Topological degree and a nonlinear Dirichlet problem

We give an extension of application of the homotopy invariance of the topological degree to a problem with jumping nonlinearities of type -u(t) = g(u(t)) - lambdaf(t), u(0) = u(1) = 0. To do this, we define sets Z(k) which are a generalization of sets S-k introduced by Rabinowitz, and calculate a priori estimates. We define a compact operator depending on parameters lambda > 0 and theta is an element of [0, 1], which is related to the equation -u = g(u)theta + (1 - theta)u(2) - lambda(f(t)theta + 1 - 0). Then using the homotopy invariance of the topological degree with respect to theta, on open sets of Z(k), for lambda large enough, we deduce multiple solutions from the problem -u(t) = u(2)(t) - lambda studied by Ammar Khodja. (C) 2003 Elsevier Science Ltd. All rights reserved.