Existence of bounded uniformly continuous mild solutions on $\mathbb{R}$ of evolution equations and their asymptotic behaviour

Abstract

We prove that $u^{\prime }=Au+\varphi$ has on $\mathbb{R}$ a mild solution $u_{\varphi }\in BUC(\mathbb{R},X)$ (that is bounded and uniformly continuous), where $A$ is the generator of a $C_0$-semigroup on the Banach space $X$ with resolvent satisfying $||R(it,A)||=O(|t{|}^{-\theta })$, $|t|\to \infty$, with some $\theta >1/2$, $\varphi \in L^{\infty }(\mathbb{R},X)$ and $isp(\varphi )\cap \sigma (A)=\varnothing$. As a consequence it is shown that if ${\Cal F}$ is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or certain classes of recurrent functions and $\varphi$ as above is such that $M_h ϕ:=(1/h)\int_0^h ϕ(\cdot+s)\, ds \in \Cal {F}$ for each $h>0$, then $u_ϕ\in \Cal {F}\cap BUC (\Bbb{R},X)$. These results seem new and strengthen several recent theorems.

arXiv comment17 pages