Singular discrete and continuous mixed boundary value problems

For each n ∈ N, n ≥ 2 we prove the existence of a positive solution of the singular discrete problem 1 h2 ∆uk−1 + f(tk, uk) = 0, k = 1, . . . , n− 1, ∆u0 = 0, un = 0, where T ∈ (0,∞), h = T n , tk = hk, f : [0, T ] × (0,∞) is continuous and has a singularity at x = 0. We prove that for n → ∞ 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.