Abstract
In this paper, we introduce a new explicit iterative algorithm for finding a solution for a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the variational inequality. Our result improves and extends the corresponding results announced by many others. At the end of the paper, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings.
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1 Introduction
Let H be a real Hilbert space with inner product and norm . Throughout this paper, we always assume that T is a nonexpansive operator on H. The fixed point set of T is denoted by , i.e., . The typical problem is to minimize a quadratic function on a real Hilbert space H:
where C is a nonempty closed convex subset of H, u is a given point in H and A is a strongly positive bounded linear operator on H.
In 2003, Xu [1] introduced the following iterative scheme:
where u is some point of H and is a sequence in . He proved that the sequence converges strongly to the unique solution of the minimization problem (1.1) with .
In 2006, Marino and Xu [2] considered the viscosity method on the iterative scheme (1.2), and they gave the following general iterative method:
where f is a contraction on H. They proved the above sequence 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 ).
In 2001, Yamada [3] considered the following hybrid iterative method:
where F is L-Lipschitzian continuous and η-strongly monotone operator with , and . Under some appropriate conditions, the sequence generated by (1.4) converges strongly to the unique solution of the variational inequality
Combining (1.3) and (1.4), Tian [4] considered the following general viscosity type iterative method:
Improving and extending the corresponding results given by Marino et al., he proved that the sequence generated by (1.5) converges strongly to the unique solution of the variational inequality
In [5], Tian generalized the iterative method (1.5) replacing the contraction operator f with a Lipschitzian continuous operator V to solve the following variational inequality:
On the other hand, let be a finite family of nonexpansive self-mappings of H. Assume . In [1], Xu also defined the following sequence :
where and the mod function takes values in . He found that the sequence generated by (1.7) converges strongly to the unique solution of the minimization problem (1.1) with under suitable conditions on and the following additional condition on :
In fact, there are many nonexpansive mappings which do not satisfy (1.8).
In 1999, Atsushiba and Takahashi [6] defined the -mappings generated by and as follows:
From [[6], Lemma 3.1], we know that .
In 2006, Yao [7] introduced the following iterative method:
Without the condition (1.8), he proved that the sequence generated by (1.9) converges strongly to the unique solution of the following variational inequality:
which is the optimality condition for the minimization problem
where and h is a potential function for γf (i.e., ).
Shang et al. [8] introduced the following scheme:
Under certain appropriate conditions, without (1.8), they proved that defined by (1.12) converges strongly to the unique solution of (1.10) which is also the optimality condition for (1.11).
Recently, combining the Krasnoselskii-Mann type algorithm and the steepest-descent method, Buong and Duong [9] introduced a new explicit iterative algorithm:
where for , , and F is an L-Lipschitz continuous and η-strongly monotone mapping. Under some appropriate assumptions, they proved that the sequence converges strongly to the unique solution of the following variational inequality:
Very recently, Zhou and Wang [10] proposed a simpler iterative algorithm than the iterative algorithm (1.13) given by Buong and Duong:
They proved that the sequence defined by (1.15) converges strongly to the unique solution of the variational inequality (1.14) in a faster rate of convergence.
Motivated and inspired by the results of Zhou et al., in this paper, we consider a new iterative algorithm to solve the class of variational inequalities (1.6). The iterative algorithm improves and extends the results of Yao et al., and the corresponding results announced by many others. At the end of this paper, we extend our iterative algorithm to the more broad family of λ-strictly pseudo-contractive mappings.
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. 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 results, we collect some necessary conceptions and lemmas in this section.
Definition 2.1 A mapping is said to be an averaged mapping if there exists some number such that
where is the identity mapping and is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged.
Lemma 2.1 ([11])
-
(i)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then both and are α-averaged, where .
-
(ii)
If the mappings are averaged and have a common fixed point, then
In particular, if , we have .
Lemma 2.2 ([12])
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 .
Lemma 2.3 ([5])
Assume V is a contraction on a Hilbert space H with coefficient , and is an L-Lipschitzian continuous and η-strongly monotone operator with , . Then, for , is strongly monotone with coefficient .
