# American Institute of Mathematical Sciences

January  2018, 23(1): 487-492. doi: 10.3934/dcdsb.2018033

## On homoclinic solutions for a second order difference equation with p-Laplacian

 Institute of Mathematics, Lodz University of Technology, Wolczanska 215, 90-924 Lodz, Poland

Received  July 2016 Revised  September 2016 Published  January 2018

In this paper, we obtain conditions under which the difference equation
 $-Δ ≤ft( a(k)φ _{p}(Δ u(k-1))) +b(k)φ_{p}(u(k))=λ f(k, u(k)), \;\;k∈\mathbb{Z},$
has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. Some known results in the literature are extended and complemented.
Citation: Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033
##### References:
 [1] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009 (2009), Art. ID 670675, 20 pp. [2] A. Iannizzotto and S. Tersian, Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory, J. Math. Anal. Appl., 403 (2013), 173-182.  doi: 10.1016/j.jmaa.2013.02.011. [3] L. Kong, Homoclinic solutions for a second order difference equation with p-Laplacian, Appl. Math. Comput., 247 (2014), 1103-1121.  doi: 10.1016/j.amc.2014.09.069. [4] L. Kong, Homoclinic solutions for a higher order difference equation with p-Laplacian, Indag. Math., 27 (2016), 124-146.  doi: 10.1016/j.indag.2015.08.007. [5] S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016. [6] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1. [7] R. Stegliński,, On sequences of large homoclinic solutions for a difference equations on the integers, Adv. Difference Equ., 2016 (2016), 11pp.  doi: 10.1186/s13662-017-1344-6. [8] R. Stegliński, On sequences of large homoclinic solutions for a difference equations on the integers involving oscillatory nonlinearities, Electron. J. Qual. Theory Differ. Equ., 35 (2016), 1-11. [9] G. Sun and A. Mai, Infinitely many homoclinic solutions for second order nonlinear difference equations with p-Laplacian The Scientific World Journal 2014 (2014), Article ID 276372, 6 pages. doi: 10.1155/2014/276372.

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##### References:
 [1] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009 (2009), Art. ID 670675, 20 pp. [2] A. Iannizzotto and S. Tersian, Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory, J. Math. Anal. Appl., 403 (2013), 173-182.  doi: 10.1016/j.jmaa.2013.02.011. [3] L. Kong, Homoclinic solutions for a second order difference equation with p-Laplacian, Appl. Math. Comput., 247 (2014), 1103-1121.  doi: 10.1016/j.amc.2014.09.069. [4] L. Kong, Homoclinic solutions for a higher order difference equation with p-Laplacian, Indag. Math., 27 (2016), 124-146.  doi: 10.1016/j.indag.2015.08.007. [5] S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016. [6] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1. [7] R. Stegliński,, On sequences of large homoclinic solutions for a difference equations on the integers, Adv. Difference Equ., 2016 (2016), 11pp.  doi: 10.1186/s13662-017-1344-6. [8] R. Stegliński, On sequences of large homoclinic solutions for a difference equations on the integers involving oscillatory nonlinearities, Electron. J. Qual. Theory Differ. Equ., 35 (2016), 1-11. [9] G. Sun and A. Mai, Infinitely many homoclinic solutions for second order nonlinear difference equations with p-Laplacian The Scientific World Journal 2014 (2014), Article ID 276372, 6 pages. doi: 10.1155/2014/276372.
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