Almost Periodic Solutions for Stepanov-Almost Periodic Differential Equations

Abstract

In this paper, we study the existence of almost periodic solutions and also the nonexistence of non-uniformly continuous Stepanov-almost periodic solutions of the $n$th-order differential equation $u^{(n)}(t)=Au(t)+f(t),\;n\in \mathbb N ,\;t\in \mathbb R ,$ in a Banach space $\mathbb X ,$ where $A:\mathbb X \rightarrow \mathbb X$ is a nonzero bounded linear operator and $f:\mathbb R \rightarrow \mathbb X$ is a Stepanov-almost periodic continuous function. We also study the existence of an almost periodic solution to $u^{(n)}(t)=Au(t)+f(t,u(t)),$ where $f:\mathbb R \times \mathbb X \rightarrow \mathbb X$ is a Stepanov-almost periodic continuous function satisfying a suitable Lipschitz condition.