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Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions
Fixed Point Theory and Applications volume 2010, Article number: 948529 (2010)
Abstract
We introduce and study some new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. General results for asymptotically strict pseudocontractions are established. The main result extends the previous results.
1. Introduction
Let be a real Hilbert space,
a nonempty closed convex subset of
,
a self-mapping of
and
Recall that a mapping is called to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ1_HTML.gif)
is called to be asymptotically nonexpansive [1] if there exists a sequence
with
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ2_HTML.gif)
is called to be an asymptotically
-strict pseudocontraction, if there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ3_HTML.gif)
for all and all integers
As , asymptotically
-strict pseudocontraction
is asymptotically nonexpansive.
In [2], Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.
Theorem 1 A.
Let be a nonempty closed convex subset of a Hilbert space
and let
be a nonexpansive mapping of
into itself such that
. Suppose
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ4_HTML.gif)
where is the metric projection from C onto
and
is chosen so that
Then,
converges strongly to
, where
is the metric projection from C onto
Such algorithm in (1.4) is referred to be the (CQ) algorithm in [3], due to the fact that each iterate is obtained by projecting
onto the intersection of the suitably constructed closed convex sets
and
It is known that the (CQ) algorithm in (1.4) is of independent interest, and the (CQ) algorithm has been extended to various mappings by many authors (cf., e.g., [3–11]).
Very recently, by extending the (CQ) algorithm, Takahashi et al. [9] studied a family of nonexpansive mappings and gave some good strong convergence theorems. Kim and Xu [5] extended the (CQ) algorithm to study asymptotically -strict pseudocontractions and established the following interesting result with the help of some boundedness conditions.
Theorem 1 B.
Let be a closed convex subset of a Hilbert space
and let
be an asymptotically
-strict pseudocontractions for some
Assume that the fixed point set
of
is nonempty and bounded. Let
be the sequence generated by the following (CQ) algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ5_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ6_HTML.gif)
Assume that control sequence is chosen so that
Then
converges strongly to
.
It is our purpose in this paper to try to obtain some new fixed point theorems for asymptotically strict pseudocontractions without the boundedness conditions as in Theorem B. Motivated by Nakajo and Takahashi [2], Takahashi et al. [9], and Kim and Xu [5], we introduce and study certain new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. Our results improve essentially the corresponding results of [5].
2. Results and Proofs
Throughout this paper,
(i) means that
converges weakly to
(ii) means that
converges strongly to
(iii), that is, the weak
-limit set of
(iv).
(v) is the set of nonnegative integers.
The following lemmas are basic (cf., e.g., [6] for Lemma 2.1, and [5] for Lemmas 2.2-2.3).
Lemma 2.1.
Let be a closed convex subset of a real Hilbert space
. Given
. Then
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ7_HTML.gif)
where is the unique point in
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ8_HTML.gif)
Lemma 2.2.
Let be a closed convex subset of a real Hilbert space
,
, and
. Suppose that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ9_HTML.gif)
and . Then
Lemma 2.3.
Let be a closed convex subset of a Hilbert space
and
an asymptotically
-strict pseudocontraction. Then
for each
,
satisfies the Lipschitz condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ10_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ11_HTML.gif)
if
is a sequence in
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ12_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ13_HTML.gif)
In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ14_HTML.gif)
is closed and convex so that the projection
is well defined.
Theorem 2.4.
Let be a closed convex subset of a Hilbert space
,
an asymptotically
-strict pseudocontraction for some
, and
Let
be the sequence generated by the following CQ-type algorithm with variable coefficients:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ15_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ16_HTML.gif)
the sequence is chosen so that
, the positive real number
is chosen so that
, and
is as in (1.3). Then
converges strongly to
.
Proof.
We divide the proof into five steps.
Step 1.
We prove that is nonempty, convex and closed.
Clearly, both and
are convex and closed, so is
. Since
is an asymptotically
-strict pseudocontraction, we have by (1.3),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ17_HTML.gif)
for all ,
, and all integers
By (2.9) and (2.11), we deduce that for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ18_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ19_HTML.gif)
Next, we prove by induction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ20_HTML.gif)
Obviously, , that is, (2.14) holds for
. Assume that
for some
Then, (2.13) implies that
and
is well defined.
By Lemma 2.1, we get In particular, for each
we have
This together with the definition of
, the inequality (2.14) holds for
. So (2.14) is true.
Step 2.
We prove that
By the definition of and Lemma 2.1, we get
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ21_HTML.gif)
Denoting , we have
for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ22_HTML.gif)
where The definition of
shows that
, that is,
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ23_HTML.gif)
Thus is increasing. Since
is bounded,
exists and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ24_HTML.gif)
Step 3.
We prove that
The definition of shows that
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ25_HTML.gif)
By (2.19) and the definition of in (2.9), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ26_HTML.gif)
Further, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ27_HTML.gif)
Thus, (2.19) and (2.21) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ28_HTML.gif)
Noticing , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ29_HTML.gif)
From and (2.22), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ30_HTML.gif)
Step 4.
We prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ31_HTML.gif)
By Lemma 2.3 and the definition of , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ32_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ33_HTML.gif)
By (2.18), (2.24), and (2.26), we know that (2.25) holds.
Step 5.
Finally, by Lemma 2.3 and (2.25), we have . Furthermore, it follows from (2.16) and Lemma 2.2 that the sequence
converges strongly to
Remark 2.5.
Theorem 2.4 improves [5, Theorem ] since the condition that
is satisfied and the boundedness of
is dropped off.
Theorem 2.6.
Let be a closed convex subset of a Hilbert space
,
an asymptotically
-strict pseudocontraction for some
, and
be nonempty and bounded. Let
the sequence generated by the following CQ-type algorithm with variable coefficients:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ34_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F948529/MediaObjects/13663_2009_Article_1370_Equ35_HTML.gif)
the sequence is chosen so that
and
is as in (1.3). Then
converges strongly to
.
Proof.
It is easy to see that in Theorem 2.6. Following the reasoning in the proof of Theorem 2.4 and using
instead of
, we deduce the conclusion of Theorem 2.6.
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Acknowledgments
The authors are very grateful to the referee for his/her valuable suggestions and comments. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). This work is dedicated to W. Takahashi.
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Ge, CS., Liang, J. & Xiao, TJ. Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions. Fixed Point Theory Appl 2010, 948529 (2010). https://doi.org/10.1155/2010/948529
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DOI: https://doi.org/10.1155/2010/948529