Exact multiplicity of solutions for discrete second order Neumann boundary value problems

where h : [,T]Z → R, u(t) = u(t + ) – u(t) and T >  is a given positive integer. Our purpose is to find the exact number of solutions and positive solutions of (.). In these last years, the existence andmultiplicity of solutions for nonlinear discrete problems subject to various boundary value conditions have been widely studied by using different abstract methods such as critical point theory, fixed point theorems, lower and upper solutions method, and Brower degree (see, e.g., [–] and the references therein). All these results are about the unique solution, or the minimum amount of solutions, and positive solutions. To the best of our knowledge, there is no report on the exact number of solutions for discrete boundary value problems. For BVPs of differential equations, there are many papers concerned with the bifurcation values and exact multiplicities of solutions and positive solutions by bifurcation theory, quadraturemethod, time-map analysis and otherwise. See [–] and the references therein. For difference equations, however, the loss of continuity puts somemethods used well in differential equations, such as the quadrature method and its time-map analysis, out of action. Therefore, it is very meaningful to study the exact number of solutions for