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A general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 138 (2013)
Abstract
In this paper, we introduce a new general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to a common point of the sets of fixed points, which solves a variational inequality. Our results improve and extend the corresponding results announced by many others. As applications, at the end of the paper, we apply our results to the split common fixed point problem.
1 Introduction
Let H be a real Hilbert space with the inner product and the norm . Let T be a nonexpansive operator. The set of fixed points of T is denoted by . In 2000, Moudafi [1] introduced the viscosity approximation method for a nonexpansive operator and considered the sequence by
where f is a contraction on H and is a sequence in . In 2004, Xu [2] proved that under some conditions on , the sequence generated by (1.1) strongly converges to in which is the unique solution of the variational inequality
It is well known that iterative methods for nonexpansive operators have been used to solve convex minimization problems; see, e.g., [3, 4]. A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive operator T on a real Hilbert space H:
where b is a given point in H and A is a strongly positive bounded linear operator. In [3], Xu proved that the sequence defined by the following iterative method:
converges strongly to the unique solution of the minimization problem (1.2). In [5], Marino and Xu combined the iterative method (1.3) and the viscosity method (1.1) and considered the following general iterative method:
They proved that the sequence generated by (1.4) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for ).
On the other hand, Yamada [4] in 2001 introduced the following hybrid iterative method:
where F is a k-Lipschitzian and η-strongly monotone operator with , and . Under some appropriate conditions, he proved that the sequence generated by (1.5) converges strongly to the unique solution of the variational inequality
Recently, combining (1.4) and (1.5), Tian [6] considered the following general iterative method:
Improving and extending the corresponding results given by Marino, Xu and Yamada, he proved that the sequence generated by (1.6) converges strongly to the unique solution of the variational inequality
Based on the above results of Marino, Xu, Yamada and Tian, much generalization work has been made by the corresponding authors; for instance, [7–23]. The problem of finding an element in the intersection of the fixed point sets of an infinite family of nonexpansive operators has attracted much attention because of its extraordinary utility and broad applicability in many branches of mathematical science and engineering. For example, if the nonexpansive operators are projection onto some closed convex sets () in a real Hilbert space H, then such a fixed point problem becomes the convex feasibility problem of finding a point in . Many previous results [24–31] and many results not cited here considered the common fixed point about an infinite family of nonexpansive operators by -mappings.
Motivated and inspired by the above results, we consider the following iterative algorithm without -mappings:
where is a sequence in and is a strictly decreasing sequence in . Under some appropriate conditions, we proved the sequence generated by (1.7) converges strongly to the unique solution of the variational inequality:
Our results improve and extend the corresponding results announced by many others. As applications, at the end of the paper, we apply our results to the split common fixed point problem.
2 Preliminaries
Throughout this paper, we write and to indicate that converges weakly to x and converges strongly to x, respectively.
An operator is said to be nonexpansive if for all . It is well known that is closed and convex. It is known that A is called strongly positive if there exists a constant such that for all . The operator F is called η-strongly monotone if there exists a constant such that
for all .
In order to prove our main results, we collect the following lemmas in this section.
Lemma 2.1 (Demiclosedness principle [32])
Let H be a Hilbert space, C be a closed convex subset of H, and be a nonexpansive operator with . If is a sequence in C weakly converging to and converges strongly to , then . In particular, if , then .
Lemma 2.2 [2]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(i)
,
-
(ii)
or .
Then .
Lemma 2.3 [33]
Let H be a real Hilbert space, let be an L-Lipschitzian operator with , and let be a k-Lipschitzian continuous operator and η-strongly monotone operator with , . Then, for , is strongly monotone with coefficient .
Lemma 2.4 [34]
Let C be a closed convex subset of a real Hilbert space H, given and . Then if and only if the following inequality holds:
for every .
3 Main results
Lemma 3.1 Let be an infinite family of nonexpansive operators, let be a k-Lipschitzian and η-strongly monotone operator with and , and let be an L-Lipschitzian operator. Let and . Assume that , where is a strictly decreasing sequence with and . Consider the following mapping on H defined by
where is a sequence in . Then is a contraction.
Proof Observe that
Since , is a contraction. This completes the proof. □
Since is a contraction, using the Banach contraction principle, has a unique fixed point such that
For simplicity, we denote for without confusion.
Now we state and prove our main results in this paper.
Theorem 3.2 Let be an infinite family of nonexpansive self-mappings of a real Hilbert space H, let F be a k-Lipschitzian and η-strongly monotone operator on H with and , and let V be an L-Lipschitzian operator. Suppose that is nonempty. Suppose that is generated by the following algorithm:
where and with . If the following conditions are satisfied:
-
(i)
is a sequence in and ;
-
(ii)
is a strictly decreasing sequence in and .
Then converges strongly to , which solves the variational inequality:
Equivalently, we have .
Proof We proceed with the following steps:
Step 1: First we show that is bounded.
In fact, let , then for every , . Observe that
Thus it follows that
Then we have
which implies that is bounded. Hence we can obtain , , and are bounded. Note that
we immediately obtain that
Step 2: We show .
Since , we note that
which implies that
Thus
Then we immediately obtain . Since is strictly decreasing, it follows that
for every . Since , thus
It shows that
Step 3: We show that there exists a subsequence of such that .
