Existence and approximation of a solution for a two point nonlinear Dirichlet problem

The existence of at least one positive solution to a second-order nonlinear two-point boundary value problem, is established. Combining difference methods with Brouwer fixed point and Ascoli-Arzelà theorems, we get a solution as the limit of an appropriate sequence of piecewise linear interpolations. Furthermore, a priori bounds on the infinite norm of a solution and its derivatives are pointed out. Some examples are also discussed to illustrate our results.