# American Institute of Mathematical Sciences

2015, 2015(special): 533-539. doi: 10.3934/proc.2015.0533

## Existence of homoclinic solutions for second order difference equations with $p$-laplacian

 1 Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States 2 Equifax Inc., Alpharetta, GA 30005, United States

Received  August 2014 Revised  December 2014 Published  November 2015

Using the variational method and critical point theory, the authors study the existence of infinitely many homoclinic solutions to the difference equation \begin{equation*} -\Delta \big(a(k)\phi_p(\Delta u(k-1))\big)+b(k)\phi_p(u(k))=\lambda f(k,u(k))),\quad k\in\mathbb{Z}, \end{equation*} where $p>1$ is a real number, $\phi_p(t)=|t|^{p-2}t$ for $t\in\mathbb{R}$, $\lambda>0$ is a parameter, $a, b:\mathbb{Z}\to (0,\infty)$, and $f: \mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ is continuous in the second variable. Related results in the literature are extended.
Citation: John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533
##### References:
 [1] 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.   Google Scholar [2] M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory,, Springer, (2011).   Google Scholar [3] J. R. Graef, L. Kong, and M. Wang, Infinitely many homoclinic solutions for second order difference equations with $p$-Laplacian,, Commun. Appl. Anal., ().   Google Scholar [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.   Google Scholar [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.   Google Scholar [6] M. Ma and Z. Guo, Homoclinic orbits for second order self-adjoint difference equations,, J. Math. Anal. Appl., 323 (2006), 513.   Google Scholar [7] M. Mihăilescu, V. Rădulescu, and S. Tersian, Homoclinic solutions of difference equations with variable exponents,, Topol. Methods Nonlinear Anal., 38 (2011), 277.   Google Scholar [8] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, Appl. Math. Sci., 74 (1989).   Google Scholar [9] X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems,, J. Math. Anal. Appl. 373 (2011), 373 (2011), 59.   Google Scholar [10] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III,, Springer, (1985).   Google Scholar [11] Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 4125.   Google Scholar

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##### References:
 [1] 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.   Google Scholar [2] M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory,, Springer, (2011).   Google Scholar [3] J. R. Graef, L. Kong, and M. Wang, Infinitely many homoclinic solutions for second order difference equations with $p$-Laplacian,, Commun. Appl. Anal., ().   Google Scholar [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.   Google Scholar [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.   Google Scholar [6] M. Ma and Z. Guo, Homoclinic orbits for second order self-adjoint difference equations,, J. Math. Anal. Appl., 323 (2006), 513.   Google Scholar [7] M. Mihăilescu, V. Rădulescu, and S. Tersian, Homoclinic solutions of difference equations with variable exponents,, Topol. Methods Nonlinear Anal., 38 (2011), 277.   Google Scholar [8] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,, Appl. Math. Sci., 74 (1989).   Google Scholar [9] X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems,, J. Math. Anal. Appl. 373 (2011), 373 (2011), 59.   Google Scholar [10] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III,, Springer, (1985).   Google Scholar [11] Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 4125.   Google Scholar
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