Method of locally linear approximation of nonlinear difference operators by weakly regular operators
type: Item
instance of: scholarly article
title: Method of locally linear approximation of nonlinear difference operators by weakly regular operators
zbMATH Open document ID: 1272.39006
published in: Journal of Mathematical Sciences (New York)
publication date: 2013-04-24T00:00:00Z
review text: The author studies the difference operator \[ (\mathcal{F}\mathbf{x})_n:=x_n+f_n(x_{n-1},\dots,x_{n-p}),\quad n\in\mathbb{Z},\;p\in\mathbb{N}, \] acting in the Banach space \(\ell_{\infty}(\mathbb{Z},E)\) of two-sided sequences \(\mathbf{x}=(x_n)_{n\in\mathbb{\scriptstyle{Z}}}\), \(x_n\in E\), with the norm \[ \|\mathbf{x}\|_{\ell_{\infty}(\mathbb{\scriptstyle{Z}},E)}:=\sup_{n\in\mathbb{\scriptstyle{Z}}}\|x_n\|_E, \] where \(E\) denotes a finite-dimensional Banach space with the norm \(\|\cdot\|_E\). Moreover, it is assumed that the mappings \(f_n:E^p\to E\), \(n\in\mathbb Z\), are continuous and for each \(r\in(0,\infty)\) satisfying the inequality \[ \sup_{n\in\mathbb{\scriptstyle{Z}},\|y_1\|_E\leq r,\dots,\|y_p\|_E\leq r} \|f_n(y_1,\dots,y_p)\|_E<\infty. \] In the main result, see Theorem 1, sufficient conditions under which the equation \(\mathcal{F}\mathbf{x}=\mathbf{h}\) has at least one solution \(\mathbf{x}\in\ell_{\infty}(\mathbb{Z},E)\) for any \(\mathbf{h}\in\ell_{\infty}(\mathbb{Z},E)\), are established. The proof is based on the method of a local linear approximation of \(\mathcal{F}\) by weakly regular operators. Some applications and illustrating examples of the main result are also provided.
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 6157084
zbMATH Keywords: nonlinear difference equation
zbMATH Keywords: bounded solution
zbMATH Keywords: method of locally linear approximation
zbMATH Keywords: difference operator
author: V. E. Slyusarchuk
reviewed by: Petr Zemánek
MaRDI profile type: Publication
full work available at URL: s10958-012-1078-7
OpenAlex ID: W2031797126
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DOI: 10.1007/S10958-012-1078-7
description: scientific article; zbMATH DE number 6157084