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Iterative methods for hierarchical common fixed point problems and variationalinequalities
Fixed Point Theory and Applications volume 2013, Article number: 299 (2013)
Abstract
The purpose of this paper is to deal with the problem of finding hierarchically acommon fixed point of a sequence of nearly nonexpansive self-mappings defined ona closed convex subset of a real Hilbert space which is also a solution of someparticular variational inequality problem. We introduce two explicit iterativeschemes and establish strong convergence results for sequences generatediteratively by the explicit schemes under suitable conditions. Our strongconvergence results include the previous results as special cases, and can beviewed as an improvement and refinement of several corresponding known resultsfor hierarchical variational inequality problems.
MSC: 47J20, 49J40.
1 Introduction
Variational inequality problems were initially studied by Stampacchia [1] in 1964. Since then, many kinds of variational inequalities have beenextended and generalized in several directions using novel and innovative techniques(see [2–5] and the references therein).
The classical variational inequality problem or the Stampacchia variationalinequality problem in a Hilbert space is defined as follows:
Let C be a nonempty closed convex subset of a real Hilbert space H,and let be a nonlinear mapping. Then the classicalvariational inequality problem is a problem of finding such that
Problem (1.1) is denoted by . A point is a solution of if and only if is a fixed point of , where is a constant, I is the identity mappingfrom C into itself, and is the metric projection from H onto aclosed convex subset C of H. The set of solutions of (1.1) isdenoted by , that is,
The variational inequality is called a monotone variational inequality if theoperator T is a monotone operator. The problem of kind (1.1) is connectedwith the convex minimization problem, complementarity problem, saddle point problem,fixed point problem, Nash equilibrium problem, the problem of finding a point satisfying and so on, and it has several applications indifferent branches of natural sciences, social sciences, management and engineering(see [6–11] and the references therein). In this context, we discuss the variationalinequality problem over the set of fixed points of a mapping which is known as ahierarchical variational inequality problem or a hierarchical fixed point problem.
The hierarchical variational inequality problem over the set of fixed points of anonexpansive mapping is defined as follows:
Let C be a nonempty closed convex subset of a real Hilbert space H,and let be two nonexpansive mappings. Then the hierarchicalvariational inequality problem is given as follows:
Problem (1.2) is denoted by . The set of solutions of (1.2) is denoted by, that is,
It is easy to observe that is equivalent to the fixed point problem, that is, is a fixed point of the nonexpansive mapping. Of course, if , then the set of solutions of is .
Firstly, Moudafi and Maingé [12] introduced an implicit iterative algorithm to solve problem (1.2) andproved weak and strong convergence results, and after that, many iterative methodshave been developed for solving hierarchical problem (1.2) by several authors (see,e.g., [13–17]).
Very recently, Gu et al.[18] motivated and inspired by the results of Marino and Xu [15], Yao et al.[17] introduced and studied two iterative schemes for solving hierarchicalvariational inequality problem and proved the corresponding strong convergenceresults for the generated sequences in the context of a countable family ofnonexpansive mappings under suitable conditions on parameters.
In this paper, inspired by Gu et al.[18] and Sahu et al.[19], we introduce two explicit iterative schemes which generate sequences viaiterative algorithms. We prove that the generated sequences converge strongly to theunique solutions of particular variational inequality problems defined over the setof common fixed points of a sequence of nearly nonexpansive mappings. Our results,in one sense, extend the results of Gu et al.[18] to the sequence of nonexpansive mappings and, in another sense, to thesequence of nearly nonexpansive mappings, which is a wider class of sequence ofnonexpansive mappings. Our results also generalize the results of Cianciaruso etal.[13], Yao et al.[17], Moudafi [20], Xu [21] and many other related works.
2 Preliminaries
Let C be a nonempty subset of a real Hilbert space H with the innerproduct and the norm , respectively. A mapping is called
-
(1)
monotone if
-
(2)
η-strongly monotone if there exists a positive real number η such that
-
(3)
k-Lipschitzian if there exists a constant such that
-
(4)
ρ-contraction if there exists a constant such that
-
(5)
nonexpansive if
-
(6)
nearly nonexpansive [22, 23] with respect to a fixed sequence in with if
Throughout this paper, we denote by I the identity mapping of H.Also, we denote by → and ⇀ the strong convergence and weak convergence,respectively. The symbol ℕ stands for the set of all natural numbers and denotes the set of all weak subsequential limits of.
Let C be a nonempty closed convex subset of H. Then, for any, there exists a unique nearest point in C,denoted by , such that
The mapping is called the metric projection fromH onto C.
It is observed that is a nonexpansive and monotone mapping fromH onto C (see Agarwal et al.[22] for other properties of projection operators).
Let C be a nonempty subset of a real Hilbert space H, and let be two mappings. We denote by the collection of all bounded subsets of C.The deviation between and on [24], denoted by , is defined by
In what follows, we shall make use of the following lemmas and proposition.
Lemma 2.1 ([11])
Let C be a nonempty closed convex subset of a real Hilbert space H. Ifand, thenif and only if the following inequality holds:
Lemma 2.2 ([25])
Letbe a λ-contraction mapping andbe nonexpansive. Then the following hold:
-
(a)
The mapping is -strongly monotone, i.e.,
-
(b)
The mapping is monotone, i.e.,
Lemma 2.3 ([22])
Let T be a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. Thenis demiclosed at zero, i.e., ifis a sequence in C weakly converging to someand the sequencestrongly converges to 0, then.
Lemma 2.4 ([26])
Let and be the sequences of nonnegative real numbers such that
whereis a real number sequence inandis a real number sequence. Assume that the following conditionshold:
-
(i)
;
-
(ii)
and .
Then.
Let C be a nonempty closed convex subset of a real Hilbert space H and () such that. Letbe nonexpansive mappings withand let. Then T is nonexpansive from C into itself and.
Let C be a nonempty subset of a real Hilbert space H. Let be a sequence of mappings from C intoitself. We denote by the set of common fixed points of the sequence, that is,. Fix a sequence in with , and let be a sequence of mappings from C intoH. Then the sequence is called a sequence of nearly nonexpansivemappings[19] with respect to a sequence if
One can observe that the sequence of nonexpansive mappings is essentially a sequenceof nearly nonexpansive mappings.
We now introduce the following:
Let C be a nonempty closed convex subset of a Hilbert space H. Let be a sequence of nearly nonexpansive mappings fromC into itself with sequence such that . Let be a mapping such that for all with . Then is said to satisfy condition (G) iffor each sequence in C with and for all imply .
Proposition 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a sequence of nonexpansive mappings from C into itself. Thensatisfies condition (G).
Proposition 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a sequence of nearly nonexpansive mappings from C into itself with sequencesuch that. Then
for alland.
Proof Let and . Then
Since , we have
Therefore,
□
Proposition 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a sequence of nearly nonexpansive mappings from C into itself with sequence. Then
Proof Let . Then
□
3 Main result
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a λ-contraction, and letbe a sequence of nonexpansive mappings from C into itself. Let S be a nonexpansive mapping from C into itself such thatfor all. Letbe a sequence of uniformly continuous nearly nonexpansive mappings from C into itself with sequencesuch that. Let T be a mapping from C into itself defined byfor all. Suppose that. For arbitrary, consider the sequencegenerated by the following iterative process:
for all, where, is a strictly decreasing sequence inandis a sequence insatisfying the conditions:
-
(i)
, ;
-
(ii)
, and ;
-
(iii)
;
-
(iv)
there exists a constant such that ;
-
(v)
, where , and either or for each .
Then the sequenceconverges strongly to a point, which is the unique solution of the followingvariational inequality:
Proof It was proved in [17] that variational inequality problem (3.2) has the unique solution. Let. We now break the proof into the following steps.
Step 1. is bounded.
From (3.1), we have
It follows that
Note that and , so there exists a constant such that
Thus, we have
Hence is bounded. So, , , and are bounded.
Step 2. as .
Set for all . Set . From (3.1) we have
Set . Now, from (3.1) we have
Now, using (3.4) in (3.3), we obtain that
Thus, by using conditions (i), (v), , and applying Lemma 2.4, we conclude that
Step 3. We claim for all .
