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

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August 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

Jean Mawhin ^1,2,3,

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

Abstract
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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.

Keywords: Difference equations, variational methods, $\phi$-laplacian, periodic solutions, curvature oscillator, relativistic oscillator..

Mathematics Subject Classification: Primary: 39A11, 47J20; Secondary: 49J40, 83A0.

Citation: Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065

References:

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[3]	Dynamics Reported, 3 (1994), 1-24.
[4]	Ann. Inst. Henri-Poincaré. Anal. non Linéaire, 5 (1989), 259-281.
[5]	in "Variational Problems" (eds. H. Berestycki, J. M. Coron and I. Ekeland) Birkhäuser, Basel, (1990), 95-104.
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[21]	ANZIAM J., 47 (2005), 89-102. doi: 10.1017/S1446181100009792.

show all references

References:

[1]	J. Difference Equations Applic., 14 (2008), 1099-1118. doi: 10.1080/10236190802332290.
[2]	J. Math. Anal. Appl., 330 (2007), 1002-1015. doi: 10.1016/j.jmaa.2006.07.104.
[3]	Dynamics Reported, 3 (1994), 1-24.
[4]	Ann. Inst. Henri-Poincaré. Anal. non Linéaire, 5 (1989), 259-281.
[5]	in "Variational Problems" (eds. H. Berestycki, J. M. Coron and I. Ekeland) Birkhäuser, Basel, (1990), 95-104.
[6]	J. London Math. Soc. (2), 68 (2003), 419-430. doi: 10.1112/S0024610703004563.
[7]	Nonlinear Anal., 55 (2003), 969-983. doi: 10.1016/j.na.2003.07.019.
[8]	Springer, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.
[9]	J. Differential Equations, 82 (1989), 372-385. doi: 10.1016/0022-0396(89)90139-3.
[10]	Le Matematiche, 65 (2010), 97-107.
[11]	Discrete Continuous Dynamical Systems, 32 (2012), 89-111. doi: 10.3934/dcds.2012.32.4015.
[12]	Nonlinear Anal., 75 (2012), 4672-4687. doi: 10.1016/j.na.2011.11.018.
[13]	Springer, New York, 1989.
[14]	Commun. Contemp. Math., 13 (2011), 863-883. doi: 10.1142/S0219199711004488.
[15]	CBMS 65, American Math. Soc., Providence RI, 1986.
[16]	Trans. Amer. Math. Soc., 310 (1988), 303-311. doi: 10.2307/2001123.
[17]	Manuscripta Math., 7 (1972), 387-411.
[18]	Gordon and Breach, New York, 1969.
[19]	Nonlinear Anal., 15 (1990), 725-739. doi: 10.1016/0362-546X(90)90089-Y.
[20]	Discrete Continuous Dynamical Systems, 15 (2006), 939-950. doi: 10.3934/dcds.2006.15.939.
[21]	ANZIAM J., 47 (2005), 89-102. doi: 10.1017/S1446181100009792.

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