Stepanov pseudo almost periodic functions and applications

TL;DR

This paper explores Stepanov pseudo almost periodic functions with measures, establishing new composition results and applying them to prove existence and uniqueness of solutions for certain semilinear differential equations in Banach spaces.

Contribution

It introduces a new composition theorem for Stepanov pseudo almost periodic functions with measures under minimal assumptions and applies it to differential equations.

Findings

Proved a new composition result for $$-pseudo almost periodic functions.

Established existence and uniqueness of solutions for semilinear fractional inclusions.

Provided examples illustrating theoretical results.

Abstract

In this work, we present basic results and applications of Stepanov pseudo almost periodic functions with measures. Using only the continuity assumption, we prove a new composition result of $\mu$-pseudo almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions in Banach spaces with weak regular forcing terms. We prove the existence and uniqueness of $\mu$-pseudo almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations, respectively, provided that the nonlinear forcing terms are only Stepanov $ \mu $-pseudo almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Some examples illustrating our theoretical results are also presented.