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Uniqueness of the fixed point of nonexpansive semidifferentiable maps

Authors: Marianne Akian, Stéphane Gaubert and Roger Nussbaum
Journal: Trans. Amer. Math. Soc. 368 (2016), 1271-1320
MSC (2010): Primary 47H07, 47H09, 47H10, 47J10; Secondary 91A20
Published electronically: February 19, 2015
MathSciNet review: 3430364
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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.

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Additional Information

Marianne Akian
Affiliation: INRIA and CMAP, École Polytechnique, 91128 Palaiseau Cedex, France

Stéphane Gaubert
Affiliation: INRIA and CMAP, École Polytechnique, 91128 Palaiseau Cedex, France

Roger Nussbaum
Affiliation: Mathematics Department, Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

Keywords: Nonlinear eigenvector, Hilbert's metric, Thompson's metric, nonexpansive maps, AM-space with unit, semidifferentiability, nonlinear spectral radius, geometric convergence, zero-sum stochastic games, value iteration
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.
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