April  2013, 6(4): 1065-1076. doi: 10.3934/dcdss.2013.6.1065

Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential

1. 

Institut de Recherche en Mathmatique et Physique

2. 

Universit Catholique de Louvain, chemin du Cyclotron, 2

3. 

B-1348 Louvain-la-Neuve

Received  August 2011 Published  December 2012

T-periodic solutions of systems of difference equations of the form\begin{eqnarray*}\Delta \phi[\Delta q(n-1)] = \nabla_q F[n,q(n)] + h(n) \quad (n \in \mathbb{Z})\end{eqnarray*}where $\phi = \nabla \Phi$, with $\Phi$ strictly convex, is a homeomorphism of $\mathbb{R}^N$ onto the ball $B_a \subset \mathbb{R}^N$, or a homeomorphism of the ball $B_{a} \subset \mathbb{R}^N$ onto $\mathbb{R}^N$, are considered when $F(n,u)$ is periodic in the $u_j$. The approach is variational.
Citation: Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065
References:
[1]

C. Bereanu and J. Mawhin, Boundary value problems for second order nonlinear difference equations with singular $\phi$,, J. Difference Equations Applic., 14 (2008), 1099. doi: 10.1080/10236190802332290. Google Scholar

[2]

C. Bereanu and H. B. Thompson, Periodic solutions of second order nonlinear difference equations with discrete $\phi$-Laplacian,, J. Math. Anal. Appl., 330 (2007), 1002. doi: 10.1016/j.jmaa.2006.07.104. Google Scholar

[3]

G. Fournier, D. Lupo, M. Ramos and M. Willem, Limit relative category and critical point theory,, Dynamics Reported, 3 (1994), 1. Google Scholar

[4]

G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation,, Ann. Inst. Henri-Poincaré. Anal. non Linéaire, 5 (1989), 259. Google Scholar

[5]

G. Fournier and M. Willem, Relative category and the calculus of variations,, in, (1990), 95. Google Scholar

[6]

Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations,, J. London Math. Soc. (2), 68 (2003), 419. doi: 10.1112/S0024610703004563. Google Scholar

[7]

Z. M. Guo and J. S. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,, Nonlinear Anal., 55 (2003), 969. doi: 10.1016/j.na.2003.07.019. Google Scholar

[8]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer, (1984). doi: 10.1007/978-3-642-69409-7. Google Scholar

[9]

J. Q. Liu, A generalized saddle point theorem,, J. Differential Equations, 82 (1989), 372. doi: 10.1016/0022-0396(89)90139-3. Google Scholar

[10]

J. Mawhin, Periodic solutions of the forced pendulum : classical vs relativistic,, Le Matematiche, 65 (2010), 97. Google Scholar

[11]

J. Mawhin, Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities,, Discrete Continuous Dynamical Systems, 32 (2012), 89. doi: 10.3934/dcds.2012.32.4015. Google Scholar

[12]

J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: A variational approach,, Nonlinear Anal., 75 (2012), 4672. doi: 10.1016/j.na.2011.11.018. Google Scholar

[13]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989). Google Scholar

[14]

F. Obersnel and P. P. Omari, Multiple bounded variation solutions of a periodically perturbed sine-curvature equation,, Commun. Contemp. Math., 13 (2011), 863. doi: 10.1142/S0219199711004488. Google Scholar

[15]

P. Rabinowitz, "Minimax Methods in Critical Point Theory and Applications to Differential Equations,", CBMS 65, 65 (1986). Google Scholar

[16]

P. Rabinowitz, On a class of functionals invariant under a $Z_n$ action,, Trans. Amer. Math. Soc., 310 (1988), 303. doi: 10.2307/2001123. Google Scholar

[17]

M. Reeken, Stability of critical points under small perturbations. Part I : Topological theory,, Manuscripta Math., 7 (1972), 387. Google Scholar

[18]

J. T. Schwartz, "Nonlinear Functional Analysis,", Gordon and Breach, (1969). Google Scholar

[19]

A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals,, Nonlinear Anal., 15 (1990), 725. doi: 10.1016/0362-546X(90)90089-Y. Google Scholar

[20]

