Bounded solutions and asymptotic stability of nonlinear difference equations in the complex plane. II.
type: Item
instance of: scholarly article
title: Bounded solutions and asymptotic stability of nonlinear difference equations in the complex plane. II.
zbMATH Open document ID: 1080.39015
DOI: 10.1016/S0898-1221(01)00168-7
publication date: 2002-06-13T00:00:00Z
review text: The authors investigate asymptotic properties of solutions of a certain \(m\)-th order nonlinear difference equation (which is rather complicated to be given here explicitly). The paper is a continuation of the earlier paper of the same authors [Arch. Math., Brno 36, No. 2, 139--158 (2000; Zbl 1053.39016)]. Comparing with that paper, the nonlinearity in the studied equation of the reviewed paper is allowed to be more general. The investigated equation is transformed into an operator equation in a suitable Banach space and the main results of the paper, the existence and uniqueness statements for solutions in \(l_1\), are proved using the fixed point theorem for holomorphic maps due to \textit{C. J. Earle} and \textit{R. S. Hamilton} [Global Analysis, Proc. Sympos. Pure Math. 16, 61--65 (1970; Zbl 0205.14702)]. The results of the paper are illustrated by a number of examples which seem to be quite nontrivial.
reviewed by: Ondřej Došlý
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1755265
zbMATH Keywords: difference equations
zbMATH Keywords: bounded solutions
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Banach space
zbMATH Keywords: fixed point theorem
author: Eugenia N. Petropoulou
author: Panayiotis D. Siafarikas
MaRDI profile type: Publication
cites work: On the existence of positive solutions of higher order difference equations
cites work: On the convergence of power series those coefficients satisfy a poincaré type linear and nonlinear difference equation
cites work: Q4814549
cites work: Q5604772
cites work: Recent Advances in Liapunov Stability Theory
cites work: Q4381431
cites work: Invariants boundedness and persistence of nonautonomous difference equations of rational form.
cites work: Q4317046
cites work: Global stability in some population models
cites work: Global attractivity and convergence to a two-cycle in a difference equation
cites work: Global attractivity in a nonlinear second‐order difference equation
cites work: On the recursive sequence \(x_{n+1}=\alpha+x_{n-1}/x_n\)
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1755265