Maximum norm analysis for nonlinear two-point boundary value problems

In this paper, we propose a new approach for the finite difference approxi- mation on non-uniform mesh of the nonlinear two-point boundary value problem-(p(x)u')' = f(x, u); a <x <b; u(a) = u(b) = 0. Under a realistic assump-tion on the nonlinearity and a C3;1[a; b] regularity of the solution, we show that the approximation is O(h²) accurate in the maximum norm, making use of the Banach's fixed point principle.