J. S. Yu, H. H. Bin and Z. M. Guo, Periodic solutions for discrete convex Hamiltonian systems via Clarke duality,, Discrete Continuous Dynamical Systems, 15 (2006), 939. doi: 10.3934/dcds.2006.15.939. Google Scholar

[21]

Z. Zhou, J. S. Yu and Z. M. Guo, The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems,, ANZIAM J., 47 (2005), 89. doi: 10.1017/S1446181100009792. Google Scholar

show all references

References:
[1]

C. Bereanu and J. Mawhin, Boundary value problems for second order nonlinear difference equations with singular $\phi$,, J. Difference Equations Applic., 14 (2008), 1099. doi: 10.1080/10236190802332290. Google Scholar

[2]

C. Bereanu and H. B. Thompson, Periodic solutions of second order nonlinear difference equations with discrete $\phi$-Laplacian,, J. Math. Anal. Appl., 330 (2007), 1002. doi: 10.1016/j.jmaa.2006.07.104. Google Scholar

[3]

G. Fournier, D. Lupo, M. Ramos and M. Willem, Limit relative category and critical point theory,, Dynamics Reported, 3 (1994), 1. Google Scholar

[4]

G. Fournier and M. Willem, Multiple solutions of the forced double pendulum equation,, Ann. Inst. Henri-Poincaré. Anal. non Linéaire, 5 (1989), 259. Google Scholar

[5]

G. Fournier and M. Willem, Relative category and the calculus of variations,, in, (1990), 95. Google Scholar

[6]

Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations,, J. London Math. Soc. (2), 68 (2003), 419. doi: 10.1112/S0024610703004563. Google Scholar

[7]

Z. M. Guo and J. S. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,, Nonlinear Anal., 55 (2003), 969. doi: 10.1016/j.na.2003.07.019. Google Scholar

[8]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer, (1984). doi: 10.1007/978-3-642-69409-7. Google Scholar

[9]

J. Q. Liu, A generalized saddle point theorem,, J. Differential Equations, 82 (1989), 372. doi: 10.1016/0022-0396(89)90139-3. Google Scholar

[10]

J. Mawhin, Periodic solutions of the forced pendulum : classical vs relativistic,, Le Matematiche, 65 (2010), 97. Google Scholar

[11]

J. Mawhin, Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities,, Discrete Continuous Dynamical Systems, 32 (2012), 89. doi: 10.3934/dcds.2012.32.4015. Google Scholar

[12]

J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: A variational approach,, Nonlinear Anal., 75 (2012), 4672. doi: 10.1016/j.na.2011.11.018. Google Scholar

[13]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989). Google Scholar

[14]

F. Obersnel and P. P. Omari, Multiple bounded variation solutions of a periodically perturbed sine-curvature equation,, Commun. Contemp. Math., 13 (2011), 863. doi: 10.1142/S0219199711004488. Google Scholar

[15]

P. Rabinowitz, "Minimax Methods in Critical Point Theory and Applications to Differential Equations,", CBMS 65, 65 (1986). Google Scholar

[16]

P. Rabinowitz, On a class of functionals invariant under a $Z_n$ action,, Trans. Amer. Math. Soc., 310 (1988), 303. doi: 10.2307/2001123. Google Scholar

[17]

M. Reeken, Stability of critical points under small perturbations. Part I : Topological theory,, Manuscripta Math., 7 (1972), 387. Google Scholar

[18]

J. T. Schwartz, "Nonlinear Functional Analysis,", Gordon and Breach, (1969). Google Scholar

[19]

A. Szulkin, A relative category and applications to critical point theory for strongly indefinite functionals,, Nonlinear Anal., 15 (1990), 725. doi: 10.1016/0362-546X(90)90089-Y. Google Scholar

[20]

J. S. Yu, H. H. Bin and Z. M. Guo, Periodic solutions for discrete convex Hamiltonian systems via Clarke duality,, Discrete Continuous Dynamical Systems, 15 (2006), 939. doi: 10.3934/dcds.2006.15.939. Google Scholar

[21]

Z. Zhou, J. S. Yu and Z. M. Guo, The existence of periodic and subharmonic solutions to subquadratic discrete Hamiltonian systems,, ANZIAM J., 47 (2005), 89. doi: 10.1017/S1446181100009792. Google Scholar

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