Notes on Nontrivial Multiple Periodic Solutions for Second-Order Discrete Hamiltonian System
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Notes on Nontrivial Multiple Periodic Solutions for Second-Order Discrete Hamiltonian System Liang Ding1 · Jinlong Wei2 Received: 1 April 2019 / Revised: 7 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract We obtain new existence of multiple nontrivial M-periodic solutions for a class of second-order discrete Hamiltonian system with the force F(t, x) being neither supquadratic nor subquadratic growth in x. Moreover, we exhibit four instructive examples to illustrate our results. Keywords Second-order Hamiltonian system · Nontrivial M-periodic solutions · Critical points Mathematics Subject Classification 34K13 · 47J25 · 34L30
1 Introduction and Main Results In this paper, we are interested in the existence of multiple nontrivial M-periodic solutions for the following discrete Hamiltonian system with the case of F(n, u n ) being neither supquadratic nor subquadratic growth in u n : 2 u n−1 + f (n, u n ) = 0, n ∈ Z,
(1.1)
Communicated by Shangjiang Guo.
B
Jinlong Wei [email protected] Liang Ding [email protected]
1
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China
2
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
123
L. Ding, J. Wei
where u n = u(n), u n = u n+1 − u n , 2 u n = (u n ), f (n, u n ) = ∇u n F(n, u n ), F : Z × R → R, F(t, x) is continuously differentiable in x for every t ∈ Z and M-periodic (0 < M ∈ N) in t for all x ∈ R, and ∇x F(t, x) is the gradient of F(t, x) in x. The background of system (1.1) comes from discrete heat equations [5]. Consider the temperature distribution of a “very long” rod, and assume that the rod is so long that it can be laid on top of the set of integers Z. Let u n (t) be the temperature at the integral time t and integral position n of the rod. At time t, if the temperature u n−1 (t) is higher than u n (t), heat will flow from the point n − 1 to n. The amount of temperature increase is u n (t + 1) − u n (t), and it is reasonable to postulate that the increase is proportional to the difference u n−1 (t) − u n (t), i.e., u n (t + 1) − u n (t) = r (u n−1 (t) − u n (t)), where r > 0 is a positive diffusion rate constant. Similarly, heat will flow from n + 1 to n if u n+1 (t) > u n (t). Suppose that the temperature u n (t + M) = u n (t) for any fixed integral M > 0. Then, it is reasonable that the total effect is u n (t + 1) − u n (t) = r (u n−1 (t) − u n (t)) + r (u n+1 (t) − u n (t)), n ∈ Z, t ∈ N, (1.2) where u n (t) = u n (t + M) ∈ R, M ∈ N+ , and it can be regard as a discrete Newton law of cooling. If a reaction term f is introduced, (1.2) may be generalized to yield the following nonlinear reaction diffusion equation: u n (t + 1) − u n (t) = r (u n−1 (t) − u n (t)) + r (u n+1 (t) − u n (t)) + f (n, u n (t)), n ∈ Z, t ∈ N,
(1.3)
where u n (t) = u n (t + M), f (n, ·) = f (n + M, ·), M ∈ N+ . After a very long time, the temperature at every integral position n of the rod is stable, and t
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