Lemma 2.4 ([13])
Let H be a Hilbert space, C a closed convex subset of H, and a nonexpansive mapping with . If is a sequence in C weakly converging to and converges strongly to , then . In particular, if , then .
Lemma 2.5 ([14])
Let and be bounded sequences in a Banach space X and be a sequence in which satisfies the following condition:
Suppose for all integers and
Then .
Lemma 2.6 ([1])
Assume 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 2.7 ([15])
Assume S is a λ-strictly pseudo-contractive mapping on a Hilbert space H. Define a mapping T by for all and . Then T is a nonexpansive mapping such that .
3 Main results
Now we state and prove our main results in this paper.
Theorem 3.1 Let be N nonexpansive mappings of a real Hilbert space H such that , F be an L-Lipschitzian continuous and η-strongly monotone operator on H with and , V be an α-Lipschitzian on H with . Suppose and . Define a sequence as follows:
where with and for . Suppose and for some . If the following conditions are satisfied:
-
(i)
;
-
(ii)
;
-
(iii)
for .
Then the sequence converges strongly to the unique solution of the variational inequality:
Equivalently, we have .
Proof Since our methods easily deduce the general case, we prove Theorem 3.1 for .
First, we show is bounded. In fact, for some point , by (3.1) we have
Therefore, is bounded. Hence we also see that , , and are all bounded. From (3.1), it follows that
We next show that . Noting that and are -averaged and -averaged, respectively, by Lemma 2.1, we find that is -averaged for every k, where . Set and . It is easy to deduce that for all k and
Since for every k, is -averaged, we can find a family of nonexpansive mappings on H such that
Substituting (3.4) into (3.1) yields
Define a sequence by , so
Now, we claim that
To this end, we observe that
and
where M is a fixed constant satisfying
Note that
Since for , and and are bounded, we easily obtain
Similarly,
from which it follows that
Using (3.4), (3.9), and (3.10), from (3.8) we have
Since and , combining (3.7) and (3.11) we get
By Lemma 2.5, we conclude that , which implies that by (3.6). Thus from (3.3), it is true that
From [[8], Theorem 3.2], we know that the solution of the variational inequality (3.2) is unique. We use to denote the unique solution of (3.2). Since is bounded, there exists a subsequence of such that as and
Since is bounded for , we can assume that as , where for . Define (). Then we have for . Note that
Hence, we deduce that
where D is an arbitrary bounded subset of H.
Since and is -averaged for , by Lemma 2.1, we know that . Combining (3.12) and (3.13), we obtain
where is a bounded subset including and is a bounded subset including . Hence . From Lemma 2.4, we have . It follows that
Finally, we show that as . From (3.1), we have
where is a constant satisfying
Consequently, according to the conditions (i) and (ii), (3.14), and Lemma 2.6, we conclude that as . This completes the proof. □
4 An extension of our result
In this section, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings. Now let us recall that a mapping is said to be λ-strictly pseudo-contractive if there exists a constant such that
Let be a family of -strictly pseudo-contractive self-mappings of H with . For , define
where . By virtue of Lemma 2.7, we know that is a family of nonexpansive mappings. Thus we extend Theorem 3.1 to the family of -strictly pseudo-contractions.
Theorem 4.1 Let H be a real Hilbert space, be an L-Lipschitizian continuous and η-strongly monotone operator on H with and , V be an α-Lipschitzian continuous on H with . Let be N -strictly pseudo-contractive mappings on H such that . Suppose , with , , for some and for . If the conditions (i)-(iii) of Theorem 3.1 are satisfied, the sequence defined by (3.1) with replaced by (4.1), converges strongly to the unique solution of the following variational inequality:
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Acknowledgements
This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. 3122013k004).
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Zhang, C., Yang, C. A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings. Fixed Point Theory Appl 2014, 60 (2014). https://doi.org/10.1186/1687-1812-2014-60
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DOI: https://doi.org/10.1186/1687-1812-2014-60