Pseudo-almost periodic and pseudo-almost automorphic solutions to some evolution equations involving theoretical measure theory

Toka Diagana; Khalil Ezzinbi; Mohsen Miraoui

doi:10.4067/S0719-06462014000200001

Authors

Toka Diagana Howard University, 2441 6th Street N.W., Washington, D.C. 20059, USA.
Khalil Ezzinbi Université Cadi Ayyad, Faculté des Sciences Semlalia, Département de Mathématiques, BP 2390, Marrakesh, Morocco.
Mohsen Miraoui Institut supérieur des Études technologiques de Kairouan, Rakkada-3191 Kairouan, Tunisie.

DOI:

https://doi.org/10.4067/S0719-06462014000200001

Keywords:

Evolution family, exponential dichotomy, Acquistapace–Terreni conditions, pseudo-almost periodic, pseudo-almost automorphic, evolution equation, nonautonomous equation, doubly-weighted pseudo-almost periodic, doubly-weighted pseudo-almost automorphy, (Î¼, Î½)-pseudo-almost periodicity, (Î¼, Î½)-pseudo-almost automorphy, neutral systems, positive measure

Abstract

Motivated by the recent works by the first and the second named authors, in this paper we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively, doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic properties of these new spaces are studied. To illustrate our work, we study, under Acquistapace–Terreni conditions and exponential dichotomy, the existence of (Î¼,Î½)- pseudo almost periodic (respectively, (Î¼,Î½)-pseudo almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces. A few illustrative examples will be discussed at the end of the paper.