Stability properties and existence theorems of pseudo almost periodic solutions of linear Volterra difference equations
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Stability properties and existence theorems of pseudo almost periodic solutions of linear Volterra difference equations Yoshihiro Hamaya Correspondence: [email protected]. ac.jp Department of Information Science, Okayama University of Science, 1-1 Ridai-chyo, Kitaku, Okayama 700-0005, Japan
Abstract The purpose of this article is to discuss the existence of pseudo almost periodic solutions of linear Volterra equation: x(n + 1) = A(n)x(n) + ns=−∞ F(n, s)x(s) + p(n) , n Î Z, by using an exponentially stable of the zero solution, which is equivalent to the exponential behaviors of the resolvent matrix G(n, m) as n ® ∞ and of some summability of the kernel. AMS (MOS) 2000 Subject classifications: 39A11.
1 Introduction For the difference equations and functional difference equations, the existence of almost periodic solutions of almost periodic systems has been studied by many authors. One of the most popular method is to assume the certain stability properties ([1-5]; T Itokazu and Y Hamaya, unpublished work). Recently, [6-8] have shown the existence of pseudo almost periodic solutions of difference equations, differential equations, and abstract differential equations. On the other hand, in the case of almost periodic solutions of linear Volterra systems, [3] has shown that if the zero solution of the linear Volterra equation is uniformly asymptotically stable, then the system has a unique almost periodic solution. In this article, we shall give some characterizations for the exponentially asymptotically stable of the zero solution of equation and in order to obtain the existence theorem for a pseudo almost periodic solutions of linear Volterra difference equations, we discusse to improve Hamanaka and Hamaya’s result [4], for Volterra integro differential equations, to theorems for pseudo almost periodic linear Volterra difference equations. To be best of author’s observation, no article has been published regarding the investigation of pseudo almost periodicity of a linear Volterra difference equation with infinite delay. Thus, our results are presents a new originality for Volterra type difference equations. Let Rl denotes the l-dimensional Euclidean space, Z is the set of integers, Z+, Z- are the set of nonnegative and nonpositive integers, respectively, and |x| will denote the norm of x in R l. For any interval J ⊂ Z, we denote by BS = BS(J : Rl) the set of all bounded functions mapping J into Rl and set |j|J = sup{|j(s)| : s Î J} for j Î BS. In this article, all interval are discrete, e.g., [0, n0] = {0, 1, 2,..., n0}, n0 Î Z+. © 2012 Hamaya; 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.
Hamaya Advances in Difference Equations 2012, 2012:58 http://www.advancesindifferenceequations.com/content/2012/1/58
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We first introduce an almost
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