Convolutions in (µ, ν)-pseudo-almost periodic and (µ, ν)-pseudo-almost automorphic function spaces and applications to solve integral equations

Abstract

In this paper we give sufficient conditions on $k\in L^1(\mathbb{R})$ and the positive measures $\mu$, $\nu$ such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo-almost automorphic) function spaces are invariant by the convolution product $\zeta f=k\ast f$. We provide an appropriate example to illustrate our convolution results. As a consequence, we study under Acquistapace-Terreni conditions and exponential dichotomy, the existence and uniqueness of $\left( \mu,\nu\right)$- pseudo-almost periodic (respectively, $\left( \mu,\nu\right)$- pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.

Keywords

• measure theory
• μ-ν-ergodic
• μ-ν -pseudo almost periodic and automorphic functions
• evolution families
• nonautonomous equations
• neutral systems