# Almost automorphic and asymptotically almost automorphic type functions in Lebesgue spaces with variable exponents $L^{p(x)}$

## TL;DR

This paper explores a new class of almost automorphic functions with variable exponents in Lebesgue spaces, extending existing concepts and applying them to Volterra integro-differential inclusions in Banach spaces.

## Contribution

It introduces and analyzes (asymptotically) Stepanov almost automorphic functions with variable exponents, expanding the theoretical framework and applications in differential equations.

## Key findings

- Established properties of variable exponent almost automorphic functions
- Connected these functions to solutions of Volterra integro-differential inclusions
- Provided applications demonstrating the theoretical results

## Abstract

The paper introduces and studies the class of (asymptotically) Stepanov almost automorphic functions with variable exponents. Any function belonging this class needs to be (asymptotically) Stepanov almost automorphic. A few relevant applications to abstract Volterra integro-differential inclusions in Banach spaces is presented.