Since for all and , we get
Noticing and fixing , from (3.1) we have
Hence,
where R is a positive constant such that for all and
Set . From Proposition 2.2, using (3.7), we obtainthat
Using (3.6), condition (i), and , we have
Since for all and is strictly decreasing, we have
Step 4. as for all .
Noticing that and using condition (i), we have as . Therefore, we obtain that
So that for all , we have
Since each is uniformly continuous, from (3.8) and (3.9), wehave
for all .
Step 5. and .
From (3.5) we obtain that
We observe that
Set
From (3.10) we obtain that
Using conditions (i), (iii), (v) and applying Lemma 2.4, we have
Step 6. Set and for all . Then, for any , we can calculate .
From (3.1) we have
which gives that
Then we conclude that
Noticing that . For any , we have
By using Lemma 2.2, we obtain that
Also, from Lemma 2.1 we get that
Substituting (3.12), (3.13), (3.14) and (3.15) into (3.11), we have
Step 7. , where is a strong cluster point of the sequence.
From (3.16) we observe that
Noticing that , , and for all , as . It is easy to see that a weak cluster point of is also a strong cluster point. Note that thesequence is bounded, then there exists a subsequence of which converges to a point . We also have, for all , as . By using condition (G), we get that. Thus, for all , we obtain that
Now, letting , we have
Now, since variational inequality problem (3.2) has the unique solution, then we getthat . Note that every weak cluster point of the sequence is also a strong cluster point. Then we have. □
Recently, Marino et al.[28] used a different approach to obtain the convergence of a more generaliterative method that involves an equilibrium problem. We now present the result ofGu et al. [[18], Theorem 3.5] as a corollary.
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a λ-contraction. Letbe a nonexpansive mapping, and letbe a countable family of nonexpansive mappings from C into itself such that. For arbitrary, consider the sequencegenerated by the following iterative process:
for all, where, is a strictly decreasing sequence inandis a sequence insatisfying conditions (i)-(iv) of Theorem 3.1. Then thesequenceconverges strongly to a point, which is the unique solution of the followingvariational inequality:
Again, we present the result of Yao et al. [[17], Theorem 3.2] as a corollary.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a λ-contraction. Letbe a nonexpansive mapping, and letbe a nonexpansive mapping such that. For arbitrary, consider the sequencegenerated by the following iterative process:
for all, whereandare two sequences insatisfying conditions (i)-(iv) of Theorem 3.1. Then thesequenceconverges strongly to a point, which is the unique solution of the followingvariational inequality:
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a λ-contraction, and letbe a sequence of nonexpansive mappings from C into itself. Let S be a nonexpansive mapping from C into itself such thatfor all. Letsuch that. Letbe nonexpansive mappings such that, and assume that. For arbitrary, consider the sequencegenerated by the following iterative process:
for all, whereandare two sequences insatisfying conditions (i)-(iv) of Theorem 3.1. Then thesequenceconverges strongly to a point, which is the unique solution of the followingvariational inequality:
Proof The proof follows from Lemma 2.5 andCorollary 3.2. □
4 Numerical example
We present an example to show the effectiveness and convergence of the sequencegenerated by the considered iterative scheme.
Example 4.1 Let and . Let T be a self-mapping defined by for all . Let be a contraction mapping defined by for all , and let be a sequence of nonexpansive mappings fromC into itself defined by such that for all , where S is a nonexpansive mapping definedby for all . Define sequences and in by . Without loss of generality, we may assume that for all . For each , define by
It was shown in [19] that is a sequence of nearly nonexpansive mappings fromC into itself such that and for all , where T is a nonexpansive mapping.
One can observe that all the assumptions of Theorem 3.1 are satisfied, and thesequence generated by (3.1) converges to a unique solution of variational inequality (3.2) over.
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Sahu, D., Kang, S.M. & Sagar, V. Iterative methods for hierarchical common fixed point problems and variationalinequalities. Fixed Point Theory Appl 2013, 299 (2013). https://doi.org/10.1186/1687-1812-2013-299
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DOI: https://doi.org/10.1186/1687-1812-2013-299