Lyusternik-Graves theorem and fixed points

Authors:
Asen L. Dontchev and Hélène Frankowska

Journal:
Proc. Amer. Math. Soc. **139** (2011), 521-534

MSC (2010):
Primary 49J53; Secondary 47J22, 49J40, 49K40, 90C31

Published electronically:
July 16, 2010

MathSciNet review:
2736335

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Abstract | References | Similar Articles | Additional Information

Abstract: For set-valued mappings and acting in metric spaces, we present local and global versions of the following general paradigm which has roots in the Lyusternik-Graves theorem and the contraction principle: if is metrically regular with constant and is Aubin (Lipschitz) continuous with constant such that , then the distance from to the set of fixed points of is bounded by times the infimum distance between and . From this result we derive known Lyusternik-Graves theorems, a recent theorem by Arutyunov, as well as some fixed point theorems.

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Additional Information

**Asen L. Dontchev**

Affiliation:
Mathematical Reviews and the University of Michigan, Ann Arbor, Michigan 48109. On leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria

Email:
ald@ams.org

**Hélène Frankowska**

Affiliation:
Combinatoire & Optimisation, CNRS, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France

Email:
frankowska@math.jussieu.fr

DOI:
https://doi.org/10.1090/S0002-9939-2010-10490-2

Keywords:
Metric regularity,
openness,
Lyusternik-Graves theorem,
fixed point,
Ekeland principle

Received by editor(s):
December 24, 2009

Received by editor(s) in revised form:
March 8, 2010

Published electronically:
July 16, 2010

Additional Notes:
The first author was supported by National Science Foundation grant DMS 1008341.

Communicated by:
Nigel J. Kalton

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.