INFINITELY MANY HOMOCLINIC SOLUTIONS FOR SECOND ORDER DIFFERENCE EQUATIONS WITH p-LAPLACIAN

John R. Graef; LINGJU KONG; MIN WANG

Title	INFINITELY MANY HOMOCLINIC SOLUTIONS FOR SECOND ORDER DIFFERENCE EQUATIONS WITH p-LAPLACIAN
Publication Type	Journal Article
Year of Publication	2015
Authors	Graef, JR, KONG, LINGJU, WANG, MIN
Volume	19
Issue	1
Start Page	95
Pagination	8
Date Published	2015
ISSN	1083-2564
AMS	37C20, 39A10, 58E0
Abstract	By using critical point theory, the authors study the existence of infinitely many homoclinic solutions to the difference equation $${ −∆a(k)\phi_p(∆u(k − 1)) + b(k)\phi_p(u(k)) = λf(k, u(k))), \ \ k ∈ \mathbb{Z} }$$, where ${p > 1}$ is a real number, ${ \phi_p(t) = \|t\|^{p−2} t}$ for ${ t ∈ \mathbb{R},\ \ λ > 0}$ is a parameter, ${ a, b : \mathbb{Z} → (0, ∞), }$ and ${ f : \mathbb{Z} × \mathbb{R} → \mathbb{R} }$ is a continuous function in the second variable. Some known work in the literature is extended.
URL	http://www.acadsol.eu/en/articles/19/1/7.pdf
Refereed Designation	Refereed
Full Text	REFERENCES [1] R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods, and Applications, Second Edition, Marcel Dekker, New York, 2000. [2] P. Chen, X. Tang, and R. P. Agarwal, Existence of homoclinic solutions for p(n)-Laplacian Hamiltonian systems on Orlicz sequence spaces, Math. Comput. Modelling 55 (2012), 989–1002. [3] M. Fabian, P. Habala, P. H´ajek, V. Montesinos, and V. Zizler, Banach Space Theory, Springer, New York, 2011. [4] A. Iannizzotto and S. A. Tersian, Multiple homoclinic solutions for the discrete p-Laplacian via critical point theory, J. Math. Anal. Appl. 403 (2013), 173–182. [5] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352–370. [6] W. G. Kelly and A. C. Peterson, Difference Equations, an Introduction with Applications, Second Edition, Academic Press, New York, 2001. [7] M. Ma and Z. Guo, Homoclinic orbits for second order self-adjoint difference equations, J. Math. Anal. Appl. 323 (2006), 513–521. [8] M. Mih˘ailescu, V. R˘adulescu, and S. Tersian, Homoclinic solutions of difference equations with variable exponents, Topol. Methods Nonlinear Anal. 38 (2011), 277–289. [9] X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl. 373 (2011), 59–72. [10] Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal. 71 (2009), 4125–4130.