Existence of positive solutions for a class of discrete Dirichlet boundary value problems

Let x=f(x)x=f(x), where x∈Rx∈R. Clearly, if there exists b>a>0b>a>0 such that f∈C[a,b]f∈C[a,b] and either f(a)≤af(a)≤a and f(b)≥bf(b)≥b or f(a)≥af(a)≥a and f(b)≤bf(b)≤b, then there is x∗∈[a,b]x∗∈[a,b] such that x∗=f(x∗)x∗=f(x∗), that is, the function f(x)f(x) has a fixed point x∗∈[a,b]x∗∈[a,b]. By using the above main idea and famous Guo–Krasnosel’skii fixed point theorem, existence of positive solutions for a nonlinear second order difference equation and a discrete second order system with the Dirichlet boundary conditions will be considered. The new existence results will be obtained. In particular, the main idea is also valid for the partial difference problems or the general nonlinear algebraic system.