Generalized almost periodic and generalized asymptotically almost periodic type functions in Lebesgue spaces with variable exponents $L^{p(x)}$

TL;DR

This paper introduces and analyzes generalized almost periodic functions with variable exponents in Lebesgue spaces, extending classical concepts and applying them to Volterra integro-differential equations in Banach spaces.

Contribution

It defines new classes of generalized almost periodic functions with variable exponents and explores their properties and applications in differential equations.

Findings

Defined new classes of generalized almost periodic functions with variable exponents

Analyzed properties of these functions in Lebesgue spaces with variable exponents

Applied the concepts to abstract Volterra integro-differential equations

Abstract

In the paper under review, we introduce the notions of various types of generalized (asymptotical) almost periodicity with variable exponents. We define and thoroughly analyze an important subclass of (asymptotically) Stepanov almost periodic functions which contains all (asymptotically) almost periodic functions. We provide a great number of relevant applications to abstract Volterra integro-differential equations in Banach spaces.