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Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 852030 (2010)
Abstract
We consider two new iterative methods for a countable family of nonexpansive mappings in Hilbert spaces. We proved that the proposed algorithms strongly converge to a common fixed point of a countable family of nonexpansive mappings which solves the corresponding variational inequality. Our results improve and extend the corresponding ones announced by many others.
1. Introduction
Let be a real Hilbert space and let
be a nonempty closed convex subset of
. Recall that a mapping
is said to be nonexpansive if
. We use
to denote the set of fixed points of
. A mapping
is called
-Lipschitzian if there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ1_HTML.gif)
is said to be
-strongly monotone if there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ2_HTML.gif)
Let be a strongly positive bounded linear operator on
, that is, there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ3_HTML.gif)
A typical problem is that of minimizing a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ4_HTML.gif)
where is a given point in
.
Remark 1.1.
From the definition of , we note that a strongly positive bounded linear operator
is a
-Lipschitzian and
-strongly monotone operator.
Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative algorithms for finding fixed points of nonexpansive mappings have received vast investigation (cf. [1, 2]) since these algorithms find applications in variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [3–8]. One classical way to find the fixed point of a nonexpansive mapping is to use a contraction to approximate it. More precisely, take
and define a contraction
by
, where
is a fixed point. Banach's Contraction Mapping Principle guarantees that
has a unique fixed point
in
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ5_HTML.gif)
The strong convergence of the path has been studied by Browder [9] and Halpern [10] in a Hilbert space.
Recently, Yao et al. [11] considered the following algorithms:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ6_HTML.gif)
and for arbitrarily,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ7_HTML.gif)
They proved that if and
satisfying appropriate conditions, then the
defined by (1.6) and
defined by (1.7) converge strongly to a fixed point of
.
On the other hand, Yamada [12] introduced the following hybrid iterative method for solving the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ8_HTML.gif)
where is a
-Lipschitzian and
-strongly monotone operator with
,
,
. Then he proved that
generated by (1.8) converges strongly to the unique solution of variational inequality
,
.
In this paper, motivated and inspired by the above results, we introduce two new algorithms (3.3) and (3.13) for a countable family of nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to which solves the variational inequality:
,
.
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
. For the sequence
in
, we write
to indicate that the sequence
converges weakly to
.
implies that
converges strongly to
. For every point
, there exists a unique nearest point in
, denoted by
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ9_HTML.gif)
The mapping is called the metric projection of
onto
. It is well know that
is a nonexpansive mapping. In a real Hilbert space
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ10_HTML.gif)
In order to prove our main results, we need the following lemmas.
Lemma 2.1 (see [13]).
Let be a Hilbert space,
a closed convex subset of
, and
a nonexpansive mapping with
, if
is a sequence in
weakly converging to
and if
converges strongly to
, then
.
Lemma 2.2 (see [14]).
Let and
be bounded sequences in Banach space
and
a sequence in
which satisfies the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ11_HTML.gif)
Suppose that ,
and
. Then
.
Let be a sequence of nonnegative real numbers satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ12_HTML.gif)
where ,
, and
satisfy the following conditions: (i)
and
, (ii)
or
, (iii)
. Then
.
Lemma 2.4 (see [17, Lemma ]).
Let C be a nonempty closed convex subset of a Banach space E. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ13_HTML.gif)
Then, for each ,
converges strongly to some point of C. Moreover, let T be a mapping of C into itself defined by
, for all
. Then
.
Lemma 2.5.
Let be a
-Lipschitzian and
-strongly monotone operator on a Hilbert space
with
and
. Then
is a contraction with contraction coefficient
.
Proof.
From (1.1), (1.2), and (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ14_HTML.gif)
for all . From
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ15_HTML.gif)
where . Hence
is a contraction with contraction coefficient
.
3. Main Results
Let be a
-Lipschitzian and
-strongly monotone operator on
with
and
a nonexpansive mapping. Let
and
; consider a mapping
on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ16_HTML.gif)
It is easy to see that is a contraction. Indeed, from Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ17_HTML.gif)
for all . Hence it has a unique fixed point, denoted
, which uniquely solves the fixed point equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ18_HTML.gif)
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping such that
. Let
be a
-Lipschitzian and
-strongly monotone operator on
with
. For each
, let the net
be generated by (3.3). Then, as
, the net
converges strongly to a fixed point
of
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ19_HTML.gif)
Proof.
We first show the uniqueness of a solution of the variational inequality (3.4), which is indeed a consequence of the strong monotonicity of . Suppose
and
both are solutions to (3.4); then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ20_HTML.gif)
Adding up (3.5) gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ21_HTML.gif)
The strong monotonicity of implies that
and the uniqueness is proved. Below we use
to denote the unique solution of (3.4).
Next, we prove that is bounded. Take
; from (3.3) and using Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ22_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ23_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ24_HTML.gif)
From , we may assume, without loss of generality, that
. Thus, we have that
is continuous, for all
. Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ25_HTML.gif)
From (3.8) and (3.10), we have that is bounded and so is
.
