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A relaxed Picard iteration process for set-valued operators of the monotone type

Author: J. C. Dunn
Journal: Proc. Amer. Math. Soc. 73 (1979), 319-327
MSC: Primary 47H10; Secondary 65J05
MathSciNet review: 518512
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Abstract: The fixed points $ \bar x$ of set-valued operators, $ T:X \to {2^X}$, satisfying a condition of the monotonicity type on convex subsets X of a Hilbert space are approximated by a relaxation process, $ {x_{n + 1}} = {x_n} + {\omega _n}(T{x_n} - {x_n})$, in which $ \tilde T$ is a single-valued branch of T and the relaxation parameter $ {\omega _n} \in [0,1]$ is made to depend in a certain way on the prior history of the process. If $ \tilde T$ is bounded on bounded subsets of X, then $ \left\Vert {{x_n} - \bar x} \right\Vert$ converges to 0 like $ O({n^{ - 1/2}})$. If $ \tilde T$ is also continuous at $ \bar x$ and if $ \bar x = \tilde T\bar x$, then $ \left\Vert {{x_n} - \bar x} \right\Vert = o({n^{ - 1/2}})$. If $ \tilde T$ satisfies a condition of the Lipschitz type at $ \bar x$, then $ \left\Vert {{x_n} - \bar x} \right\Vert = O({\mu ^{n/2}})$ for some $ \mu \in [0,1)$.

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Keywords: Set-valued Hilbert space operators, fixed points, relaxed Picard iterations, residual-dependent step lengths, continuity conditions and convergence rates
Article copyright: © Copyright 1979 American Mathematical Society

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