Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions

This paper investigates the solvability of discrete Dirichlet boundary value problems by the lower and upper solution method. Here, the second-order difference equation with a nonlinear right hand side f is studied and f(t,u,v) can have a superlinear growth both in u and in v. Moreover, the growth conditions on f are one-sided. We compute a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution. It is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations.