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On Fast Homoclinic Solutions for Second-Order Damped Difference Equations

Abstract

By means of critical point theory, we study the existence and multiplicity of homoclinic solutions of the damped second-order difference equation

$$\begin{aligned} \Delta ^{2}u(n-1)-c\Delta u(n-1)-a(n)u(n)+f(n,u(n))=0 ,\quad n\in {\mathbb {Z}}, \end{aligned}$$

where \(c>-1\) is a constant, \(a: {\mathbb {Z}}\rightarrow (0,+\infty )\) and \(f: {\mathbb {Z}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuous with respect to the second variable and satisfies some additional assumptions. The proofs of our results are based on variational methods in some weighted Hilbert space of sequences. Some recent results in the literature are extended even in the case of \(c=0\).

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Correspondence to Adel Daouas.

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Pham Huu Anh Ngoc.

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Daouas, A., Guefrej, A. On Fast Homoclinic Solutions for Second-Order Damped Difference Equations. Bull. Malays. Math. Sci. Soc. 43, 3125–3142 (2020). https://doi.org/10.1007/s40840-019-00858-x

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Keywords

  • Homoclinic solution
  • Fast solution
  • (PS) condition
  • Mountain pass theorem
  • Difference equations

Mathematics Subject Classification

  • 34C37
  • 37J45
  • 39A10