On -stability of a class of singular difference equations
© Wang et al.; licensee Springer 2013
Received: 25 February 2013
Accepted: 4 June 2013
Published: 25 June 2013
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.
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 .
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 .
(H1) , ,
(H2) , , where , .
For the next discussion, the following lemma from  is needed.
the matrix is nonsingular;
where γ is an arbitrary vector in and is a fixed nonnegative integer.
Theorem 2.1 
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 
Definition 2.6 The trivial solution of (2.1) is said to be
(S2) A-uniformly -stable (P-uniformly -stable) if δ in () is independent of .
(S4) A-uniformly asymptotically -stable if and N in (S3) are independent of .
where is any solution of system (2.1).
3 Main results
Then (2.1) is A-uniformly -stable.
the proof of Lemma 3.1 is complete. □
Similar to the proof of Lemma 3.1, we have the following.
- (i), , is locally Lipschitzian in r relative to K, and for each ,
is quasimonotone in relative to K;
for some and , .
Then the trivial solution of SDEs (2.1) is A--stable.
Then the trivial solution of (2.1) is A--stable. The proof of Theorem 3.1 is complete. □
Then the trivial solution of SDEs (2.1) is A-asymptotically -stable.
Then the trivial solution of (2.1) is A-asymptotically -stable. The proof of Theorem 3.2 is complete. □
, , and some ,
for any solution of (2.1).
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 .
which leads to a contradiction. The proof of Theorem 3.3 is complete. □
, , ;
for any solution of (2.1).
Proof The proof of the necessity part is similar to the corresponding part of Theorem 3.3.
Then the trivial solution of (2.1) is P-uniformly -stable. The proof of Theorem 3.4 is complete. □
According to Theorem 3.4, the trivial solution of (2.1) is P-uniformly -stable.
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).
- Anh PK, Du NH, Loi LC: Singular difference equations: an overview. Vietnam J. Math. 2007, 35: 339-372.MathSciNetMATHGoogle Scholar
- Anh PK, Loi LC: On the solvability of initial-value problems for nonlinear implicit difference equations. Adv. Differ. Equ. 2004, 3: 195-200.Google Scholar
- Anh PK, Loi LC: On multipoint boundary-value problems for linear implicit non-autonomous systems of difference equations. Vietnam J. Math. 2001, 29: 281-286.MathSciNetMATHGoogle Scholar
- Agarwal RP Monographs and Textbooks in Pure and Applied Mathematics 228. In Difference Equations and Inequalities, Theory, Methods and Applications. 2nd edition. Dekker, New York; 2000.Google Scholar
- Elaydi SN: An Introduction to Difference Equations. 3rd edition. Springer, London; 2005.MATHGoogle Scholar
- Halanay A, Rásvan VRV: Stability and Stable Oscillations in Discrete Time Systems. Gordon & Breach Science, Taipei; 2002.Google Scholar
- Martynjuk DI: Lectures on the Qualitative Theory of Difference Equations. “Naukova Dumka”, Kiev; 1972.Google Scholar
- Diblík J, Khusainov DY, Grytsay IV, Šmarda Z: Stability of nonlinear autonomous quadratic discrete systems in the critical case. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 539087 10.1155/2010/539087Google Scholar
- Anh PK, Hoang DS: Stability of a class of singular difference equations. Int. J. Differ. Equ. 2006, 1: 181-193.MATHGoogle Scholar
- Lakshmikantham V, Leela S: Cone-valued Liapunov functions. Nonlinear Anal. 1977, 1: 215-222. 10.1016/0362-546X(77)90031-1MathSciNetView ArticleMATHGoogle Scholar
- Akpan EP, Akineyle O:On the -stability of comparison differential systems. J. Math. Anal. Appl. 1992, 164: 307-324. 10.1016/0022-247X(92)90116-UMathSciNetView ArticleMATHGoogle Scholar
- EL-Sheikh MMA, Soliman AA: On stability of nonlinear systems of functional differential equations. Appl. Math. Comput. 2000, 107: 81-93. 10.1016/S0096-3003(98)10049-8MathSciNetView ArticleMATHGoogle Scholar
- Wang PG, Geng FJ:On stability of difference equations. Appl. Math. Lett. 2005, 18: 139-147. 10.1016/j.aml.2004.05.002MathSciNetView ArticleMATHGoogle Scholar
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