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

TL;DR

This paper proves the existence of bounded, uniformly continuous mild solutions for certain evolution equations on , , and their asymptotic behavior, extending previous results with new conditions on the resolvent and forcing term.

Contribution

It establishes the existence of bounded, uniformly continuous solutions for evolution equations with specific resolvent decay and spectral conditions, and links solutions to classes of recurrent functions.

Findings

Existence of mild solutions under resolvent decay condition

Solutions belong to classes of almost periodic or recurrent functions

Strengthens previous theorems on evolution equations

Abstract

We prove that $u'= A u + \phi $ has on $\Bbb{R}$ a mild solution $u_{\phi}\in BUC (\Bbb{R},X)$ (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 $\theta > 1/2$, $\phi\in L^{\infty} (\Bbb{R},{X})$ and $i\,sp (\phi)\cap \sigma (A)=\emptyset$. 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 $\phi$ as above is such that $M_h \phi:=(1/h)\int_0^h \phi (\cdot+s)\, ds \in \Cal {F}$ for each $h >0$, then $u_{\phi}\in \Cal {F}\cap BUC (\Bbb{R},X)$. These results seem new and strengthen several recent theorems.