### Counting solutions of nonlinear abstract equations

DOI: http://dx.doi.org/10.12775/TMNA.2004.030

#### 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.

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.

#### Keywords

Bifurcation theory; counting the number of solutions; fine topological structure

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