Singular discrete and continuous mixed boundary value problems

For each n epsilon N, n >= 2, we prove the existence of a solution. (u(0),..., u(n)) epsilon Rn+1 of the singular discrete problem 1/h(2)Delta(2)u(k-1) + f (t(k), u(k)) = 0, k = 1,..., n - 1 Delta u(0) = 0, u(n) = 0, where u(k) > 0 for k = 0,..., n - 1. Here T epsilon (0, infinity), h = T/n, t(k) = hk, f(t,x): [0, T] x (0, infinity) -> R is continuous and has a singularity at x = 0. We prove that for n -> infinity the sequence of solutions of the above discrete problems converges to a solution y of the corresponding continuous boundary value problem y '' (t) + f (t, y(t)) - 0, y' (0) = 0, y(T) = 0. (C) 2008 Elsevier Ltd. All rights reserved.