Existence of periodic solutions for a class of discrete systems with classical or bounded (\(\phi_1,\phi_2\))Laplacian
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: \[
\begin{cases}
\Delta\phi_1(\Delta u_1(t1))+\nabla_{u_1}F(t,u_1(t),u_2(t))=0,\\
\Delta\phi_2(\Delta u_2(t1))+\nabla_{u_2}F(t,u_1(t),u_2(t))=0.
\end{cases}
\]
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.
Keywords
 Discrete systems
 periodic solutions
 saddle point theorem
 (\(\phi_1،\phi_2\))Laplacian
 the least action principle.
MSC
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