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

Jean Mawhin

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


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


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

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