An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions to difference equations in Banach spaces
type: Item
instance of: scholarly article
title: An application of Krasnoselskii fixed point theorem to the asymptotic behavior of solutions to difference equations in Banach spaces
zbMATH Open document ID: 0962.39007
author: Cristóbal González
author: Antonio Jiménez-Melado
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-06-14T00:00:00Z
review text: Let \(X\) be a Banach space, let \(f:X\to X\) be a Lipschitz mapping and let \(g:X\to X\) be a compact operator. Under some additional assumptions, for each \(x\in X\), there exists a solution \((x_n)\in l_\infty(X)\) to the difference equation \[ \Delta x=\sum^\infty_{i=0} a^i_n f(x_{n+i}) +\sum^n_{i=0} b^i_n g(x_{n+i}) +y_n, \] which is asymptotically equal to \(x\). The proof of this is based on the Krasnoselskii fixed point theorem. Next this result is applied to obtain a similar result for a second order difference equation \[ \Delta(q_n \Delta x_n)+ r_nf(x_n)+ s_ng(x_n)+ z_n=0. \]
reviewed by: Dobiesław Bobrowski
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 1497303
zbMATH Keywords: Banach space
zbMATH Keywords: compact operator
zbMATH Keywords: difference equation
zbMATH Keywords: Krasnoselskii fixed point theorem
zbMATH Keywords: second order difference equation
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6877
OpenAlex ID: W2047056613
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DOI: 10.1006/JMAA.2000.6877
description: scientific article; zbMATH DE number 1497303