Solvability of Discrete Two-point Boundary Value Problems

We study the discrete approximation to solutions of first-order system arising from applying the trapezoidal rule to a second-order scalar ordinary differential equation. In the trapezoidal rule the finite difference approximation are Dyk = (zk + zk−1)/2, Dzk = ( fk + fk−1)/2, for k = 1, 2, .., n, and tk = kh for k = 0, ..., n, 0 = G ( (y0, yn), (z0 + z1)/2, (zn−1 + zn)/2) ) , where fi ≡ f (ti, yi, zi) and G = (g0, g1) are continuous and fully nonlinear. We assume there exist strict discrete lower and strict discrete upper solutions and impose additional conditions on fk andG which are known to yield a priori bounds on, and to guarantee the existence of solutions of the discrete approximation for sufficiently small grid size. We use the homotopy to compute the solutions of the discrete approximations. In this paper we study the first-order system of difference equations that arise when one applies the trapezoidal rule to approximate solutions of the second-order scalar ordinary differential equation.