Multiplicity results for a class of second order superlinear difference systems

Bull. Korean Math. Soc. 2006 Vol. 43, No. 4, 693-701 Printed December 1, 2006

Guoqing Zhang and Sanyang Liu University of Shanghai, Xidian University

Abstract : Using Minimax principle and Linking theorem in critical point theory, we prove the existence of two nontrivial solutions for the following second order superlinear difference systems \[ (P)\left \{ \begin{array}{ll} \Delta^{2}x(k-1)+g(k,y(k))=0,&{k\in [1,T],}\\ \Delta^{2}y(k-1)+f(k,x(k))=0,&{k\in [1,T],}\\ x(0)=y(0)=0,x(T+1)=y(T+1)=0, \end{array} \right. \] where $T$ is a positive integer, [1,T] is the discrete interval $\{1,2,\ldots,$ $T\}, \Delta x(k)=x(k+1)-x(k)$ is the forward difference operator and $\triangle^{2}x(k)=\triangle(\triangle x(k))$.

Keywords : difference systems, multiple, critical point theory, super-linear