Green functions and positive solutions for boundary value problems of third-order difference equations

In this paper, we investigate the boundary value problems for the third-order nonlinear difference equation Δ3u(t−2)+a(t)f(u(t))=0,t∈[2,T+2], satisfying five kinds of different boundary value conditions. The Green functions are constructed carefully and their bounds are evaluated. Some verifiable criteria for the existence of at least one positive solution and two positive solutions are obtained by using the Green functions and the evaluations of bounds. Some examples are given to demonstrate the applications of the theorems obtained.