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Uniqueness of the fixed point of nonexpansive semidifferentiable maps

Marianne Akian 1, 2, 3 Stephane Gaubert 1, 2, 3 Roger Nussbaum 4
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
3 TROPICAL - TROPICAL
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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 the 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 non-linear 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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https://hal.inria.fr/hal-00783682
Contributor : Marianne Akian <>
Submitted on : Friday, February 1, 2013 - 2:55:13 PM
Last modification on : Friday, April 30, 2021 - 10:03:53 AM

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Marianne Akian, Stephane Gaubert, Roger Nussbaum. Uniqueness of the fixed point of nonexpansive semidifferentiable maps. Transactions of the American Mathematical Society, American Mathematical Society, 2016, 368 (2), ⟨10.1090/S0002-9947-2015-06413-7⟩. ⟨hal-00783682⟩

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