Existence and Stability Property of Almost Periodic Solutions in Discrete Almost Periodic Systems

Abstract

In this paper, we consider an almost periodic system which includes a system of the type , where <i>k</i> is a positive integer, a_ij are almost periodic in n and satisfy a_ij(n)≥0 for i≠j, for 1≤j≤m. In the special case where a_ij(n) are constant functions, above system is a mathematical model of gas dynamics and was treated by T. Carleman and R. D. Jenks for differential systems. In the main theorem, we show that if the m X m matrix (a_ij(n)) is irreducible, then there exists a positive almost periodic solution which is unique and has some stability. Moreover, we can see that this result gives R. D. Jenks’ result for differential model in the case where a _ij (n) are constant functions. In Section 3, we consider the linear system with variable cofficients . Even in nonlinear problems, this linear system plays an important role, as their variational equations, and it is requested to determine the uniform asymptotically stability of the zero solution from the information about A(n). In order to obtain the existence of almost periodic solutions of both linear and nonlinear almost periodic discrete systems: above linear system and for 1≤i≤m, respectively, we shall consider between certain stability properties, which are referred to as uniformly asymptotically stable, and the diagonal dominance matrix condition.