Let X be an arbitrary Banach space. This work deals with the asymptotic behavior, the continuity and the compactness properties of solutions of the non-linear Volterra difference equation in X described by u(n + 1) = λ n j=−∞ a(n − j)u(j) + f (n, u(n)), n ∈ Z, for λ in a distinguished subset of the complex plane, where a(n) is a complex summable sequence and the perturbation f is a non-Lipschitz nonlinearity. Concrete applications to control systems and integro-difference equations are given.
C. Cuevas et al. / Asymptotic analysis for Volterra difference equationssubmitted to a non-linear perturbation f , where λ is a complex number and a(n) is a C-valued summable function, that is ∞ n=0 |a(n)| < +∞. Such a theory for non-linear equations like Eq. (1.1) is largely nonexistent at this time 1 and consequently should be widely investigated, so that to produce a progress in the qualitative theory of non-linear Volterra difference equations. 2 When the perturbation of Eq. (1.1) is a Lipschitz nonlinearity it induces a contraction operator (see (3.6)) and consequently we can get uniqueness of solutions. Investigations when the perturbations are not necessarily globally Lipschitz are technically more complicated, because the perturbation induces an operator which left to be a contraction (so we loss uniqueness of solutions) but in many concrete situations it is a compact operator. To understand this situation is crucial to deal with more general fixed point theorems than the contraction principle. The usefulness of the fixed point methods for applications has increased enormously due to the development of accurate and efficient techniques for computing fixed points, making fixed point arguments a powerful weapon. Our results are a consequence of application of some theorems in classical functional analysis, specifically, we will use the Schauder's fixed point theorem, the Leray-Schauder alternative theorem, the Krasnosel'skiȋ theorem (Theorem 3.4) and a Mönch's fixed point theorem (Theorem 3.6). We remark that to use the latter four results, some compactness assumptions must to be imposed on the perturbations. We emphasize that the implementation of this approach is a priori not trivial as the reader will find out from this work. Anticipating a wide interest in the subject, this paper contributes in filling this important gap.Volterra difference equations can be considered as natural generalization of difference equations. During the last few years Volterra difference equations have emerged vigorously in several applied fields and nowadays there is a wide interest in developing the qualitative theory for such equations. 3 In recent years, there has been an increasing interest in the study of the asymptotic behavior of the solutions of Volterra difference equations. One of the central interest in the asymptotic behavior subject is to find conditions in order that the solutions to be stable, unstable and especially to have an exponential dichotomy (see [2,8,20,26,27,32,51,52] and the references therein). Elaydi et al. [20...
