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.