### Leray-Schauder degree: a half century of extensions and applications

DOI: http://dx.doi.org/10.12775/TMNA.1999.029

#### Abstract

The Leray-Schauder degree is defined for mappings of the form $I-C$, where

$C$ is a compact mapping from the closure of an open bounded subset of a

Banach space $X$ into $X$. Since the fifties, a lot of work has been devoted

in extending this theory to the same type of mappings on some nonlinear

spaces, and in extending the class of mappings in the frame of Banach spaces

or manifolds. New applications of Leray-Schauder theory and its extensions

have also been given, specially in bifurcation theory, nonlinear boundary

value problems and equations in ordered spaces. The paper surveys those

developments.

$C$ is a compact mapping from the closure of an open bounded subset of a

Banach space $X$ into $X$. Since the fifties, a lot of work has been devoted

in extending this theory to the same type of mappings on some nonlinear

spaces, and in extending the class of mappings in the frame of Banach spaces

or manifolds. New applications of Leray-Schauder theory and its extensions

have also been given, specially in bifurcation theory, nonlinear boundary

value problems and equations in ordered spaces. The paper surveys those

developments.

#### Keywords

Leray-Schauder degree; fixed point index; fixed point theorems; continuation theorems; bifurcation

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