Strongly damped wave equations with Stepanov type nonlinear forcing term

Edgardo Alvarez,Tovias Castro,John Moreno

doi:10.1186/s13662-018-1770-0

Edgardo Alvarez, Tovias Castro + Show 1 more

Open Access

https://doi.org/10.1186/s13662-018-1770-0

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Abstract

In this paper we investigate the existence and uniqueness of weighted pseudo almost automorphic mild solution for a class of strongly damped wave equations where the semilinear forcing term is a Stepanov weighted pseudo almost automorphic function.

Highlights

Let X be a Banach reflexive space, let us assume that A : D(A) ⊂ X → X is a closed densely defined operator, η > 0, and θ ∈ [1/2, 1]
The main purpose of this paper is the study of existence and uniqueness of weighted pseudo almost automorphic mild solutions of the abstract Cauchy problem
We present some definitions and main results of Stepanov weighted pseudo almost automorphic functions

Summary

Introduction

Let X be a Banach reflexive space, let us assume that A : D(A) ⊂ X → X is a closed densely defined operator, η > 0, and θ ∈ [1/2, 1]. Xia et al in [20] worked with two suitable measures and they investigated the existence and uniqueness of (μ, ν)-pseudo almost automorphic mild solutions to a semilinear fractional differential equation with Riemann–Liouville derivative in a Banach space, where the nonlinear perturbation is of (μ, ν)-pseudo almost automorphic type or Stepanov-like (μ, ν)-pseudo almost automorphic type. To the best of our knowledge, the fact that the existence and uniqueness of weighted pseudo almost automorphic mild solutions to (1.1) with the forcing term f belong to the space of Stepanov weighted pseudo almost automorphic functions is an untreated original problem, which constitutes one of the main motivations of this work. A function f ∈ BSp(R, X) is said to be weighted Stepanov-like pseudo almost automorphic if it can be decomposed as f = g + φ, where gb ∈ AA(R, Lp([0, 1], X)) and φb ∈ PAA0(R, Lp([0, 1], X), ρ).

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