Existence of periodic solutions for a class of discrete systems with classical or bounded (\(\phi_1,\phi_2\))-Laplacian

Volume 10, Issue 2, pp 535--559 http://dx.doi.org/10.22436/jnsa.010.02.19

Publication Date: February 20, 2017 Submission Date: June 21, 2016

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Authors

Haiyun Deng - Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China. Xingyong Zhang - Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China. Hui Fang - Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan, 650500, P. R. China.

Abstract

In this paper, we investigate the existence of periodic solutions for the nonlinear discrete system with classical or bounded (\(\phi_1,\phi_2\))-Laplacian: \[ \begin{cases} \Delta\phi_1(\Delta u_1(t-1))+\nabla_{u_1}F(t,u_1(t),u_2(t))=0,\\ \Delta\phi_2(\Delta u_2(t-1))+\nabla_{u_2}F(t,u_1(t),u_2(t))=0. \end{cases} \] By using the saddle point theorem, we obtain that system with classical (\(\phi_1,\phi_2\))-Laplacian has at least one periodic solution when F has (p, q)-sublinear growth, and system with bounded (\(\phi_1,\phi_2\))-Laplacian has at least one periodic solution when \(F\) has (\(p,q\))-sublinear growth. By using the least action principle, we obtain that system with classical or bounded (\(\phi_1,\phi_2\))-Laplacian has at least one periodic solution when \(F\) has a growth like Lipschitz condition.

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Keywords

• Discrete systems
• periodic solutions
• saddle point theorem
• (\(\phi_1،\phi_2\))-Laplacian
• the least action principle.

MSC

•  37J45
•  58E50
•  34C25

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