A necessary and sufficient condition for the existence of positive solutions to a kind of singular three-point boundary value problem

In this paper, we are concerned with the following three point boundary value problem {u ''(t) + f(t, u(t)) = 0, 0 < t < 1, delta u'(0) - u(xi) = 0, u(1) = 0, where delta >= xi, 0 < xi < 1, and the nonlinear term f may be singular at t = 1 or/and t = 0. A necessary and sufficient condition for the existence of C[0, 1] as well as C-1[0, 1] positive solution is given by constructing lower and upper solutions, and with the maximal theorem. Also, the uniqueness, iterative methods, convergence rate and computational methods of the positive solutions are considered. (C) 2009 Elsevier Ltd. All rights reserved.