New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations
This paper deals with the following non-autonomous evolution equation: \[ u'(t)=A(t)u(t)+f(t,u(t)),\,\, t\in \mathbb{R},\tag{1} \] where \(A(t)\) satisfies the ``Acquistapace-Terreni'' condition, and generates an evolution family \((U(t,s))_{t\geq s}\) with exponential dichotomy. The function \(f:\mathbb{R}\times \mathbb{X}\to \mathbb{X}\) is a Stepanov-like weighted pseudo almost automorphic function. The authors start by proving some new composition theorems of Stepanov-like weighted pseudo almost automorphic function. Then under suitable conditions on \(f\), they prove the existence and uniqueness of solution of (1). The results are obtained by means of Banach fixed point theorem. An example is also given to illustrate the main results.
Stepanov-like weighted pseudo almost automorphic
scientific article; zbMATH DE number 6118825