Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space
© Yi-An Chen. 2009
Received: 23 June 2009
Accepted: 12 October 2009
Published: 1 November 2009
We introduce a modified Ishikawa iterative process for approximating a fixed point of two infinitely nonexpansive self-mappings by using the hybrid method in a Hilbert space and prove that the modified Ishikawa iterative sequence converges strongly to a common fixed point of two infinitely nonexpansive self-mappings.
Construction of fixed points of nonexpansive mappings via Mann's iteration  has extensively been investigated in literature (see, e.g., [2–5] and reference therein). But the convergence about Mann's iteration and Ishikawa's iteration is in general not strong (see the counterexample in ). In order to get strong convergence, one must modify them. In 2003, Nakajo and Takahashi  proposed such a modification for a nonexpansive mapping .
Consider the algorithm,
where denotes the metric projection from onto a closed convex subset of . They prove the sequence generated by that algorithm (1.1) converges strongly to a fixed point of provided that the control sequence is chosen so that .
Such a mapping is called the -mapping generated by and ; see .
In this paper, motivated by , for any given ( is a fixed number), we will propose the following iterative progress for two infinitely nonexpansive mappings and in a Hilbert space :
We will use the notation:
In this paper, we need some facts and tools which are listed as lemmas below.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
Let be a sequence of nonexpansive self-mappings on , where is a nonempty closed convex subset of a strictly convex Banach space . Given a sequence in , one defines a sequence of self-mappings on by (1.2). Then one has the following results.
Lemma 2.3 (see ).
Let be a nonempty closed convex subset of a strictly convex Banach space , a sequence of nonexpansive self-mappings on such that and let be a sequence in for some . Then, for every and the limit exists.
Lemma 2.6 (see ).
3. Strong Convergence Theorem
This completes the proof.
This work is supported by Grant KJ080725 of the Chongqing Municipal Education Commission.
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