A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems

We study the nonlinear boundary value problem (phi(u ''))'' = f(t, u, u ', u ''), t epsilon (0, 1), u((2i))(0) = u((2i))(1) = 0, i = 0, 1, and obtain a necessary and sufficient condition for the existence of symmetric positive solutions. We also discuss the application of our result to the special case where f is a power function of u and its derivatives. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our analysis mainly relies on the lower and upper solution method. (c) 2006 Elsevier Ltd. All rights reserved.