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

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

John R. Graef 1, , Lingju Kong 1, and  Min Wang 2,

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.
Keywords: homoclinic solutions, Diff erence equations, infi nitely many homoclinic solutions, variational method..
Mathematics Subject Classification: Primary: 39A10, 37C20, 58E0.
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), 989-1002.

[2]

M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory, Springer, New York, 2011.

[3]

J. R. Graef, L. Kong, and M. Wang, Infinitely many homoclinic solutions for second order difference equations with $p$-Laplacian, Commun. Appl. Anal., to appear.

[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]

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

[7]

M. Mihăilescu, V. Rădulescu, and S. Tersian, Homoclinic solutions of difference equations with variable exponents, Topol. Methods Nonlinear Anal., 38 (2011), 277-289.

[8]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci., Springer, New York, 74 1989.

[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]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Springer, New York, 1985.

[11]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130.

show all references

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), 989-1002.

[2]

M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory, Springer, New York, 2011.

[3]

J. R. Graef, L. Kong, and M. Wang, Infinitely many homoclinic solutions for second order difference equations with $p$-Laplacian, Commun. Appl. Anal., to appear.

[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]

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

[7]

M. Mihăilescu, V. Rădulescu, and S. Tersian, Homoclinic solutions of difference equations with variable exponents, Topol. Methods Nonlinear Anal., 38 (2011), 277-289.

[8]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci., Springer, New York, 74 1989.

[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]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. III, Springer, New York, 1985.

[11]

Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems, Nonlinear Anal., 71 (2009), 4125-4130.

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