Composition and convolution theorems for mu-Stepanov pseudo almost periodic functions and applications to fractional integro-differential equations

Abstract

In this article we establish new convolution and composition theorems for $\mu$-Stepanov pseudo almost periodic functions. We prove that the space of vector-valued $\mu$-Stepanov pseudo almost periodic functions is a Banach space. As an application, we prove the existence and uniqueness of $\mu$-pseudo almost periodic mild solutions for the fractional integro-differential equation $$ D^\alpha u(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)\,ds+f(t,u(t)), $$ where A generates an $\alpha$-resolvent family $\{S_\alpha(t)\}_{t\geq 0}$ on a Banach space X, $a\in L^1_{\rm loc}(\mathbb{R}_+)$, $\alpha>0$, the fractional derivative is understood in the sense of Weyl and the nonlinearity f is a $\mu$-Stepanov pseudo almost periodic function.

Keywords

• mu-Stepanov pseudo almost periodic
• mild solutions,
• fractional integro-differential equations
• composition
• convolution