Quasi-asymptotically almost periodic functions and applications

TL;DR

This paper introduces and studies quasi-asymptotically almost periodic functions in Banach spaces, extending existing classes and analyzing their invariance properties, with applications to differential equations.

Contribution

It defines new classes of quasi-asymptotically almost periodic functions and explores their invariance under convolution, applying these results to nonautonomous differential equations.

Findings

Invariance of quasi-asymptotically almost periodic functions under convolution.

Extension of classical almost periodic function classes.

Application to abstract differential equations.

Abstract

The main aim of this paper is to consider the classes of quasi-asymptotically almost periodic functions and Stepanov quasi-asymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions, Stepanov asymptotically almost periodic functions and S-asymptotically $\omega$-periodic functions with values in Banach spaces. We investigate the invariance of introduced properties under the action of finite and inifinite convolution products, providing also an illustrative application to abstract nonautonomous differential equations of first order.