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Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space
Fixed Point Theory and Applications volume 2009, Article number: 962303 (2009)
Abstract
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.
1. Introduction
Let be a nonempty closed convex subset of a Hilbert space
,
a self-mapping of
. Recall that
is said to be nonexpansive if
for all
.
Construction of fixed points of nonexpansive mappings via Mann's iteration [1] 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 [6]). In order to get strong convergence, one must modify them. In 2003, Nakajo and Takahashi [7] proposed such a modification for a nonexpansive mapping .
Consider the algorithm,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ1_HTML.gif)
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
.
Let be a sequence of nonexpansive self-mappings of
,
a sequence of nonnegative numbers in
. For each
, defined a mapping
of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ2_HTML.gif)
Such a mapping is called the
-mapping generated by
and
; see [8].
In this paper, motivated by [9], for any given (
is a fixed number), we will propose the following iterative progress for two infinitely nonexpansive mappings
and
in a Hilbert space
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ3_HTML.gif)
and prove, converges strongly to a fixed point of
and
.
We will use the notation:
for weak convergence and
for strong convergence.
denotes the weak
-limit set of
.
2. Preliminaries
In this paper, we need some facts and tools which are listed as lemmas below.
Lemma 2.1 (see [10]).
Let be a Hilbert space,
a nonempty closed convex subset of
, and
a nonexpansive mapping with Fix
. If
is a sequence in
weakly converging to
and if
converges strongly to
, then
.
Lemma 2.2 (see [11]).
Let be a nonempty bounded closed convex subset of a Hilbert space
. Given also a real number
and
. Then the set
is closed and convex.
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 [8]).
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.
Remark 2.4.
It can be known from Lemma 2.3 that if is a nonempty bounded subset of
, then for
there exists
such that
for all
.
Remark 2.5.
Using Lemma 2.3, we can define a mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ4_HTML.gif)
for all Such a
is called the
-mapping generated by
and
Since
is nonexpansive mapping,
is also nonexpansive. Indeed, observe that for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ5_HTML.gif)
If is a bounded sequence in
, then we put
. Hence, it is clear from Remark 2.4 that for
there exists
such that for all
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ6_HTML.gif)
Lemma 2.6 (see [8]).
Let be a nonempty closed convex subset of a strictly convex Banach space
. Let
be a sequence of nonexpansive self-mappings on
such that
and let
be a sequence in
for some
. Then,
.
3. Strong Convergence Theorem
Theorem 3.1.
Let be a closed convex subset of a Hilbert space
and let
and
be defined as (1.2). Assume that
for all
and for some
and
for all
and
. If
then
generated by (1.3) converges strongly to
.
Proof.
Firstly, we observe that is convex by Lemma 2.2. Next, we show that
for all
.
Indeed, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ7_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ8_HTML.gif)
That is for all
. Next we show that
for all
.
We prove this by induction. For , we have
Assume that
for all
since
is the projection of
onto
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ9_HTML.gif)
As by the induction assumption, the last inequality holds, in particular, for all
. This together with definition of
implies that
. Hence
for all
.
Notice that the definition of implies
. This together with the fact
further implies
for all
The fact asserts that
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ10_HTML.gif)
We now claim that and
. Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ11_HTML.gif)
since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ12_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ13_HTML.gif)
We now show . Let
be any subsequence of
. Since
is a bounded subset of
, there exists a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ14_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ15_HTML.gif)
it follows that . By (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ16_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ17_HTML.gif)
Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ18_HTML.gif)
Using (3.1) again, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ19_HTML.gif)
This imply that . For the arbitrariness of
, we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ20_HTML.gif)
Thus, by (3.4), (3.7) and (3.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ21_HTML.gif)
Since and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ22_HTML.gif)
Thus, using (3.16), Lemma 2.1, and the boundedness of , we get that
. Since
and
, we have
where
. By the weak lower semicontinuity of the norm, we have
for all
. However, since
, we must have
for all
. Hence
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F962303/MediaObjects/13663_2009_Article_1195_Equ23_HTML.gif)
That is, converges to
.
This completes the proof.
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Acknowledgment
This work is supported by Grant KJ080725 of the Chongqing Municipal Education Commission.
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Chen, YA. Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space. Fixed Point Theory Appl 2009, 962303 (2009). https://doi.org/10.1155/2009/962303
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DOI: https://doi.org/10.1155/2009/962303