On Fast Homoclinic Solutions for Second-Order Damped Difference Equations

  • Adel DaouasEmail author
  • Ameni Guefrej


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 \(c=0\).


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

Mathematics Subject Classification

34C37 37J45 39A10 



  1. 1.
    Agarwal, R.P.: Difference Equations and Inequalities. Theory, Methods and Applications, 2nd edn. Marcel Dekker, New York (2000)zbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., Wong, P.J.Y.: Advanced Topics in Difference Equations. Kluwer, Dordrecht (1997)CrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Daouas, A., Boujlida, M.: Existence of positive homoclinic solutions for damped differential equations. Positivity 21, 1353–1367 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Elaydi, S.N.: An Introduction to Difference Equations. Springer, New York (1996)CrossRefGoogle Scholar
  7. 7.
    Fabian, M., Habala, P., Hàjek, P., Montesinos, V., Zizler, V.: Banach Space Theory. Springer, New York (2011)CrossRefGoogle Scholar
  8. 8.
    Hartman, P.: Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. Am. Math. Soc. 246, 1–30 (1978)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kelley, W.G., Peterson, A.C.: Difference Equations. An Introduction with Applications, 2nd edn. Elsevier, Amsterdam (2001)zbMATHGoogle Scholar
  11. 11.
    Kong, L.: Homoclinic solutions for a second order difference equation with p-Laplacian. Appl. Math. Comput. 247, 1113–1121 (2014)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetGoogle Scholar
  14. 14.
    Ma, M., Guo, Z.: Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 323, 513–521 (2006)MathSciNetCrossRefGoogle 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)Google Scholar
  16. 16.
    Thandapani, E., Lalli, B.S.: Oscillation criteria for a second order damped difference equation. Appl. Math. Lett. 8(1), 1–6 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.High School of Sciences and TechnologyMaPSFA Laboratory, Sousse UniversityH. SousseTunisia

Personalised recommendations