Since is bounded, there exist a point and a subsequence of such that . By Lemma 2.1 and (3.4), we obtain for any . This shows that . On the other hand, we note that
Hence we obtain
Then it follows that
In particular,
From and (3.3), it follows that .
Step 4: We show that solves the variational inequality (3.2).
Observe that
Hence, we conclude that
Since is nonexpansive, we have that is monotone. Note that for any , . Then we deduce
Now, replacing n with in the above inequality, and letting , by (3.5) we have
That is, for every . It follows that is a solution of the variational inequality (3.2). Since is -strongly monotone and -Lipschitzian, the variational inequality (3.2) has a unique solution. So, we conclude that as . The variational inequality (3.2) can be written as
By Lemma 2.4, we have . □
Theorem 3.3 Let be an infinite family of nonexpansive self-mappings of a real Hilbert space H, let F be a k-Lipschitzian and η-strongly monotone operator on H with and , and let V be an L-Lipschitzian operator. Suppose that is nonempty. Suppose that , and with . Let , be a sequence in , and let be a strictly decreasing sequence in . If the following conditions are satisfied:
-
(i)
;
-
(ii)
;
-
(iii)
;
-
(iv)
.
Then generated by (1.7) converges strongly to , which solves the variational inequality
Equivalently, we have .
Proof We proceed with the following steps:
Step 1: First show that there exists such that .
In fact, by Lemma 2.3, is strongly monotone. So, the variational inequality (3.6) has only one solution. We set to indicate the unique solution of (3.6). The variational inequality (3.6) can be written as
So, by Lemma 2.4, it is equivalent to the fixed point equation
Step 2: Now we show that is bounded.
Let , then for every , . Observe that
Thus it follows that
Therefore, is bounded. Hence we can obtain that , , and are bounded.
Step 3: we show .
We observe that
It follows that
We have
Combining (3.7) and (3.8), we obtain that
Using (ii), (iii), (iv) and Lemma 2.2, we have .
Step 4: We show for all .
Since , we note that
which implies that
From (1.7) and (3.9), we deduce
Using (1.7), we can have
Noting that and , we immediately obtain
Since is strictly decreasing, it follows that for every ,
Step 5: Show , where .
Since is bounded, there exist a point and a subsequence of such that
and . Now, applying (3.10) and Lemma 2.1, we conclude that for every . Hence, . Since Ω is closed and convex, by Lemma 2.4, we get
Step 6: Show .
Since , we have for every . Using (1.7), we have
which implies that
Consequently, according to (3.11) and Lemma 2.2, we deduce that converges strongly to . This completes the proof. □
Corollary 3.4 Let T be a nonexpansive self-mapping of a real Hilbert space H, let F be a k-Lipschitzian and η-strongly monotone operator on H with and , and let V be an L-Lipschitzian operator. Suppose that is nonempty. Suppose that and that is generated by the following algorithm:
where and with . Let be in . If the following conditions are satisfied:
-
(i)
;
-
(ii)
;
-
(iii)
.
Then converges strongly to , which solves the variational inequality
Equivalently, we have .
Proof Set to be the sequences of operators defined by for all in Theorem 3.3. Then by Theorem 3.3, we obtain the desired result. □
4 Application in the split common fixed point problem
Let and be Hilbert spaces, let be a bounded linear operator. The split common fixed point problem (SCFPP) is to find a point satisfied with
where () and () are nonlinear operators. The concept of the SCFPP in finite-dimensional Hilbert spaces was firstly introduced by Censor and Segal in [35]. Now we consider a generalized split common fixed point problem (GSCFPP) which is to find a point
We know that if for all i and j, and are nonexpansive operators, the GSCFPP is equivalent to the following common fixed point problem:
where with for every (see [36]). The solution set of GSCFPP (4.1) is denoted by S.
Theorem 4.1 Let and be sequences of nonexpansive operators on real Hilbert spaces and , respectively. Let F be a k-Lipschitzian and η-strongly monotone operator on with and . Let V be an L-Lipschitzian operator. Suppose that S is nonempty. Suppose that and that is generated by the following algorithm:
where and with . Let be in , be a strictly decreasing sequence in and . If the following conditions are satisfied:
-
(i)
;
-
(ii)
;
-
(iii)
;
-
(iv)
.
Then converges strongly to , which solves the variational inequality
Equivalently, we have .
Proof Set to be the sequences of operators defined by for all in Theorem 3.3. By Theorem 3.3, we can obtain
in Step 4. But it does not imply that the set of cluster points of the weak topology is a subset of S. In order to prove this, we only show and .
Since , . Hence, for every ,
which yields that
for every . Using (4.2), we note that
Thus
It follows that . Now we show that . Note that
Then we have for every . Then we can have . Hence, by Theorem 3.3, we obtain the desired result. □
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This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. ZXH2012K001).
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Zhang, C., He, S. A general iterative algorithm for an infinite family of nonexpansive operators in Hilbert spaces. Fixed Point Theory Appl 2013, 138 (2013). https://doi.org/10.1186/1687-1812-2013-138
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DOI: https://doi.org/10.1186/1687-1812-2013-138