The Leray-Schauder condition for continuous pseudo-contractive mappings

Morales, Claudio

doi:10.1090/S0002-9939-08-09570-1

The Leray-Schauder condition for continuous pseudo-contractive mappings
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by Claudio H. Morales

Proc. Amer. Math. Soc. 137 (2009), 1013-1020

DOI: https://doi.org/10.1090/S0002-9939-08-09570-1

Published electronically: September 24, 2008

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Abstract:

Over thirty years ago, Kirk raised the question of whether a nonexpansive mapping, defined on a convex domain with nonempty interior, has a fixed point under the Leray-Schauder condition, provided that its domain enjoys the Fixed Point Property with respect to nonexpansive self-mappings. In the present work we have found the answer to this question to be positive, even for a larger class of mappings. The result, indeed, represents a quite significant extension of a large number of theorems obtained in the last forty years. This also includes new theorems for nonexpansive mappings.

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