Abstract
By using variational methods and Morse theory, we study the multiplicity of the periodic solutions for a class of difference equations with double resonance at infinity. To the best of our knowledge, investigations on double-resonant difference systems have not been seen in the literature.
Highlights
−Δ2x k − 1 f k, x k, k ∈ Z, 1.1 xk p xk, where Δ is the forward difference operator defined by Δx k x k 1 − x k and Δ2x k Δ Δx k for k ∈ Z
We always assume that f1 f : Z × R → R is C1-differentiable with respect to the second variable and satisfies f k p, t f k, t for k, t ∈ Z × R and f k, 0 ≡ 0 for k ∈ Z
Advances in Difference Equations by a mathematical model, the feature of resonance lies in the interaction between the linear spectrum and the nonlinearity
Summary
Advances in Difference Equations by a mathematical model, the feature of resonance lies in the interaction between the linear spectrum and the nonlinearity It is known see 1 that the eigenvalue problem. It is well known that in different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, and economics, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. For this reason, in recent years the solvability of nonlinear difference equations have been extensively investigated see 1, 6–8 and the references cited therein.
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