Research

On ϕ 0 -stability of a class of singular difference equations

Peiguang Wang1*, Meng Wu2 and Yonghong Wu3

Author Affiliations

1 College of Electronic and Information Engineering, Hebei University, Baoding, 071002, China

2 Fundamental Department, Hebei Finance University, Baoding, 071051, China

3 Department of Mathematics and Statistics, Curtin University of Technology, GPO BOX U1987, Perth, WA, 6845, Australia

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Advances in Difference Equations 2013, 2013:179  doi:10.1186/1687-1847-2013-179

 Received: 25 February 2013 Accepted: 4 June 2013 Published: 25 June 2013

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

This paper investigates a class of singular difference equations. Using the framework of the theory of singular difference equations and cone-valued Lyapunov functions, some necessary and sufficient conditions on the -stability of a trivial solution of singular difference equations are obtained. Finally, an example is provided to illustrate our results.

MSC: 39A11.

Keywords:
singular difference equations; cone-valued Lyapunov functions; -stability; uniformly -stability

1 Introduction

The singular difference equations (SDEs), which appear in the Leontiev dynamic model of multisector economy, the Leslie population growth model, singular discrete optimal control problems and so forth, have gained more and more importance in mathematical models of practical areas (see [1] and references cited therein). Anh and Loi [2,3] studied the solvability of initial-value problems as well as boundary-value problems for SDEs.

It is well known that stability is one of the basic problems in various dynamical systems. Many results on the stability theory of difference equations are presented, for example, by Agarwal [4], Elaydi [5], Halanay and Rasvan [6], Martynjuk [7] and Diblík et al.[8]. Recently, Anh and Hoang [9] obtained some necessary and sufficient conditions for the stability properties of SDEs by employing Lyapunov functions. The comparison method, which combines Lyapunov functions and inequalities, is an effective way to discuss the stability of dynamical systems. However, this approach requires that the comparison system satisfies a quasimonotone property which is too restrictive for many applications because this property is not a necessary condition for a desired property like stability of the comparison system. To solve this problem, Lakshmikantham and Leela [10] initiated the method of cone and cone-valued Lyapunov functions and developed the theory of differential inequalities. By employing the method of cone-valued Lyapunov functions, Akpan and Akinyele [11], EL-Sheikh and Soliman [12], Wang and Geng [13] investigated the stability and the -stability of ordinary differential systems, functional differential equations and difference equations, respectively.

However, to the best of our knowledge, there are few results for the -stability of singular difference equations. In this paper, utilizing the framework of the theory of singular difference equations, we give some necessary and sufficient conditions for the -stability of a trivial solution of singular difference equations via cone-valued Lyapunov functions.

2 Preliminaries

The following definitions can be found in reference [10].

Definition 2.1 A proper subset K of is called a cone if

(i) , ;

(ii) ;

(iii) ;

(iv) ;

(v) ,

where and denote the closure and interior of K, respectively, and ∂K denotes the boundary of K. The order relation on induced by the cone K is defined as follows: Let , then iff and iff .

Definition 2.2 The set is said to be an adjoint cone if it satisfies the properties (i)-(v).

and

Definition 2.3 A function , is said to be quasimonotone relative to K if and implies that there exists such that

Definition 2.4 A function is said to belong to the class if , , and is strictly monotone increasing function in r.

Consider the following SDEs:

(2.1)

where and are given. Throughout this paper, we assume that the matrices are singular, and the corresponding linear homogeneous equations

(2.2)

are of index-1 [1-3], i.e., the following hypotheses hold.

(H1) , ,

(H2) , , where , .

For the next discussion, the following lemma from [9] is needed.

Lemma 2.1Suppose that the hypothesis (H1) holds. Then the hypothesis (H2) is equivalent to one of the following statements:

(i) the matrixis nonsingular;

(ii) .

Let us associate SDEs (2.1) with the initial condition

(2.3)

where γ is an arbitrary vector in and is a fixed nonnegative integer.

Theorem 2.1[9]

Letbe a Lipschitz continuous function with a sufficiently small Lipschitz coefficient, i.e.,

where

Then IVP (2.1), (2.3) has a unique solution.

Set . If is any solution of IVP (2.1), (2.3), then obviously ().

Definition 2.5[9]

The trivial solution of (2.1) is said to be A-stable (P-stable) if for each and any , there exists a such that

Definition 2.6 The trivial solution of (2.1) is said to be

(S1) A--stable (P--stable) if for each and any , there exists a such that for some

(S2) A-uniformly -stable (P-uniformly -stable) if δ in () is independent of .

(S3) A-asymptotically -stable (P-asymptotically -stable) if for any there exist positive numbers and such that for some ,

(S4) A-uniformly asymptotically -stable if and N in (S3) are independent of .

Let K be a cone in , . is continuous in the second variable, we define

where is any solution of system (2.1).

