Topological Methods in Nonlinear Analysis

Nonexpansive mappings on Hilbert's metric spaces

Bas Lemmens

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Abstract

This paper deals with the iterative behavior of nonexpansive mappings on Hilbert's metric spaces $(X,d_X)$. We show that if $(X,d_X)$ is strictly convex and does not contain a hyperbolic plane, then for each nonexpansive mapping, with a fixed point in $X$, all orbits converge to periodic orbits. In addition, we prove that if $X$ is an open $2$-simplex, then the optimal upper bound for the periods of periodic points of nonexpansive mappings on $(X,d_X)$ is $6$. The results have applications in the analysis of nonlinear mappings on cones, and extend work by Nussbaum and others.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 38, Number 1 (2011), 45-58.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461184806

Mathematical Reviews number (MathSciNet)
MR2893623

Zentralblatt MATH identifier
1244.54079

Citation

Lemmens, Bas. Nonexpansive mappings on Hilbert's metric spaces. Topol. Methods Nonlinear Anal. 38 (2011), no. 1, 45--58. https://projecteuclid.org/euclid.tmna/1461184806


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References

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