Stepanov-almost periodic solutions of non-autonomous neutral functional differential equations with functional delay
type: Item
instance of: scholarly article
title: Stepanov-almost periodic solutions of non-autonomous neutral functional differential equations with functional delay
zbMATH Open document ID: 1431.34082
author: Md. Maqbul
published in: Mediterranean Journal of Mathematics
publication date: 2018-10-02T00:00:00Z
review text: The paper deals with the following non-autonomous neutral functional differential equation in a Banach space \(\mathbb{X}\): \[\frac{d}{dt}\left[u(t)-F(t,u(t-\alpha(t)))\right]= A(t)\left[u(t)-F(t,u(t-\alpha(t)))\right] + G(t,u(t),u(t-\alpha(t))),\quad t\in \mathbb{R}\tag{1} \] where \(A(t), \, t\in \mathbb{R}\) is a family of linear operator. The operator \((\gamma I -A(t))^{-1}\) and the real function \(\alpha\) defined on \( \mathbb{R}\) are almost periodic. \(F: \mathbb{R}\times \mathbb{X}\times \mathbb{X} \to \mathbb{X}\) and \(G: \mathbb{R}\times \mathbb{X}\times \mathbb{X} \to \mathbb{X}\) are Stepanov-almost periodic continuous functions satisfying Lipschitz conditions. Under the assumption that family of linear operator \(A(t), \, t\in \mathbb{R}\) satisfies the `Acquistapace-Terreni' condition and the evolution family generated by \(A(t), \, t\in \mathbb{R}\) is exponentially stable, the author proves the existence and uniqueness of Stepanov-almost periodic mild soulution to (1). The result is achieved by means of Banach's fixed point theorem.
reviewed by: Gisèle M. Mophou
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K40
Mathematics Subject Classification ID: 34K14
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 6946427
zbMATH Keywords: Stepanov-almost periodic function
zbMATH Keywords: almost periodic function
zbMATH Keywords: evolution family
zbMATH Keywords: neutral functional differential equation
zbMATH Keywords: contraction mapping principle
MaRDI profile type: Publication
full work available at URL: s00009-018-1224-7
OpenAlex ID: W2884013696
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Wikidata QID: Q129507236
DOI: 10.1007/S00009-018-1224-7
description: scientific article; zbMATH DE number 6946427