On variational and topological methods in nonlinear difference equations
type: Item
instance of: scholarly article
title: On variational and topological methods in nonlinear difference equations
zbMATH Open document ID: 1398.39004
author: Zalman Balanov
author: C. García-Azpeitia
author: Wieslaw Krawcewicz
published in: Communications on Pure and Applied Analysis
publication date: 2018-08-23T00:00:00Z
review text: The authors first survey the recent progress in the application of critical point theory to study the existence of multiple periodic and subharmonic solutions of second-order difference equations and discrete Hamiltonian systems with a variational structure. Next, they propose a new topological method, based on the application of the equivariant version of the Brouwer degree, to study difference equations without the extra assumption of an existing variational structure. A new result on the existence of multiple periodic solutions for non-variational difference systems satisfying a Nagumo-type condition is obtained. In Section 2 the authors outline some variational methods for discrete systems. In Section 3 they focus on the progress related to the existence of multiple periodic solutions for second-order difference equations, including second-order discrete Hamiltonian systems and second-order self-adjoint difference equations. Section 4 mainly deals with solutions of boundary value problems for difference equations. In Section 5 recent results on discrete Hamiltonian systems are discussed. Recent progresses on a discrete \(p\)-Laplace equation and higher-order difference equations are presented in Section 6. In Section 7 the authors present their original results on multiple periodic solutions for a system of second-order difference equations without the extra assumption on a variational structure. Finally, in Section 8, they put forward a new direction for further investigations. For the convenience of the reader, they also added two appendices: the first one containing the main results of critical point theory, and the second one providing basic information on equivariant degree theory.
reviewed by: Eszter Gselmann
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A23
Mathematics Subject Classification ID: 58E50
zbMATH DE Number: 6923969
zbMATH Keywords: difference equation
zbMATH Keywords: discrete Hamiltonian system
zbMATH Keywords: periodic solution
zbMATH Keywords: variational method
zbMATH Keywords: symmetries
zbMATH Keywords: equivariant degree
MaRDI profile type: Publication
full work available at URL: cpaa.2018133
OpenAlex ID: W2806480970
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On variational and topological methods in nonlinear difference equations (Wikidata QID)
Wikidata QID: Q129686691

On variational and topological methods in nonlinear difference equations (DOI)
DOI: 10.3934/CPAA.2018133

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