Local existence of Lipschitz-continuous solutions of systems of nonlinear functional equations with iterated deviations
type: Item
instance of: scholarly article
title: Local existence of Lipschitz-continuous solutions of systems of nonlinear functional equations with iterated deviations
zbMATH Open document ID: 1251.39013
author: Stevo Stević
published in: Applied Mathematics and Computation
publication date: 2012-09-13T00:00:00Z
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 65Q20
zbMATH DE Number: 6081851
zbMATH Keywords: nonlinear functional equations
zbMATH Keywords: unique existence
zbMATH Keywords: local solution
zbMATH Keywords: Lipschitz-continuous solutions
zbMATH Keywords: deviating argument, successive approximations
zbMATH Keywords: Weierstrass criterion.
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
full work available at URL: j.amc.2012.04.016
OpenAlex ID: W2049907453
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DOI: 10.1016/J.AMC.2012.04.016
review text: The following functional equation is studied\newline NEWLINE\[NEWLINE(E):\quad X(t)=F[t,\big(X(\alpha_j t +\varphi_j(t,X(\beta_j t+\psi_j(t,X(t)))))\big)_{j=1}^{k}],NEWLINE\]NEWLINE where \(\alpha_j\) and \(\beta_j\) are given real constants, \(1\leq j\leq k\), \(\varphi_j\) and \(\psi_j\) are real continuous functions defined in certain domain of \(\mathbb{R}^{n+1}\) containing the origin \((0,\mathbf{0})\in\mathbb{R}\times\mathbb{R}^n\), and \(F\) is a real vector continuous function defined in an appropriate neighbourhood of \((0,\mathbf{0},\ldots,\mathbf{0})\in\mathbb{R}\times(\mathbb{R}^n)^k\). Here the unknown \(X\) is a real vector function defined in an open interval \(J\) centered at the origin with values in a subset of \(\mathbb{R}^n\). A similar study was done by \textit{G. P. Pelyukh} [Ukrainian Math. J. 54, 75--90 (2001; Zbl 0985.39021)] in the case for which \(\beta_j=1\) and \(\psi_j\equiv 0,\) \(j=1,\ldots,k.\)NEWLINENEWLINEThe author imposes some sufficient conditions on the functions \(F, \varphi_j\) and \(\psi_j\) and on the coefficients \(\alpha_j,\beta_j\) in order to guarantee that Equation~\((E)\) has a unique Lipschitz continuous solution \(X(t)\) defined in a suitable neighborhood of the origin. It is interesting to notice that the conditions imposed to the real functions are of Lipschitzian type and that for constructing the solution the author employs an iterative method analogous to that of the successive approximations considered in the classical proof of the first order differential equation \(Z'=f(t,Z)\) being \(f\) a Lipschitz continuous map.
description: scientific article; zbMATH DE number 6081851