Uniqueness of the fixed point of nonexpansive semidifferentiable maps

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2016-02, Vol.368 (2), p.1271-1320
Main Authors: Akian, Marianne, Gaubert, Stéphane, Nussbaum, Roger
Format: Article
Language:English
Subjects:
Functional Analysis
Mathematics
Optimization and Control
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2015-06413-7