Existence of periodic solutions for nonautonomous second-order discrete Hamiltonian systems

Abstract

In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz's saddle point theorem. MSC: 34C25; 58E50

Periodic solutions for second-order difference equations with quadratic–supquadratic condition

“…Moreover, the existence of one nontrivial solution for general nonlinear difference equations, that is, discrete φ-Laplacian equations with quadraticsupquadratic condition, is considered in [13,15,24]. For other related works, we refer to [14,16,17,[25][26][27][28][29][30][31]. Note that in [21], for F with quadratic-supquadratic condition with respect to X at infinity, by introducing the smallest and largest eigenvalues of the matrix…”

Periodic solutions for discrete p ( k ) $p(k)$ -Laplacian systems with partially periodic potential

“…Our main results are the following theorems. Comparing with the results in [6][7][8][9][10][11][13][14][15], Theorem 1.1 is a different result even in the case p(k) = 2. For example,…”

“…During the past decade, periodic solutions, subharmonic solutions, and homoclinic orbits for second order discrete Hamiltonian systems have captured special attention, and some solvability conditions have been given under distinct hypotheses on potential function [6][7][8][9][10][11][12][13].…”

“…it is easy to see that all the conditions of Theorem 1.1 hold, but F is not covered by the results in [6][7][8][9][10][11][12][13].…”