Counting solutions of nonlinear abstract equations

Abstract

In this paper we use the topological degree to estimate the minimal number of solutions of the sections (defined by fixing a parameter) of the semi-bounded components of a general class of one-parameter abstract nonlinear equations by means of the {\it signature} of the semi-bounded component. A semi-bounded component is, roughly speaking, a component that is bounded along one direction of the parameter. The signature consists of the set of bifurcation values from the trivial state of the component together with their associated parity indices. The parity is a local invariant measuring the change of the local index of the trivial state.