Existence result for a singular nonlinear boundary value problem at resonance

We study the existence of solutions of the second-order boundary value problems u('')(t) + pi(2)u(t) + a(t)g(u(t)) = h(t), a.e.t is an element of (0,1), u' is an element of AC(loc) (0,1), u(0) = u(t) = 0, where g : R -> R is a,h is an element of {z is an element of L-loc(1) (0,1) vertical bar f(0)(1) t vertical bar z(t)vertical bar dt < infinity}. The proof of the main result is based upon the Lyapunov-Schmidt procedure and the connectivity properties of the Solution set of parametrized families of compact vector fields. (c) 2006 Elsevier Ltd. All rights reserved.