3.2) implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M200">View MathML</a>

by Property 2.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M201">View MathML</a>

By (3.2), this implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M202">View MathML</a>

which implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M203">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a> is uniformly stable, Lemma 2.1 implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M206">View MathML</a>

Therefore, the zero solution is uniformly stable.

To prove uniform asymptotic stability, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M207">View MathML</a>, choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M208">View MathML</a> corresponding to uniform stability. Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M88">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110">View MathML</a> implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M211">View MathML</a>

(3.3)

Next, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M117">View MathML</a>, we wish to show that there is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M213">View MathML</a> such that any solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M214">View MathML</a> of (2.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110">View MathML</a> satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M216">View MathML</a>

(3.4)

Suppose that (3.4) is not true, then there is a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M217">View MathML</a> of (2.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M219">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M220">View MathML</a>) for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39">View MathML</a>. Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M222">View MathML</a>, from (2.4) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M223">View MathML</a>

(3.5)

Choose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M224">View MathML</a> so that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M225">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M226">View MathML</a>, from (3.5) we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M227">View MathML</a>

Then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M228">View MathML</a>

Therefore, there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M229">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M230">View MathML</a>

(3.6)

Now consider a sequence of intervals <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M231">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M232">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M233">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M234">View MathML</a> . By (3.5) and (3.6), there must be some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M235">View MathML</a> in each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M231">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M237">View MathML</a>

Then there exists a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M238">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M239">View MathML</a>, as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M240">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M241">View MathML</a>

Without loss of generality, we may assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M242">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M243">View MathML</a>. It follows from equation (3.1) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M244">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M245">View MathML</a>

Then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M246">View MathML</a>

Since f takes bounded sets into bounded sets, there is a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M247">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M248">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M39">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M250">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M251">View MathML</a>

Hence, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M252">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M253">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M254">View MathML</a>.

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M255">View MathML</a>

Then there exists some positive integer <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M256">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M257">View MathML</a>

(3.7)

By (3.2), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M258">View MathML</a>

By (3.2), (3.7) and Property 2.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M259">View MathML</a>

(3.8)

which is a contradiction. Then there must be some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M260">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M261">View MathML</a>. Therefore, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M110">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M263">View MathML</a>

This proves the uniform asymptotic stability of the zero solution of (2.1). □

Remark 3.1 From the proof of inequality (3.8), we can know that the analysis on stability of fractional differential equations is more complex than that of classical differential equations, since fractional derivatives are nonlocal and have weakly singular kernels.

Remark 3.2 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M69">View MathML</a> is linear in ϕ and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M265">View MathML</a>, Theorem 3.1 is just the same as Theorem 4.1 in [21].

Remark 3.3 If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M266">View MathML</a>, the conclusions of Theorem 3.1 are just the same as the corresponding conclusions of Theorem in [15].

Theorem 3.2Suppose that the assumptions in Theorem 3.1 are satisfied except replacing<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M267">View MathML</a>by<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M268">View MathML</a>, then one has the same result for uniform stability and uniform asymptotic stability.

Proof By using Property 2.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M269">View MathML</a>

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M270">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/379/mathml/M271">View MathML</a>. Then we can obtain the same result for uniform stability and uniform asymptotic stability. □

4 Conclusions

In this paper, we have studied the stability of nonlinear fractional order neutral systems. We introduce the Lyapunov-Krasovskii approach for the nonlinear fractional neutral systems, which enrich the knowledge of both the system theory and the fractional calculus. We partly extend the application of Caputo fractional systems by using Lyapunov-Krasovskii approach. By using Caputo and Riemann-Liouville derivatives and Lyapunov-Krasovskii technique, uniform stability criteria are obtained for the nonlinear fractional neutral systems. The obtained conclusions generalize the corresponding conclusions in [15,21].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11371027), the Fundamental Research Funds for the Central Universities (2013HGXJ0226) and the Fund of Anhui University Graduate Academic Innovation Research (10117700004).

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