Abstract
By applying the least action principle and minimax methods in critical point theory, we prove the existence of periodic solutions for a class of difference systems withp-Laplacian and obtain some existence theorems.
Highlights
Consider the following p-Laplacian difference system:Δ |Δu t − 1 |p−2Δu t − 1 ∇F t, u t, t ∈ Z, 1.1 where Δ is the forward difference operator defined by Δu t u t 1 − u t, Δ2u t Δ Δu t, p ∈ 1, ∞ such that 1/p 1/q 1, t ∈ Z, u ∈ RN, F : Z × RN → R, and F t, x is continuously differentiable in x for every t ∈ Z and T -periodic in t for all x ∈ RN.When p 2, 1.1 reduces to the following second-order discrete Hamiltonian system: Δ2u t − 1 ∇F t, u t, t ∈ Z.Difference equations provide a natural description of many discrete models in real world
The lower bounds and the upper bounds of our theorems are more accurate than the existing results in the literature
Let the Sobolev space ET be defined by ET u : Z −→ RN | utTut, t ∈ Z
Summary
Difference equations provide a natural description of many discrete models in real world. Abstract and Applied Analysis control, it is of practical importance to investigate the solutions of difference equations. In some recent papers 4–18 , the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Motivated by the above papers, we consider the existence of periodic solutions for problem 1.1 by using the least action principle and minimax methods in critical point theory
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