Strong convergence of a Halpern-type algorithm for common solutions of fixed point and equilibrium problems
© Zhang; licensee Springer 2014
Received: 5 May 2014
Accepted: 16 July 2014
Published: 21 August 2014
In this article, fixed points of nonexpansive mappings and equilibrium problems based on a Halpern-type algorithm are investigated. Strong convergence theorems for common solutions of the two problems are obtained in the framework of real Hilbert spaces.
Keywordsnonexpansive mapping equilibrium problem fixed point variational inequality
The study of equilibrium problems is an important branch of optimization theory and nonlinear functional analysis. Numerous problems in physics, optimization, transportation, signal processing, and economics are reduced to find a solution to equilibrium problems, which cover fixed point problems, variational inequalities, saddle problems, inclusion problems, and so on. A closely related subject of current interest is the problem of finding common elements in the fixed point set of nonlinear operators and in the solution set of monotone variational inequalities; see [1–15] and the references therein. The motivation for this subject is mainly due to its possible applications to mathematical modeling of concrete complex problems. The aim of this paper is to investigate a common element problem based on a Halpern-type algorithm. Strong convergence of the algorithm is obtained in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a Halpern-type algorithm is proposed and analyzed. Strong convergence theorems for common solutions of two problems are established in the framework of Hilbert spaces. In Section 4, applications of the main results are provided.
Let H be a real Hilbert space with inner product and norm . Let C be a nonempty, closed, and convex subset of H and let be the metric projection from H onto C.
It is well known that the fixed point set of nonexpansive mappings is nonempty provided that the subset C is bounded, convex, and closed.
For such a case, A is also said to be α-inverse-strongly monotone.
In this paper, we use to denote the solution set of (2.1). It is well known that is a solution of the variational inequality (2.1) iff x is a solution of the fixed point equation , where is a constant.
Recall that a set-valued mapping is said to be monotone iff, for all , and imply . M is maximal iff the graph of R is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if, for any , , for all implies .
For a maximal monotone operator M on H, and , we may define the single-valued resolvent , where denotes the domain of M. It is known that is firmly nonexpansive, and , where , and .
In this paper, we use to denote the solution set of the generalized equilibrium problem (2.2).
If , then problem (2.2) is reduced to the classical variational inequality (2.1).
- (II)If , the zero mapping, then problem (2.2) is reduced to the following equilibrium problem:(2.3)
In this paper, we use to denote the solution set of the equilibrium problem (2.3).
for all ;
F is monotone, i.e., for all ;
- (A3)for each ,
for each , is convex and weakly lower semi-continuous.
Recently, many authors have studied fixed point problems of nonexpansive mappings and solution problems of the equilibrium problems (2.2) and (2.3); for more details, see [16–25] and the references therein. In this paper, motivated and inspired by the research going on in this direction, we consider common element problems based on a mean iterative process. Strong convergence of the iterative process is obtained in the framework of real Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Hao , Qin et al. , Chang et al. .
In order to prove our main results, we need the following lemmas.
Lemma 2.1 
Lemma 2.2 
- (2)is firmly nonexpansive, i.e., for any ,
is closed and convex.
Such a mapping is nonexpansive from C to C and it is called a W-mapping generated by and ; see  and the references therein.
Lemma 2.3 
is nonexpansive and , for each ;
for each and for each positive integer k, the limit exists;
- (3)the mapping defined by(2.5)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 2.4 
Let be a mapping and let be a maximal monotone operator. Then , where r is some positive real number.
Lemma 2.5 
Throughout this paper, we always assume that , .
Lemma 2.6 
Then W is maximal monotone and if and only if .
Lemma 2.7 
3 Main results
Then the sequence converges strongly to .
Using Lemma 2.1, we find that . This completes the proof. □
for all . Note that the class of k-strict pseudo-contractions strictly includes the class of nonexpansive mappings. Put , where is a k-strict pseudo-contraction. Then A is -inverse-strongly monotone. Now, we are in a position to state a results on fixed points of strict pseudo-contractions.
Then the sequence converges strongly to .
Proof Taking , wee see that is a α-strict pseudo-contraction with and . Using Theorem 3.1, we find the desired conclusion immediately. □
From Rockafellar , we know that ∂g is maximal monotone. It is not hard to verify that if and only if .
Since is a proper lower semi-continuous convex function on H, we see that the subdifferential of is a maximal monotone operator. It is clear that , . Notice that . Now, we are in a position to state the result on variational inequalities.
Then the sequence converges strongly to .
The author is grateful to Prof. Tian Cheng and the reviewers’ suggestions which improved the contents of the article.
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