By Andrzej Cegielski
Iterative tools for locating fastened issues of non-expansive operators in Hilbert areas were defined in lots of guides. during this monograph we attempt to provide the tools in a consolidated means. We introduce a number of sessions of operators, learn their houses, outline iterative equipment generated by way of operators from those sessions and current common convergence theorems. in this foundation we talk about the stipulations below which specific tools converge. a wide a part of the consequences awarded during this monograph are available in quite a few types within the literature (although numerous effects awarded listed here are new). we've got attempted, in spite of the fact that, to teach that the convergence of a giant classification of generation tools follows from common homes of a few sessions of operators and from a few normal convergence theorems.
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Extra info for Iterative Methods for Fixed Point Problems in Hilbert Spaces
Sample text
0; 1. , S is a nonexpansive operator. i0 > 0, for some i0 2 I . , S is a contraction. 0; 1. 0; 1/ and S is a contraction. 13. Let Ui W X ! X be nonexpansive for all i 2 I WD f1; 2; : : : ; mg, and U WD Um Um 1 : : : U1 . X / is bounded for at least one j 2 I , then Fix U ¤ ;. Proof. X / be bounded for some j 2 I . X /. X / is closed, convex and bounded. X / Â X and X is closed and convex, we have Y Â X . The operator U jY maps a closed, convex and bounded subset Y into itself. By the Browder–G¨ohde–Kirk Fixed Point Theorem, the operator U jY has a fixed point z 2 Y .
U is quasi-nonexpansive. It is clear that the class of nonexpansive operators having a fixed point is an essential subclass of quasi-nonexpansive operators, because a quasi-nonexpansive operator needs not to be continuous. 2). In this section we present properties of the family of quasi-nonexpansive operators.
The theorem below, called the Banach fixed point theorem or the Banach theorem on contractions, is widely applied in various areas of mathematics. The theorem holds for any complete metric space, and hence, in particular, for every closed subset of a Hilbert space. 7 (Banach, 1922). Let X be a complete metric space and T W X ! X be a contraction. Then T has exactly one fixed point x 2 X . Furthermore, for any x 2 X , the orbit fT k xg1 kD0 converges to x with a rate of geometric progression. Proof.