The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces

Miroslav Bačák and Simeon Reich

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Submission date: 28. May. 2014
Pages: 14
published in: Journal of fixed point theory and applications, 16 (2014) 1-2, p. 189-202 
DOI number (of the published article): 10.1007/s11784-014-0202-3
Bibtex
MSC-Numbers: 47H20, 60J45
Keywords and phrases: dirichlet problem, fixed point, Hadamard space, heat flow, nonlinear Markov operator, proximal point algorithm, resolvent, semigroup
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Abstract:
We study a nonlinear semigroup associated to a nonexpansive mapping on a Hadamard space and establish its weak convergence to a fixed point. A discrete-time counterpart of such a semigroup, the proximal point algorithm, turns out to have the same asymptotic behavior. This complements several results in the literature – both classical and more recent ones. As an application, we obtain a new approach to heat flows in singular spaces for discrete, as well as continuous times.