Positive solutions of BVPs for third-order discrete nonlinear difference systems

This paper is concerned with the following system $$\Delta ^3u_i(k)+f_i(k,u_1(k),u_2(k),\ldots,u_n(k))=0,\quad{}k\in 0,T],\ i=1,2,\ldots,n,$$ with the Dirichlet boundary condition $$u_i(0)=u_i(1)=u_i(T+3)=0,\quad{}i=1,2,\ldots,n.$$ Some results are obtained for the existence, multiplicity and nonexistence of positive solutions to the above system by using nonlinear alternative of Leray-Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Leggett-Williams fixed point theorem. In particular, it proves that the above system has N positive solutions under suitable conditions, where N is an arbitrary integer.