- Open Access
Hybrid algorithms for equilibrium and common fixed point problems with applications
© Zhang and Wu; licensee Springer. 2014
- Received: 13 March 2014
- Accepted: 25 May 2014
- Published: 3 June 2014
In this paper, hybrid algorithms are investigated for equilibrium and common fixed point problems. Strong convergence of the algorithms is obtained in the framework of reflexive Banach spaces.
- asymptotically quasi-ϕ-nonexpansive mapping
- quasi-ϕ-nonexpansive mapping
- equilibrium problem
- fixed point
Iterative algorithms have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity and optimization; see [1–11] and the references therein. The computation of solutions of nonlinear operator equations (inequalities) is important in the study of many real-world problems. Recently, the study of the convergence of various iterative algorithms for solving various nonlinear mathematical models has formed a major part of numerical mathematics. Among these iterative algorithms, the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm are popular and much discussed. It is well known that both the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm only have weak convergence even for nonexpansive mappings in infinite-dimensional Hilbert spaces. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite-dimensional spaces. In such problems, strong convergence is often much more desirable than weak convergence. To improve the weak convergence of the Krasnoselski-Mann iterative algorithm and the Ishikawa iterative algorithm, so-called projection algorithms have been investigated in different frameworks of spaces; see [12–26] and the references therein.
The aim of this article is to investigate solutions of equilibrium and common fixed point problems in Hilbert spaces. In Section 2, we provide some necessary preliminaries. In Section 3, a projection algorithm is investigated. Strong convergence of the algorithm is obtained in a uniformly smooth and strictly convex Banach space which also has the Kadec-Klee property. In Section 4, applications are provided to support the main results of this paper.
Let E be a real Banach space with the dual . Recall that the normalized duality mapping J from E to is defined by , where denotes the generalized duality pairing. Let be the unit sphere of E. E is said to be smooth iff exists for each . It is also said to be uniformly smooth iff the above limit is attained uniformly for . E is said to be strictly convex iff for all with and . It is said to be uniformly convex iff for any two sequences and in E such that and . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that if E is (uniformly) smooth if and only if is (uniformly) convex.
In what follows, we use ⇀ and → to stand for the weak and strong convergence, respectively. Recall that a space E has the Kadec-Klee property iff for any sequence , with , and , we have as . It is known that if E is a uniformly convex Banach spaces, then E has the Kadec-Klee property.
Let F be a bifunction from to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find such that , . We use to denote the solution set of the equilibrium problem. Numerous problems in physics, optimization and economics reduce to finding a solution of the equilibrium problem.
Next, we give the following assumptions:
(A1) , ;
(A2) F is monotone, i.e., , ;
(A3) , ;
(A4) for each , is convex and weakly lower semi-continuous.
where is a sequence such that as .
Remark 2.1 The class of relatively asymptotically nonexpansive mappings, which is an extension of the class of relatively nonexpansive mappings, was first introduced in .
Remark 2.2 The class of asymptotically quasi-ϕ-nonexpansive mappings [14, 29] which is an extension of the class of quasi-ϕ-nonexpansive mappings . The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive ones do not require the restriction .
In order to prove our main results, we need the following lemmas.
Lemma 2.3 
for all and such that .
Lemma 2.4 
Lemma 2.5 
Lemma 2.6 
Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and let C be a nonempty, closed, and convex subset of E. Let be an asymptotically quasi-ϕ-nonexpansive mapping. Then is closed and convex.
is a single-valued firmly nonexpansive-type mapping, i.e., for all , ;
is closed and convex;
where is a real number sequence in for every , is a real number sequence in , where r is some positive real number and . Assume that and for every . Then the sequence converges strongly to , where is the generalized projection from E onto .
Proof Using Lemma 2.6 and Lemma 2.7, we find that is closed and convex so that is well defined. By induction, we easily find that the sets are convex and closed.
which yields , that is, .
This yields . Letting , we find from the condition (A3) that , . This implies that for every .
This yields . Since the space E has the Kadec-Klee property, we obtain . Since and T is asymptotically regular, we find that . That is, as . It follows from the closedness of that for every .
Letting , we arrive at , . In view of Lemma 2.5, we find that . This completes the proof. □
If T is quasi-ϕ-nonexpansive, we have the following result.
where is a real number sequence in for every , is a real number sequence in , where r is some positive real number. Assume that and for every . Then the sequence converges strongly to , where is the generalized projection from E onto .
Remark 3.4 Corollary 3.3 mainly improves the corresponding results in Qin et al. . Indeed, the space is extended from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space; the mapping is extended from a pari of mappings to infinite many mappings. We also remark here that the framework of the space in this paper can be applicable to , .
Finally for a single mapping and bifunction, we have the following result.
where is a real number sequence in , is a real number sequence in , where r is some positive real number and . Assume . Then the sequence converges strongly to , where is the generalized projection from E onto .
First, we give the corresponding results in the framework of Hilbert spaces.
where is a real number sequence in for every , is a real number sequence in , where r is some positive real number and . Assume that and for every . Then the sequence converges strongly to , where is the metric projection from E onto .
Proof Note that , , the identity mapping, and the generalized projection is reduced to the metric projection. In the framework of Hilbert spaces, the class of asymptotically quasi-ϕ-nonexpansive mappings is reduced to the class of asymptotically quasi-nonexpansive mappings. Using Theorem 3.1, we easily find the desired conclusion. □
Next, we give a results on common solutions of a family of equilibrium problems.
where is a real number sequence in , where r is some positive real number. Then the sequence converges strongly to , where is the metric projection from E onto .
If , then we find from Theorem 4.1 the following result.
where is a real number sequence in for every , and . Assume that and for every . Then the sequence converges strongly to , where is the metric projection from E onto .
Remark 4.4 Comparing with Theorem 2.2 in Kim and Xu , we have the following: (1) one mapping is extended to an infinite family of mappings; (2) the set is relaxed; (3) the mapping is extended from asymptotically nonexpansive mappings to asymptotically quasi-nonexpansive mappings, which may not be continuous.
From Corollary 4.3, the following result is not hard to derive.
where is a real number sequence in such that . Then the sequence converges strongly to , where is the metric projection from E onto .
Remark 4.6 Corollary 4.5 is a shrinking version of the corresponding results in Nakajo and Takahashi . It deserves mentioning that the mapping in our result is quasi-nonexpansive. The restriction of the demiclosed principal is relaxed.
This paper is dedicated to Professor Chang with respect and admiration. The authors are grateful to the editor and the three anonymous reviewers for useful suggestions which improved the contents of the article.
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