Existence of bounded uniformly continuous mild solutions on $\mathbb{R}$ of evolution equations and some applications

Abstract

We prove that there is $x_{\varphi }\in X$ for which (*)$\frac{du(t)}{dt}=Au(t)+\varphi (t)$, $u(0)=x$ has on $\r$ a mild solution $u\in C_{ub}(\mathring{,}X)$ (that is bounded and uniformly continuous) with $u(0)=x_{\varphi }$, where $A$ is the generator of a holomorphic $C_0$-semigroup $(T(t){)}_{t\ge 0}$ on $X$ with sup ${}_{t\ge 0}||T(t)||<\infty$, $\varphi \in L^{\infty }(\mathring{,}X)$ and $isp(\varphi )\cap \sigma (A)=\varnothing$. As a consequence it is shown that if $\n$ is the space of almost periodic $AP$, almost automorphic $AA$, bounded Levitan almost periodic $LA{P}_b$, certain classes of recurrent functions $RE{C}_b$ and $\varphi \in L^{\infty }(\mathring{,}X)$ such that $M_h ϕ:=(1/h)\int_0^h ϕ(\cdot+s)\, ds \in \n$ for each $h>0$, then $u\in \n\cap C_{ub}$. These results seem new and generalize and strengthen several recent Theorems.

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