Positive solutions for a system of nonpositive difference equations (Q5953387)

The authors consider the discrete boundary value problem (BVP) \[ \Delta^nu_i(k)+q_i(k,u_1(k),u_2(k),\dots,u_m(k))=0,\quad k\in\{0,1,\dots,N\}, \] \[ \Delta^ju_i(0)=0=\Delta^{p_i}u_i(N+n-p_i),\quad 0\leq j\leq n-2,\;i=1,2,\dots,m, \] where \(n\geq 2\), \(N\geq n-1\), \(1\leq p_i\leq n-1\), \(1\leq i\leq m\), and the nonlinearities \(q_i\), \(1\leq i\leq m\), can be nonpositive. They establish discrete analogues of recent results concerning BVPs for ordinary differential equations, namely sufficient conditions for the existence of a positive solution (resp. of at least two solutions) of the given discrete BVP. The proofs are based on Krasnosel'skii's fixed point theorem in a cone with the use of inequalities for a certain Green's function.