On the other hand, from (3.3), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ26_HTML.gif)
To prove that . For a given
, by (2.2) and using Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ27_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ28_HTML.gif)
From , we have
and
. Observe that, if
, we have
.
Since is bounded, we see that if
is a sequence in
such that
and
, then by (3.13), we see
. Moreover, by (3.11) and using Lemma 2.1, we have
. We next prove that
solves the variational inequality (3.4). From (3.3) and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ29_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ30_HTML.gif)
Now replacing in (3.15) with
and letting
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ31_HTML.gif)
That is is a solution of (3.4); hence
by uniqueness. In a summary, we have shown that each cluster point of
(as
) equals
. Therefore,
as
.
Setting in Theorem 3.1, we can obtain the following result.
Corollary 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping such that
. Let
be a strongly positive bounded linear operator with coefficient
. For each
, let the net
be generated by
. Then, as
, the net
converges strongly to a fixed point
of
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ32_HTML.gif)
Setting , the identity mapping, in Theorem 3.1, we can obtain the following result.
Corollary 3.3.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping such that
. For each
, let the net
be generated by (1.6). Then, as
, the net
converges strongly to a fixed point
of
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ33_HTML.gif)
Remark 3.4.
The Corollary 3.3 complements the results of Theorem in Yao et al. [11], that is,
is the solution of the variational inequality:
.
Theorem 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive mappings of
into itself such that
. Let
be a
-Lipschitzian and
-strongly monotone operator on
with
. Let
and
be two real sequences in
and satisfy the conditions:
(A1) and
;
(A2) .
Suppose that for any bounded subset
of
. Let
be a mapping of
into itself defined by
for all
and suppose that
. For given
arbitrarily, let the sequence
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ34_HTML.gif)
Then the sequence strongly converges to a
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ35_HTML.gif)
Proof.
We proceed with the following steps.
Step 1.
We claim that is bounded. From
, we may assume, without loss of generality, that
for all
. In fact, let
, from (3.19) and using Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ36_HTML.gif)
where . Then from (3.19) and (3.21), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ37_HTML.gif)
By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ38_HTML.gif)
where . Therefore,
is bounded. We also obtain that
,
, and
are bounded. Without loss of generality, we may assume that
,
,
, and
, where
is a bounded set of
.
Step 2.
We claim that . To this end, define a sequence
by
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ39_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ40_HTML.gif)
From and (3.25), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ41_HTML.gif)
By (3.26), (A2), and using Lemma 2.2, we have . Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ42_HTML.gif)
Step 3.
We claim that . Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ43_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ44_HTML.gif)
Step 4.
We claim that . Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ45_HTML.gif)
Hence, from Step 3 and using Lemma 2.4, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ46_HTML.gif)
Step 5.
We claim that , where
and
is defined by (3.3). Since
is bounded, there exists a subsequence
of
which converges weakly to
. From Step 4, we obtain
. From Lemma 2.1, we have
. Hence, by Theorem 3.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ47_HTML.gif)
Step 6.
We claim that converges strongly to
. From (3.19), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ48_HTML.gif)
where ,
, and
. It is easy to see that
,
, and
. Hence, by Lemma 2.3, the sequence
converges strongly to
. From
and Theorem 3.1, we have that
is the unique solution of the variational inequality:
Remark 3.6.
From Remark of Peng and Yao [18], we obtain that
is a sequence of nonexpansive mappings satisfying condition
for any bounded subset B of H. Moreover, let
be the
mapping; we know that
for all
and that
. If we replace
by
in the recursion formula (3.19), we can obtain the corresponding results of the so-called
mapping.
Setting and
in Theorem 3.5, we can obtain the following result.
Corollary 3.7.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping such that
. Let
be a strongly positive bounded linear operator with coefficient
. Let
and
be two real sequences in
and satisfy the conditions (A1) and (A2). For given
arbitrarily, let the sequence
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ49_HTML.gif)
Then the sequence strongly converges to a fixed point
of
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ50_HTML.gif)
Setting and
in Theorem 3.5, we can obtain the following result.
Corollary 3.8.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a nonexpansive mapping such that
. Let
and
be two real sequences in
and satisfy the conditions (A1) and (A2). For given
arbitrarily, let the sequence
be generated by (1.7). Then the sequence
strongly converges to a fixed point
of
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F852030/MediaObjects/13663_2010_Article_1354_Equ51_HTML.gif)
Remark 3.9.
The Corollary 3.8 complements the results of Theorem in Yao et al. [11], that is,
is the solution of the variational inequality:
.
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This paper is supported by the National Science Foundation of China under Grant (10771175).
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Wang, S., Hu, C. Two New Iterative Methods for a Countable Family of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 852030 (2010). https://doi.org/10.1155/2010/852030
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DOI: https://doi.org/10.1155/2010/852030