Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables
© Z. Ma and L. Xu 2009
Received: 3 June 2009
Accepted: 2 August 2009
Published: 19 August 2009
A class of impulsive infinite delay difference equations with continuous variables is considered. By establishing an infinite delay difference inequality with impulsive initial conditions and using the properties of "ϱ-cone," we obtain the attracting and invariant sets of the equations.
Difference equations with continuous variables are difference equations in which the unknown function is a function of a continuous variable . These equations appear as natural descriptions of observed evolution phenomena in many branches of the natural sciences (see, e.g., [2, 3]). The book mentioned in  presents an exposition of some unusual properties of difference equations, specially, of difference equations with continuous variables. In the recent years, the asymptotic behavior and other behavior of delay difference equations with continuous variables have received much attention due to its potential application in various fields such as numerical analysis, control theory, finite mathematics, and computer science. Many results have appeared in the literatures; see, for example, [1, 4–7].
However, besides the delay effect, an impulsive effect likewise exists in a wide variety of evolutionary process, in which states are changed abruptly at certain moments of time. Recently, impulsive difference equations with discrete variable have attracted considerable attention. In particular, delay effect on the asymptotic behavior and other behaviors of impulsive difference equations with discrete variable has been extensively studied by many authors and various results are reported [8–12]. However, to the best of our knowledge, very little has been done with the corresponding problems for impulsive delay difference equations with continuous variables. Motivated by the above discussions, the main aim of this paper is to study the asymptotic behavior of impulsive infinite delay difference equations with continuous variables. By establishing an infinite delay difference inequality with impulsive initial conditions and using the properties of " -cone," we obtain the attracting and invariant sets of the equations.
this function will be called the solution of the initial problem (2.1)–(2.3).
In what follows, we introduce some notations and recall some basic definitions. Let be the space of -dimensional (nonnegative) real column vectors, be the set of (nonnegative) real matrices, be the -dimensional unit matrix, and be the Euclidean norm of . For or , means that each pair of corresponding elements of and satisfies the inequality " ( )."Especially, is called a nonnegative matrix if , and is called a positive vector if . and .
Lemma 2.5 (La Salle ).
which is a nonempty set by Lemma 2.5, satisfying that for any scalars , , and vectors . So is a cone without vertex in , we call it a " -cone" .
3. Main Results
In this section, we will first establish an infinite delay difference inequality with impulsive initial conditions and then give the attracting and invariant sets of (2.4).
Suppose that in part (a) of Theorem 3.1, then we get [15, Lemma 3].
This implies that the conclusion of the theorem holds and the proof is complete.
Therefore, is a positive invariant set. Since , a direct calculation shows that and in Theorem 3.3. It follows from Theorem 3.3 that the set is also a global attracting set of (2.4). The proof is complete.
4. Illustrative Example
The following illustrative example will demonstrate the effectiveness of our results.
Clearly, all conditions of Corollary 3.5 are satisfied. Therefore, by Corollary 3.5, the zero solution of (4.1) is globally exponentially stable.
The work is supported by the National Natural Science Foundation of China under Grant 10671133.
- Philos ChG, Purnaras IK: An asymptotic result for some delay difference equations with continuous variable. Advances in Difference Equations 2004,2004(1):1-10. 10.1155/S1687183904310058MATHMathSciNetView ArticleGoogle Scholar
- Ladas G: Recent developments in the oscillation of delay difference equations. In Differential Equations (Colorado Springs, CO, 1989), Lecture Notes in Pure and Applied Mathematics. Volume 127. Marcel Dekker, New York, NY, USA; 1991:321-332.Google Scholar
- Sharkovsky AN, Maĭstrenko YuL, Romanenko EYu: Difference Equations and Their Applications, Mathematics and Its Applications. Volume 250. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+358.View ArticleGoogle Scholar
- Deng J: Existence for continuous nonoscillatory solutions of second-order nonlinear difference equations with continuous variable. Mathematical and Computer Modelling 2007,46(5-6):670-679. 10.1016/j.mcm.2006.11.028MATHMathSciNetView ArticleGoogle Scholar
- Deng J, Xu Z: Bounded continuous nonoscillatory solutions of second-order nonlinear difference equations with continuous variable. Journal of Mathematical Analysis and Applications 2007,333(2):1203-1215. 10.1016/j.jmaa.2006.12.038MATHMathSciNetView ArticleGoogle Scholar
- Philos ChG, Purnaras IK: On non-autonomous linear difference equations with continuous variable. Journal of Difference Equations and Applications 2006,12(7):651-668. 10.1080/10236190600652360MATHMathSciNetView ArticleGoogle Scholar
- Philos ChG, Purnaras IK: On the behavior of the solutions to autonomous linear difference equations with continuous variable. Archivum Mathematicum 2007,43(2):133-155.MATHMathSciNetGoogle Scholar
- Li Q, Zhang Z, Guo F, Liu Z, Liang H: Oscillatory criteria for third-order difference equation with impulses. Journal of Computational and Applied Mathematics 2009,225(1):80-86. 10.1016/j.cam.2008.07.002MATHMathSciNetView ArticleGoogle Scholar
- Peng M: Oscillation criteria for second-order impulsive delay difference equations. Applied Mathematics and Computation 2003,146(1):227-235. 10.1016/S0096-3003(02)00539-8MATHMathSciNetView ArticleGoogle Scholar
- Yang XS, Cui XZ, Long Y: Existence and global exponential stability of periodic solution of a cellular neural networks difference equation with delays and impulses. Neural Networks. In press
- Zhang Q: On a linear delay difference equation with impulses. Annals of Differential Equations 2002,18(2):197-204.MATHMathSciNetGoogle Scholar
- Zhu W, Xu D, Yang Z: Global exponential stability of impulsive delay difference equation. Applied Mathematics and Computation 2006,181(1):65-72. 10.1016/j.amc.2006.01.015MATHMathSciNetView ArticleGoogle Scholar
- Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge, UK; 1990:xiv+561.MATHGoogle Scholar
- LaSalle JP: The Stability of Dynamical Systems. SIAM, Philadelphia, Pa, USA; 1976:v+76.MATHView ArticleGoogle Scholar
- Zhu W: Invariant and attracting sets of impulsive delay difference equations with continuous variables. Computers & Mathematics with Applications 2008,55(12):2732-2739. 10.1016/j.camwa.2007.10.020MATHMathSciNetView ArticleGoogle Scholar
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