Almost periodic solutions in an integrodifferential equation

Abstract

SynopsisWe consider a system of integrodifferential equationswhere f(t, x) and F(t, s, x, y) are almost periodic in t uniformly for parameters, and we assume that the system has a bounded solution u(t). To discuss the existence of an almost periodic solution, we consider the relationship between the total stability of u(t) with respect to a certain metric ρ and the separation condition with respect to ρ. Moreover, we discuss a sufficient condition for the existence of a positive almost periodic solution of a model of the dynamics of an n-species system.

Stability Properties in Nonlinear Discrete Volterra Equations With Unbounded Delay

“…(1.1) by means of (K, ρ)-stability conditions. Their results are to extend results of Hamaya [4] to discrete Volterra equations. Also, Song [11] proved that if the bounded solution of (1.1) is uniformly asymptotically stable, then (1.1) has an almost periodic solution.…”

“…In [5], Hamaya studied the relationship between total stability and stability under disturbances from hull for Eq. (1.2).…”

“…Choi and Koo [2] investigated the existence of an almost periodic solution of (1.1) as a discretization of the Hamaya's results in [5]. Also, Choi et al [3] studied the total stability for the discrete Volterra equation…”

“…Hamaya [5] showed that two concepts of (K, ρ)-weak uniform asymptotic stability and (K, ρ)-uniform asymptotic stability for (1.1) are equivalent. Moreover, he obtained the existence of almost periodic solutions in (1.1) by using (K, ρ)-weak uniform asymptotic stability.…”

“…This definition is a discrete analogue of Hamaya's definition in [5]. We recall the definitions of various stabilities in [7].…”