Simpler is also better in approximating fixed points
نویسندگان
چکیده
In this paper we demonstrate that a number of fixed point iteration problems can be solved using a modified Krasnoselskij iteration process, which is much simpler to use than the other iteration schemes that have been defined. 2008 Elsevier Inc. All rights reserved. For maps with a slow enough growth rate, the Banach contraction principle provides the existence and uniqueness of the fixed point, which can be obtained by repeated function iteration, or Picard iteration. For other maps, such as nonexpansive maps, pseudocontractive maps, etc., some other iteration process is required in order to obtain weak or strong convergence to a fixed point. Let X denote a Banach space, T a self-map of X, or of some closed convex subset of X. In 1955 Krasnoselskij [12] used the averaging process x0 2 X; xnþ1 1⁄4 ðxn þ TxnÞ=2: In 1957, Schaefer [35] extended the iteration scheme of Krasnoselskij to x0 2 X; xnþ1 1⁄4 ð1 kÞxn þ kTxn; ð1Þ for some 0; < k < 1. However, this method is usually referred to as generalized Krasnoselskij, or simply Krasnoselskij. In 1953 Mann [15] defined an iteration scheme which can be written in the form x0 2 X; xnþ1 1⁄4 ð1 anÞxn þ anTxn; ð2Þ where 0 6 an 6 1 and P an < 1. Thus every Krasnoselskij iteration is a Mann iteration. However, in some applications, see, e.g., Theorem 3.2 of [4], it is also required that liman 1⁄4 0. Hence the Mann iteration scheme is a more general one. However, Hillam [9], has shown that, if f is a Lipschitz self-map of an interval [a,b], with Lipschitz constant L, then Krasnoselskij iteration, with k = 1/(L + 1), converges monotonically to a fixed point of f. Ishikawa [11] defined the iteration scheme x0 2 X; xnþ1 1⁄4 ð1 anÞxn þ anTyn; yn 1⁄4 ð1 bnÞxn þ bnTxn; ð3Þ
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ورودعنوان ژورنال:
- Applied Mathematics and Computation
دوره 205 شماره
صفحات -
تاریخ انتشار 2008