On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order
type: Item
instance of: scholarly article
title: On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order
zbMATH Open document ID: 1173.45003
author: Beata Rzepka
published in: Topological Methods in Nonlinear Analysis
publication date: 2009-04-28T00:00:00Z
review text: The author considers the quadratic Volterra integral equation of fractional order \[ x(t)=p(t)+\frac{f(t,x(t))}{\Gamma(\alpha)}\int_0^t\frac{u(t,s,x(s))}{(t-s)^{1-\alpha}}ds,\;\;t\in \mathbb{R}_+,\tag{1} \] where \(\alpha\in (0,1)\) and \(p\), \(f\) and \(u\) are continuous functions. Using the measure of noncompactness along with Schauder's fixed point theorem, she shows, under some natural -- but technical -- assumptions too involved to be reproduced here, that the problem above has at least one bounded and continuous solution \(x:\mathbb{R}_+\to \mathbb{R}\). In addition, she proves that the solutions of the equation (1) are uniformly locally attractive.
reviewed by: Ioan I. Vrabie
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45M10
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 45M05
zbMATH DE Number: 5548696
zbMATH Keywords: quadratic Volterra integral equation of fractional order
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: local attractivity
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Schauder's fixed point theorem
zbMATH Keywords: bounded and continuous solution
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5548696