3 Main results

Lemma 3.1The trivial solution of SDEs (2.1) is A-uniformly-stable (P-uniformly-stable) if and only if there exists a functionsuch that for any solutionof SDEs (2.1) and some, the following inequality holds:

(3.1)

Proof For each positive ϵ, choose such that . If is an arbitrary solution of (2.1) and , then

Then (2.1) is A-uniformly -stable.

Conversely, suppose that the trivial solution of (2.1) is A-uniformly -stable, i.e., for each positive ϵ, there exists a such that if is any solution of (2.1) which satisfies the inequality , then for all . Denote by the supremum of for the above . Obviously, if for some , then for all . Furthermore, the function is positive and increasing, and . Considering a function defined by and , it is easy to prove that and . Then the inverse of β, denoted by ψ will belong to . For some , set and consider two possibilities: (i) If , then ; (ii) If for some , in which , then , which is impossible, hence , therefore, for some ,

the proof of Lemma 3.1 is complete. □

Similar to the proof of Lemma 3.1, we have the following.

Lemma 3.2The trivial solution of SDEs (2.1) is A--stable (P--stable) if and only if there exist functionssuch that for any solutionof (2.1), each nonnegative integerand some, the following inequality holds:

(3.2)

Theorem 3.1Assume that

(i) , , is locally Lipschitzian inrrelative toK, and for each,

(ii) is quasimonotone inrelative toK;

(iii) for someand, .

Then the trivial solution of SDEs (2.1) is A--stable.

Proof Since and is continuous in r, then given , , there exists such that

For some ,

Thus

implies

It follows that

where , , . Let be any solution of (2.1) such that . Then by (i), V is nonincreasing and so

Thus implies

i.e.,

Then the trivial solution of (2.1) is A--stable. The proof of Theorem 3.1 is complete. □

Theorem 3.2Let the hypotheses of Theorem 3.1 be satisfied, except the conditionbeing replaced by

(iv) , .

Then the trivial solution of SDEs (2.1) is A-asymptotically-stable.

Proof By Theorem 3.2, the trivial solution of (2.1) is A--stable. By (iv), is a monotone decreasing function, thus the limit

exists. We prove that . Suppose , then , . Since is a monotone increasing function, then

and so from (iv), we get

Then

Thus as and for some , we have . This contradicts the condition (iii). It follows that . Thus

and so with (iii)

Thus for given , , there exist and such that

Then the trivial solution of (2.1) is A-asymptotically -stable. The proof of Theorem 3.2 is complete. □

Theorem 3.3The trivial solution of SDEs (2.1) is P--stable if and only if there exist functionsand a Lyapunov functionsuch that for some,

(i) , ;

(ii) , , and some,

(iii) for any solutionof (2.1).

ProofNecessity. Suppose that the trivial solution of (2.1) is P--stable, then, according to Lemma 3.2, there exist functions () such that for any solution of (2.1) and for some ,

(3.3)

Define the Lyapunov function

where is the unique solution of (2.1) satisfying the initial condition . Moreover, for some , , which implies and the continuity of function V in the second variable at . For each , we have

where denotes the solution of (2.1) satisfying the initial condition . Since , it follows , hence, for some ,

Further, the inequality (3.3) gives

On the other hand, for an arbitrary solution of (2.1), by the unique solvability of the initial value problem (2.1) and (2.3), we have

Thus

hence . The necessity part is proved.

Sufficiency. Assuming that the trivial solution of (2.1) is not P--stable, i.e., there exist a positive and a nonnegative integer such that for all and for some , there exists a solution of (2.1) satisfying the inequalities and for some .

Since and is continuous at , there exists a such that for all , , we have . Choosing , we can find a solution of (2.1) satisfying . However, for some . Since , we get

for some . On the other hand, using the properties (iii) of the function V, we find

which leads to a contradiction. The proof of Theorem 3.3 is complete. □

Theorem 3.4The trivial solution of SDEs (2.1) is P-uniformly-stable if and only if there exist functionsand a Lyapunov functionsuch that for some,

(i) , , ;

(ii) for any solutionof (2.1).

Proof The proof of the necessity part is similar to the corresponding part of Theorem 3.3.

For , let independent of for , and be any solution of (2.1) such that . Then by (ii), V is nonincreasing and so

Thus

i.e.,

Then the trivial solution of (2.1) is P-uniformly -stable. The proof of Theorem 3.4 is complete. □

4 Example

Consider SDEs (2.1) with the following data:

and

As , , and , , the hypotheses (H1), (H2) are fulfilled, hence SDEs (2.2) is of index-1. Clearly, the canonical projections are

therefore

hence . Further, the function is Lipschitz continuous with the Lipschitz coefficient . Moreover, and . According to Theorem 2.1, IVP (2.1), (2.3) has a unique solution. We have if and only if

Let , we get for each ,

Further, . Thus, for some ,

where and , . Suppose that is a solution of (2.1) and putting , , then we have

Using equation (2.8) in [9], we find

hence , then

According to Theorem 3.4, the trivial solution of (2.1) is P-uniformly -stable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).

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