Uniqueness of the fixed point of nonexpansive semidifferentiable maps
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- by Marianne Akian, Stéphane Gaubert and Roger Nussbaum PDF
- Trans. Amer. Math. Soc. 368 (2016), 1271-1320 Request permission
Abstract:
We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in Hilbert’s or Thompson’s metric inherited from a convex cone. We show that the global uniqueness of the fixed point of the map, as well as the geometric convergence of every orbit to this fixed point, can be inferred from the semidifferential of the map at this point. In particular, we show that the geometric convergence rate of the orbits to the fixed point can be bounded in terms of Bonsall’s nonlinear spectral radius of the semidifferential. We derive similar results concerning the uniqueness of the eigenline and the geometric convergence of the orbits to it, in the case of positively homogeneous maps acting on the interior of a cone, or of additively homogeneous maps acting on an AM-space with unit. This is motivated in particular by the analysis of dynamic programming operators (Shapley operators) of zero-sum stochastic games.References
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Additional Information
- Marianne Akian
- Affiliation: INRIA and CMAP, École Polytechnique, 91128 Palaiseau Cedex, France
- Email: marianne.akian@inria.fr
- Stéphane Gaubert
- Affiliation: INRIA and CMAP, École Polytechnique, 91128 Palaiseau Cedex, France
- Email: stephane.gaubert@inria.fr
- Roger Nussbaum
- Affiliation: Mathematics Department, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 132680
- Email: nussbaum@math.rutgers.edu
- Received by editor(s): February 15, 2012
- Received by editor(s) in revised form: February 19, 2014
- Published electronically: February 19, 2015
- Additional Notes: The first and second authors were partially supported by the Arpege programme of the French National Agency of Research (ANR), project “ASOPT”, number ANR-08-SEGI-005
The third author was partially supported by NSFDMS 0701171 and by NSFDMS 1201328. - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 1271-1320
- MSC (2010): Primary 47H07, 47H09, 47H10, 47J10; Secondary 91A20
- DOI: https://doi.org/10.1090/S0002-9947-2015-06413-7
- MathSciNet review: 3430364