Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

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

Pages: 1065 - 1076, Volume 6, Issue 4, August 2013      doi:10.3934/dcdss.2013.6.1065

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Jean Mawhin - Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, chemin du Cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium (email)

Abstract: 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, 83A05.

Received: August 2011;      Available Online: December 2012.