Iterative methods for hierarchical common fixed point problems and variationalinequalities
© Sahu et al.; licensee Springer. 2013
Received: 10 June 2013
Accepted: 10 October 2013
Published: 19 November 2013
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.
Keywordshierarchical variational inequality metric projection mapping nonexpansive mapping sequence of nearly nonexpansive mappings
Variational inequality problems were initially studied by Stampacchia  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:
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:
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  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. motivated and inspired by the results of Marino and Xu , Yao et al. 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. and Sahu et al., 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. 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., Yao et al., Moudafi , Xu  and many other related works.
- (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
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.
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. for other properties of projection operators).
In what follows, we shall make use of the following lemmas and proposition.
Lemma 2.1 ()
Lemma 2.2 ()
- (a)The mapping is -strongly monotone, i.e.,
- (b)The mapping is monotone, i.e.,
Lemma 2.3 ()
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 ()
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.
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).
3 Main result
, and ;
there exists a constant such that ;
, where , and either or for each .
Proof It was proved in  that variational inequality problem (3.2) has the unique solution. Let. We now break the proof into the following steps.
Step 1. is bounded.
Hence is bounded. So, , , and are bounded.
Step 2. as .
Step 3. We claim for all .
Step 4. as for all .
for all .
Step 5. and .
Step 6. Set and for all . Then, for any , we can calculate .
Step 7. , where is a strong cluster point of the sequence.
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. 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. [, Theorem 3.5] as a corollary.
Again, we present the result of Yao et al. [, Theorem 3.2] as a corollary.
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.
It was shown in  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.
The authors would like to thank the editor and referees for useful comments andsuggestions.
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