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New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations. (English) Zbl 1316.34064

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.

MSC:

34G20 Nonlinear differential equations in abstract spaces
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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