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November  2018, 17(6): 2813-2844. doi: 10.3934/cpaa.2018133

On variational and topological methods in nonlinear difference equations

1. 

Department of Mathematical Sciences, University of Texas at Dallas Richardson, 75080 USA

2. 

Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México DF, México

Received  December 2017 Revised  March 2018 Published  June 2018

In this paper, first we survey the recent progress in usage of the critical point theory to study the existence of multiple periodic and subharmonic solutions in second order difference equations and discrete Hamiltonian systems with variational structure. Next, we propose a new topological method, based on the application of the equivariant version of the Brouwer degree to study difference equations without an extra assumption on variational structure. New result on the existence of multiple periodic solutions in difference systems (without assuming that they are their variational) satisfying a Nagumo-type condition is obtained. Finally, we also put forward a new direction for further investigations.

Citation: Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
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