On the solvability of a Volterra integral equation of quadratic form in the class of continuous function
type: Item
instance of: scholarly article
zbMATH Open document ID: 1289.45004
author: E. M. El-Abd
author: S. A. El-Shehawy
publication date: 2013-10-18T00:00:00Z
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 6217225
zbMATH Keywords: existence of solutions
zbMATH Keywords: Darbo fixed point theorem
zbMATH Keywords: Volterra integral equation
zbMATH Keywords: Hausdorff measure of noncompactness
MaRDI profile type: Publication
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title: On the solvability of a Volterra integral equation of quadratic form in the class of continuous function
published in: Functional Analysis, Approximation and Computation
review text: By using the Darbo fixed point theorem, the authors prove the existence of solutions for the Volterra integral equation \(x(t)=1+Tx(t)\int_0^tk(t,s)\phi(s)x(s)\,ds\), \(t\in[0,a]\), in the class of continuous functions \(C([0,a])\), where \(\phi:[0,a]\to\mathbb R\) and \(k:[0,a]\times[0,a]\to\mathbb R\) are continuous, \(T:C([0,a])\to C([0,a])\) is a bounded and continuous operator satisfying the Darbo condition \(\chi(TX)\leq\alpha\chi(X)\) for any subset \(X\subset C([0,a])\) with \(\alpha\in[0,1)\), and \(\chi\) is the Hausdorff measure of noncompactness.
reviewed by: Rodica Luca
description: scientific article; zbMATH DE number 6217225