Multi-point boundary value problems of second-order differential equations (I)

We study the nonlinear boundary value problem with nonhomogeneous multi-point boundary condition u" + f (t, u, u') = 0, t is an element of (0, 1), u'(0) - Sigma(i=1)(m)a(i)u'(t(i)) = lambda(1), u(1) - Sigma(i=1)(m)b(i)u(t(i)) = lambda(2). Sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relations. Using one of the results, under some assumptions, we obtain explicit ranges of values of lambda(1) and lambda(2) with which the problem has a solution, has a positive solution, and has no solution, respectively. Furthermore, we prove that the whole plane for lambda(1) and lambda(2) can be divided into two disjoint connected regions Lambda(E) and Lambda(N) such that the problem has a solution for (lambda(1), lambda(2)) is an element of Lambda(E) and has no solution for (lambda(1), lambda(2)) is an element of Lambda(N). (C) 2004 Elsevier Ltd. All rights reserved.