- Open Access
Viscosity approximation methods for hierarchical optimization problems in spaces
© Liu and Chang; licensee Springer. 2013
- Received: 10 July 2013
- Accepted: 15 October 2013
- Published: 7 November 2013
This paper aims at investigating viscosity approximation methods for solving a system of variational inequalities in a space. Two algorithms are given. Under certain appropriate conditions, we prove that the iterative schemes converge strongly to the unique solution of the hierarchical optimization problem. The result presented in this paper mainly improves and extends the corresponding results of Shi and Chen (J. Appl. Math. 2012:421050, 2012, doi:10.1155/2012/421050), Wangkeeree and Preechasilp (J. Inequal. Appl. 2013:93, 2013, doi:10.1186/1029-242X-2013-93) and others.
- viscosity approximation method
- variational inequality
- hierarchical optimization problems
- common fixed point
The concept of variational inequalities plays an important role in various kinds of problems in pure and applied sciences (see, for example, [1–11]). Moreover, the rapid development and the prolific growth of the theory of variational inequalities have been made by many researchers.
where . It is proved that converges strongly as to such that , and converges strongly as to under certain appropriate conditions on , where is a metric projection from X onto C.
Furthermore, they also obtained that defined by (1.4) converges strongly as to under certain appropriate conditions imposed on .
(H3) either or .
We prove that iterative schemes (1.7) and (1.8) converge strongly to such that and , which is the unique solution of (1.6).
Let be a metric space. A geodesic path joining to (or, more briefly, a geodesic from x to y) is a map such that , , and for all . In particular, c is an isometry and . The image of c is called a geodesic segment joining x and y. When it is unique, this geodesic segment is denoted by . The space is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each . A subset is said to be convex if Y includes every geodesic segment joining any two of its points.
A geodesic triangle in a geodesic metric space consists of three points , , and in X (the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A comparison triangle for the geodesic triangle in is a triangle in the Euclidean plane such that for .
A geodesic space is said to be a space if all geodesic triangles satisfy the following comparison axiom.
From now on, we will use the notation for the unique point z satisfying (2.1).
We now collect some elementary facts about spaces which will be used in the proofs of our main results.
Let C be a nonempty subset of a complete space X. Recall that a self-mapping is a nonexpansion on C iff for all . A point is called a fixed point of T if . We denote by the set of all fixed points of T. A self-mapping is a contraction on C if there exists a constant such that . Banach’s contraction principle  guarantees that f has a unique fixed point when C is a nonempty closed convex subset of a complete metric space.
Fixed-point theory in spaces was first studied by Kirk (see [19, 21]). He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in spaces has been rapidly developed.
Berg and Nikolaev  introduced the concept of quasilinearization as follows.
It is easily seen that , and for all .
for all .
It is known [, Corollary 3] that a geodesically connected metric space is a space if and only if it satisfies the Cauchy-Schwarz inequality.
Recently, Dehghan and Rooin  presented a characterization of a metric projection in spaces as follows.
Lemma 2.2 Let C be a nonempty closed and convex subset of a complete space X, and . Then if and only if for all .
Lemma 2.3 
Every bounded sequence in X always has a △-convergent subsequence.
If C is a closed convex subset of X and is a nonexpansive mapping, then the conditions and imply and .
The following lemma shows a characterization of △-convergence.
Lemma 2.4 
Let X be a complete space, be a sequence in X, and . Then if and only if for all .
Lemma 2.5 
Then converges to zero as .
Lemma 2.6 
- (i)for each , one has(2.9)
for any and , letting for all , we have:
Now we are ready to give our main results in this paper.
for all . Then it is easy to verify that is a metric space, and is complete if and only if is complete.
Then is a contraction on .
Theorem 3.2 Let C be a closed convex subset of a complete space X, and let be two nonexpansive mappings such that and are nonempty. Let f, g be two contractions on C with coefficient . For each , let and be given by (1.7). Then and as such that , which is the unique solution of HOP (1.6).
Next we prove that is relatively compact as .
In fact, let be any subsequence such that as . Put and . Now we prove that contains a subsequence converging strongly to where , and it is a solution of HOP (1.6).
It follows from (3.3) that . Hence and .
Next we show that , which solves HOP (1.6).
That is, solves inequality (1.6).
Since , we have that , and so , . Hence the entire net converges to and converges to , which solves HOP (1.6). This completes the proof of Theorem 3.2. □
Theorem 3.3 Let C be a closed convex subset of a complete space X, and let be two nonexpansive mappings such that and are nonempty. Let f, g be two contractions on C with coefficient . Let and be the sequences defined by (1.8). If conditions (H1)-(H3) are satisfied, then and as , where , , which solves HOP (1.6).
for all . This implies that and are bounded, so are , , and .
and thus , .
Applying Lemma 2.6, we have . Hence and as . This completes the proof of Theorem 3.3. □
The authors would like to express their thanks to the referees for their helpful suggestions and comments. This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department (11ZA221) and the Scientific Research Fund of Science Technology Department of Sichuan Province 2011JYZ010.
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