Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions (Q884507)

The authors investigate the solvability of the discrete Dirichlet boundary value problem (BVP) by the lower and upper solution method. 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. The authors extend the ideas from their earlier paper, where the solvability of BVP has been proved provided \(f(t, u, v)\) has sublinear or linear growth in \(u\) and \(v\). By computing a priori bounds on solutions to the discrete problem, they obtain the existence of at least one solution. Also it is shown that solutions of the discrete problems converge to the solutions of corresponding ordinary differential equations. Finally an example is presented to illustrate how the new theory advances existing results from literature.