# 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
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.

### Share and Cite

##### ISRP Style

Haiyun Deng, Xingyong Zhang, Hui Fang, Existence of periodic solutions for a class of discrete systems with classical or bounded ($\phi_1,\phi_2$)-Laplacian, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 2, 535--559

##### AMA Style

Deng Haiyun, Zhang Xingyong, Fang Hui, Existence of periodic solutions for a class of discrete systems with classical or bounded ($\phi_1,\phi_2$)-Laplacian. J. Nonlinear Sci. Appl. (2017); 10(2):535--559

##### Chicago/Turabian Style

Deng, Haiyun, Zhang, Xingyong, Fang, Hui. "Existence of periodic solutions for a class of discrete systems with classical or bounded ($\phi_1,\phi_2$)-Laplacian." Journal of Nonlinear Sciences and Applications, 10, no. 2 (2017): 535--559

### Keywords

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

•  37J45
•  58E50
•  34C25

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