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A Dynamical System Associated with the Fixed Points Set of a Nonexpansive Operator

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Abstract

We study the existence and uniqueness of (locally) absolutely continuous trajectories of a dynamical system governed by a nonexpansive operator. The weak convergence of the orbits to a fixed point of the operator is investigated by relying on Lyapunov analysis. We show also an order of convergence of \(o\left( \frac{1}{\sqrt{t}}\right) \) for the fixed point residual of the trajectory of the dynamical system. We apply the results to dynamical systems associated with the problem of finding the zeros of the sum of a maximally monotone operator and a cocoercive one. Several dynamical systems from the literature turn out to be particular instances of this general approach.

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Acknowledgments

We are thankful to H. Attouch for bringing into our attention the time rescaling arguments which led to the results presented in the last section. Radu Ioan Boţ and Ernö Robert Csetnek: research partially supported by DFG (German Research Foundation), Project BO 2516/4-1.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Radu Ioan Boţ & Ernö Robert Csetnek

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  1. Radu Ioan Boţ
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  2. Ernö Robert Csetnek
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Correspondence to Radu Ioan Boţ.

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Boţ, R.I., Csetnek, E.R. A Dynamical System Associated with the Fixed Points Set of a Nonexpansive Operator. J Dyn Diff Equat 29, 155–168 (2017). https://doi.org/10.1007/s10884-015-9438-x

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  • DOI: https://doi.org/10.1007/s10884-015-9438-x

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