Pseudo-Almost Periodic and Pseudo-Almost Automorphic Solutions to Some Evolution Equations Involving Theoretical Measure Theory

Abstract

Toka DiaganaHoward University,2441 6th Street N.W.,Washington, D.C. 20059,USA.tdiagana@howard.eduKhalil EzzinbiUniversit´e Cadi Ayyad,Faculte´ des Sciences Semlalia,D´epartement deMath´ematiques,BP 2390, Marrakesh, MarocMohsen MiraouiInstitut sup´erieur des Etudestechnologiques de Kairouan,Rakkada-3191 Kairouan,Tunisie.ABSTRACTMotivated by the recent works by the first and the second named authors, in this paperwe introduce the notion of doubly-weighted pseudo-almost periodicity (respectively,doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basicproperties of these new spaces are studied. To illustrate our work,westudy,underAcquistapace–Terreni conditions and exponential dichotomy, the existence of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions tosome nonautonomous partial evolution equations in Banach spaces. A few illustrativeexamples will be discussed at the end of the paper.RESUMENMotivado por los trabajos recientes del primer y segundo autor, en este art´iculo intro-ducimos la noci´on de seudo-casi periodicidad con doble peso (seudo-casi automorf´ia condoble peso respectivamente) usando Teor´ia de la Medida. Se estudian las propiedadesbasicas de estos espacios nuevos. Para ilustrar nuestro trabajo, bajo las condiciones deAcquistapace-Terreni y dicotom´ia exponencial estudiamos la existencia de soluciones(respectivamente, (µ,ν) seudo-casi peri´odicas (µ,ν) seudo-casi automorficas) para al-gunas ecuaciones parciales de evoluci´on autonomas en espacios de Banach. Algunosejemplos ilustrativos se discutiran al final del art´iculo.Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni con-ditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomousequation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy;(µ,ν)-pseudo-almost periodicity; (µ,ν)-pseudo-almost automorphy; neutral systems; positive mea-sure.2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60;47D06; 28Axx; 58D25; 65J08.