Abstract
In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.
Highlights
Introduction and main resultDiscrete equations have been widely employed as mathematical models depicting the nature phenomena in many practical problems including computer sciences, life sciences, mathematical biology, and so on; see [1,2,3,4,5]
It is well known that by the standing wave assumptions discrete nonlinear Schrödinger (DNLS) equations can change into the following nonlinear second-order difference equation [12,13,14,15,16,17,18]:
The critical-point theory is an important tool when dealing with the existence of solutions of differential equations, and for discrete system (1.1), there are some results on the existence of periodic solutions in the last few years: especially, for
Summary
Guo and Yu [18] developed a new approach to obtain the existence and multiplicity of periodic solutions to discrete system (1.1). For F(n, X) with subquadratic growth condition with respect to X at infinity, Guo and Yu [20] proved the existence of nontrivial periodic solutions. = κ > 0, in 2004, under the assumption that κ depends on M (especially, κ(M) > 2 for even M), Zhou, Yu, and Guo [21] improved the Guo–Yu method of [18] and obtained the existence of two nontrivial M-periodic solutions for discrete system (1.1); for more details on the existence of multiple nontrivial M-periodic solutions with quadratic–supquadratic condition, we refer to [22, 23]. Xue and Tang obtained the following more general result for F with quadratic–supquadratic condition with respect to X at infinity
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