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On variational and topological methods in nonlinear difference equations. (English) Zbl 1398.39004

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.

MSC:

39A12 Discrete version of topics in analysis
39A23 Periodic solutions of difference equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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