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