Besicovitch almost periodic solutions of abstract semi-linear differential equations with delay

TL;DR

This paper introduces a new class of Besicovitch almost periodic functions, explores their properties, and applies these concepts to establish existence and uniqueness results for solutions of certain delayed semi-linear differential equations.

Contribution

It defines Besicovitch almost periodic functions via the Bohr property, investigates their properties, and applies the contraction principle to prove solution existence and uniqueness.

Findings

Established equivalence of Bohr and Bochner properties for Besicovitch almost periodic functions.

Proved existence and uniqueness of solutions for a class of semi-linear differential equations with delay.

Extended the theory of almost periodic functions to include Besicovitch type in the context of differential equations.

Abstract

In this paper, we first use the Bohr property to give a definition of Besicovitch almost periodic functions, and study some basic properties of Besicovitch almost periodic functions, including the equivalence of the Bohr property and the Bochner property. Then, as an application, we use the contraction principal to obtain the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear differential equations with delay.