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

TL;DR

This paper proves the existence of bounded, uniformly continuous mild solutions for certain evolution equations on , and explores their properties within classes like almost periodic and recurrent functions, generalizing previous results.

Contribution

It establishes new conditions for the existence of bounded, uniformly continuous solutions to evolution equations with holomorphic semigroups, extending prior theorems in the field.

Findings

Existence of mild solutions in bounded, uniformly continuous function space.

Solutions belong to classes like almost periodic, almost automorphic, and recurrent functions.

Results generalize and strengthen recent theorems in evolution equations.

Abstract

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