DISCRETIZATION PRINCIPLES FOR LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS, III

This paper extends results of Yamamoto et al. (Numer. Funct. Anal. Optimiz. 2008; 29: 213-224) to the boundary value problem {-(p(x)u\u27)\u27 + q(x)u\u27 + r (x) u = f(x), a \u3c = x \u3c = b, c(0)u(a) - c(1)u\u27(a) = d(0)u(b) + d(1)u\u27(b) = 0, where the sign of r(x) is indefinite. Let H(v)A(v)U(v) = f(v) be the finite difference equations on partitions Delta(v) : a = x(0)(v) \u3c x(1)(v) \u3c ... \u3c x(nv+ 1)(v) = b, v = 1, 2,...with h(v) = max(i)(x(i)(v) - x(i-1)(v)) - \u3e 0 as v - \u3e infinity, where H(v) and A(v) are diagonal and tridiagonal matrices, respectively, and f(v) are vectors generated by discretization of f(x). Then equivalent conditions for the boundary value problem to have a unique solution u is an element of C(2)[a, b] are given in terms of H(v)(-1) and A(v)(-1)