Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations
type: Item
instance of: scholarly article
title: Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations
zbMATH Open document ID: 1213.34089
author: Junfei Cao
author: Zaitang Huang
author: Qi-Gui Yang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-11-05T00:00:00Z
review text: The aim of this paper is to study the existence of classical solutions to the following fractional integro-differential equation \[ \frac{d^\alpha x(t)}{dt^\alpha}= Ax(t)+f(t,x(t),G x(t)),\tag{1} \] where \(\frac{d^\alpha x(t)}{dt^\alpha}\) stands for the Riemann-Liouville derivative of order \(\alpha\), \(0<\alpha<1\), \(A\) is the infinitesimal generator of an analytic semigroup \(\{Q(t)\}_{t\geq 0}\) in a Banach space \(\mathbb{X},\) \(f:\mathbb{R}\times \mathbb{X}_q\times \mathbb{X}_q\to \mathbb{X}\) is a suitable function, and \(\mathbb{X}_q\) a Banach space. \(Gx(t)\), which may be interpreted as a control of the system, is defined by \(Gx(t)=\int_{t_0}^t k(t,s,x(s))\,ds\) and \(k:D\times\mathbb{X}_q\to \mathbb{X}_q\), \(D=\{(t,s):t_0\leq s\leq t\leq T\}\). By means of the contraction mapping, the authors prove the existence and uniqueness of a classical solution of the initial value problem associated to (1). Then, they show the existence and uniqueness of an optimal mild solution among all the solutions of (1) which are bounded over \(\mathbb{R}\). Note that the notion of optimal solution was introduced by \textit{G. M. N'Guérékata} in [Riv. Mat. Univ. Parma, IV. Ser. 9, 145--151 (1983; Zbl 0547.34049)]. Finally, they study sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic classic solution. An example is also given to illustrate the abstract results.
reviewed by: Gisèle M. Mophou
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34C27
zbMATH DE Number: 5810954
zbMATH Keywords: optimal mild solutions
zbMATH Keywords: weighted pseudo-almost periodic classical solutions
zbMATH Keywords: fractional integro-differential equations
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
full work available at URL: j.na.2010.08.036
OpenAlex ID: W2026483097
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DOI: 10.1016/J.NA.2010.08.036
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description: scientific article; zbMATH DE number 5810954