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.
Keywords:singular difference equations; cone-valued Lyapunov functions; -stability; uniformly -stability
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  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 , Elaydi , Halanay and Rasvan , Martynjuk  and Diblík et al.. Recently, Anh and Hoang  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  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 , EL-Sheikh and Soliman , Wang and Geng  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.
The following definitions can be found in reference .
Consider the following SDEs:
For the next discussion, the following lemma from  is needed.
Lemma 2.1Suppose that the hypothesis (H1) holds. Then the hypothesis (H2) is equivalent to one of the following statements:
Let us associate SDEs (2.1) with the initial condition
Then IVP (2.1), (2.3) has a unique solution.
Definition 2.6 The trivial solution of (2.1) is said to be
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:
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:
Theorem 3.1Assume that
It follows that
and so from (iv), we get
and so with (iii)
Define the Lyapunov function
Further, the inequality (3.3) gives
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 .
which leads to a contradiction. The proof of Theorem 3.3 is complete. □
Proof The proof of the necessity part is similar to the corresponding part of Theorem 3.3.
Consider SDEs (2.1) with the following data:
Using equation (2.8) in , we find
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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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