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On Fast Homoclinic Solutions for Second-Order Damped Difference Equations


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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  1. 1.

    Agarwal, R.P.: Difference Equations and Inequalities. Theory, Methods and Applications, 2nd edn. Marcel Dekker, New York (2000)

    Book  Google Scholar 

  2. 2.

    Agarwal, R.P., Wong, P.J.Y.: Advanced Topics in Difference Equations. Kluwer, Dordrecht (1997)

    Book  Google Scholar 

  3. 3.

    Aprahamian, M., Souroujon, D., Tersian, S.: Decreasing and fast solutions for a second-order difference equation related to Fisher–Kolmogorov’s equation. J. Math. Anal. Appl. 363, 97–110 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cabada, A., Iannizzotto, A.: Existence of homoclinic constant sign solutions for a difference equation on the integers. Appl. Math. Comput. 224, 216–223 (2013)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Daouas, A., Boujlida, M.: Existence of positive homoclinic solutions for damped differential equations. Positivity 21, 1353–1367 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Elaydi, S.N.: An Introduction to Difference Equations. Springer, New York (1996)

    Book  Google Scholar 

  7. 7.

    Fabian, M., Habala, P., Hàjek, P., Montesinos, V., Zizler, V.: Banach Space Theory. Springer, New York (2011)

    Book  Google Scholar 

  8. 8.

    Hartman, P.: Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. Am. Math. Soc. 246, 1–30 (1978)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Iannizzotto, A., Tersian, S.: Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory. J. Math. Anal. Appl. 403, 173–183 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kelley, W.G., Peterson, A.C.: Difference Equations. An Introduction with Applications, 2nd edn. Elsevier, Amsterdam (2001)

    MATH  Google Scholar 

  11. 11.

    Kong, L.: Homoclinic solutions for a second order difference equation with p-Laplacian. Appl. Math. Comput. 247, 1113–1121 (2014)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Lin, X., Tang, X.H.: Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl. 373, 59–72 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Liu, X., Zhou, T., Shi, H.: Existence of homoclinic orbits for a class of nonlinear functional difference equations. Electron. J. Differ. Equ. 2016(315), 1–10 (2016)

    MathSciNet  Google Scholar 

  14. 14.

    Ma, M., Guo, Z.: Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 323, 513–521 (2006)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Rabinowitz, P.H.: Minimax methods in critical point theory with applications in differential equations. In: CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI (1986)

  16. 16.

    Thandapani, E., Lalli, B.S.: Oscillation criteria for a second order damped difference equation. Appl. Math. Lett. 8(1), 1–6 (1995)

    MathSciNet  Article  Google Scholar 

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Correspondence to Adel Daouas.

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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).

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

Mathematics Subject Classification

  • 34C37
  • 37J45
  • 39A10