A new approach to numerical solution of fixed-point problems and its application to delay differential equations
type: Item
instance of: scholarly article
title: A new approach to numerical solution of fixed-point problems and its application to delay differential equations
zbMATH Open document ID: 1191.65059
published in: Applied Mathematics and Computation
publication date: 2010-03-31T00:00:00Z
review text: The article deals with the fixed point problem for an operator \(A\) acting in a metric space \((L,d)\) with an additional approximation structure. The family of spaces \(S_h\) and the families of operators \(r_h:L\to S_h\) and \(p_h:S_h\to L\) define a convergent approximation of \(L\) if for any \(x \in L\) \[ d(p_nr_nx,x)\big|_{h \to 0} \to 0 \] holds. The base of the author's arguments is the following statement (Theorem 2.1): If an operator \(A\) is continuous in \(L\) and the family of spaces \(S_h\) and the families of operators \(r_h\) and \(p_h\) define a convergent approximation of \(L\), then \(A\) possesses at least one fixed point if and only if there exists a non-negative function \(\varepsilon(h)\), \(\varepsilon(h) \to 0\) for \(h \to 0\), such that the operator \(A\) possesses \(\varepsilon(h)\)-fixed-points \(p_hx_h\) and the family \(\{p_hx_h\}\) is compact with \(h \to 0\). Recall that \(x \in L\) is an \(\varepsilon\)-fixed point of \(A\) if \(f(x,Ax)\leq \varepsilon\) and the family \(x^h\in L\) is compact with \(h\to 0\) if for any sequence \(x^{h_n}\) with \(h_n\to 0\) there exists a convergent subsequence of \(x^{h_n}\). As application the author considers the following boundary value problem \[ y''(x)= f(x,y(x),y'(x),y(\alpha(x))), \quad a\leq x\leq b, \] \[ y(x)= \varphi(x), \qquad \alpha_{\min} \leq x \leq a, \qquad y(b) = 0. \] Numerical realization of the last problem for three concrete triples of functions \((f,\alpha,\varphi)\) is also presented.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L03
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34K28
zbMATH DE Number: 5688910
zbMATH Keywords: fixed points
zbMATH Keywords: approximation schemes
zbMATH Keywords: boundary value problems
zbMATH Keywords: delay differential equations
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
zbMATH Keywords: metric space
author: Zbigniew Bartoszewski
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.amc.2009.12.058
OpenAlex ID: W2039217806
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DOI: 10.1016/J.AMC.2009.12.058
description: scientific article; zbMATH DE number 5688910