- Open Access
Convergence of a regularization algorithm for nonexpansive and monotone operators in Hilbert spaces
© Yuan and Zhang; licensee Springer. 2014
Received: 28 May 2014
Accepted: 15 August 2014
Published: 2 September 2014
Variational inequality, fixed point and generalized equilibrium problems are investigated via a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solution of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by many authors.
1 Introduction and preliminaries
In this paper, we always assume that H is a real Hilbert space with inner product and induced norm for . Let C be a nonempty, closed, and convex subset of H.
For such a case, A is also said to be α-inverse-strongly monotone.
In this paper, we always use to denote the solution set of (1.1) and use denote the metric projection from H onto C. It is well known that is a solution of (1.1) iff x is a fixed point of the mapping , where is a constant, I stands for the identity mapping. If A is strongly monotone and Lipschitz continuous, the existence and uniqueness of solutions of equilibrium (1.1) is guaranteed by the Banach contraction principle.
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 .
Let be a mapping. stands for the fixed point set of S; that is, .
For such a case, S is also said to be α-contractive. We know that the mapping enjoys a unique fixed point and Picard’s algorithm can be employed to approximate its unique fixed point.
If C is a closed, bounded and convex subset of H, then is not empty; see .
Such a mapping is nonexpansive from C to C and it is called a W-mapping generated by and ; see  and the references therein.
In this paper, the set of such an is denoted by .
If , then the problem (1.3) is reduced to the classical variational inequality (1.1).
To study equilibrium problems (1.3) and (1.4), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A4) for each , is convex and weakly lower semi-continuous.
Many important problems have reformulations which require finding solutions of equilibriums (1.3) and (1.4), for instance, image recovery, inverse problems, network allocation, transportation problems and optimization problems; see [3–11] and the references therein. For solving solutions of equilibriums (1.3) and (1.4), regularization methods recently have been extensively studied; see [11–28] and the references therein.
In this paper, motivated and inspired by the research going on in this direction, we study the variational inequality (1.1), and the fixed point and equilibrium problem (1.3) based on a regularization algorithm. It is proved that the sequence generated in the regularization algorithm converges strongly to a common solutions of the three problems in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results in Chang et al. , Takahashi and Takahashi  and Hao .
The following lemmas play an important role in our paper.
Lemma 1.1 
- (2)is firmly nonexpansive, i.e., for any ,
is closed and convex.
Lemma 1.2 
Lemma 1.3 
is nonexpansive and , for each ;
for each and for each positive integer k, the limit exists.
- (3)the mapping defined by(1.5)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 1.4 
Lemma 1.5 
Throughout this paper, we always assume that , .
Lemma 1.6 
Then D is maximal monotone and if and only if .
2 Main results
In view of Lemma 1.5, we obtain from (2.15) . It follows that . Thus one derives a contradiction. Thus, we have .
Since B is Lipschitz continuous, we see that . Notice that D is maximal monotone and hence . This shows that . In the same way, we find that .
It follows from (A3) that . This proves that .
From the restriction (d), we obtain from Lemma 1.2 . This completes the proof. □
Taking with and choosing a sequence of real numbers with , we obtain the desired result by Theorem 2.1. □
that is, . This completes the proof. □
The authors are grateful to the referees for useful suggestions which improved the contents of the paper.
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