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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

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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065
References:
[1]

J. Difference Equations Applic., 14 (2008), 1099-1118. doi: 10.1080/10236190802332290.  Google Scholar

[2]

J. Math. Anal. Appl., 330 (2007), 1002-1015. doi: 10.1016/j.jmaa.2006.07.104.  Google Scholar

[3]

Dynamics Reported, 3 (1994), 1-24. Google Scholar

[4]

Ann. Inst. Henri-Poincaré. Anal. non Linéaire, 5 (1989), 259-281.  Google Scholar

[5]

in "Variational Problems" (eds. H. Berestycki, J. M. Coron and I. Ekeland) Birkhäuser, Basel, (1990), 95-104.  Google Scholar

[6]

J. London Math. Soc. (2), 68 (2003), 419-430. doi: 10.1112/S0024610703004563.  Google Scholar

[7]

Nonlinear Anal., 55 (2003), 969-983. doi: 10.1016/j.na.2003.07.019.  Google Scholar

[8]

Springer, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[9]

J. Differential Equations, 82 (1989), 372-385. doi: 10.1016/0022-0396(89)90139-3.  Google Scholar

[10]

Le Matematiche, 65 (2010), 97-107.  Google Scholar

[11]

Discrete Continuous Dynamical Systems, 32 (2012), 89-111. doi: 10.3934/dcds.2012.32.4015.  Google Scholar

[12]

Nonlinear Anal., 75 (2012), 4672-4687. doi: 10.1016/j.na.2011.11.018.  Google Scholar

[13]

Springer, New York, 1989.  Google Scholar

[14]

Commun. Contemp. Math., 13 (2011), 863-883. doi: 10.1142/S0219199711004488.  Google Scholar

[15]

CBMS 65, American Math. Soc., Providence RI, 1986.  Google Scholar

[16]

Trans. Amer. Math. Soc., 310 (1988), 303-311. doi: 10.2307/2001123.  Google Scholar

[17]

Manuscripta Math., 7 (1972), 387-411.  Google Scholar

[18]

Gordon and Breach, New York, 1969.  Google Scholar

[19]

Nonlinear Anal., 15 (1990), 725-739. doi: 10.1016/0362-546X(90)90089-Y.  Google Scholar

[20]

Discrete Continuous Dynamical Systems, 15 (2006), 939-950. doi: 10.3934/dcds.2006.15.939.  Google Scholar

[21]

ANZIAM J., 47 (2005), 89-102. doi: 10.1017/S1446181100009792.  Google Scholar

show all references

References:
[1]

J. Difference Equations Applic., 14 (2008), 1099-1118. doi: 10.1080/10236190802332290.  Google Scholar

[2]

J. Math. Anal. Appl., 330 (2007), 1002-1015. doi: 10.1016/j.jmaa.2006.07.104.  Google Scholar

[3]

Dynamics Reported, 3 (1994), 1-24. Google Scholar

[4]

Ann. Inst. Henri-Poincaré. Anal. non Linéaire, 5 (1989), 259-281.  Google Scholar

[5]

in "Variational Problems" (eds. H. Berestycki, J. M. Coron and I. Ekeland) Birkhäuser, Basel, (1990), 95-104.  Google Scholar

[6]

J. London Math. Soc. (2), 68 (2003), 419-430. doi: 10.1112/S0024610703004563.  Google Scholar

[7]

Nonlinear Anal., 55 (2003), 969-983. doi: 10.1016/j.na.2003.07.019.  Google Scholar

[8]

Springer, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[9]

J. Differential Equations, 82 (1989), 372-385. doi: 10.1016/0022-0396(89)90139-3.  Google Scholar

[10]

Le Matematiche, 65 (2010), 97-107.  Google Scholar

[11]

Discrete Continuous Dynamical Systems, 32 (2012), 89-111. doi: 10.3934/dcds.2012.32.4015.  Google Scholar

[12]

Nonlinear Anal., 75 (2012), 4672-4687. doi: 10.1016/j.na.2011.11.018.  Google Scholar

[13]

Springer, New York, 1989.  Google Scholar

[14]

Commun. Contemp. Math., 13 (2011), 863-883. doi: 10.1142/S0219199711004488.  Google Scholar

[15]

CBMS 65, American Math. Soc., Providence RI, 1986.  Google Scholar

[16]

Trans. Amer. Math. Soc., 310 (1988), 303-311. doi: 10.2307/2001123.  Google Scholar

[17]

Manuscripta Math., 7 (1972), 387-411.  Google Scholar

[18]

Gordon and Breach, New York, 1969.  Google Scholar

[19]

Nonlinear Anal., 15 (1990), 725-739. doi: 10.1016/0362-546X(90)90089-Y.  Google Scholar

[20]

Discrete Continuous Dynamical Systems, 15 (2006), 939-950. doi: 10.3934/dcds.2006.15.939.  Google Scholar

[21]

ANZIAM J., 47 (2005), 89-102. doi: 10.1017/S1446181100009792.  Google Scholar

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