Weak almost periodic solutions of differential equations

Abstract

A very important question in many practical situations is to known if there exists a mean value of a bounded solution x of a differential equation x’ =f(t, x), By the mean value we understand the limit lim,,, (l/T) ]lx(f) dr if it exists. Eberlein in [l] has found a class of functions which have the mean value-the weak almost periodic (wap) functions. A bounded continuous function x is weak almost periodic if the set of its translations (x,: s E R} is relatively compact in the weak topology of the space of bounded continuous functions on R. The purpose of this paper is to examine the properties of wap solutions of the differential equation. We demonstrate theorems on the existence of wap solution of the differential equation with the weak almost periodic right-hand side.