Hybrid methods for common solutions in Hilbert spaces with applications
© Sun; licensee Springer. 2014
Received: 26 February 2014
Accepted: 29 April 2014
Published: 13 May 2014
In this paper, hybrid methods are investigated for treating common solutions of nonlinear problems. A strong convergence theorem is established in the framework of real Hilbert spaces.
1 Introduction and preliminaries
Common solutions to variational inclusion, equilibrium and fixed point problems have been recently extensively investigated based on iterative methods; see [1–33] and the references therein. The motivation for this subject is mainly to its possible applications to mathematical modeling of concrete complex problems, which use more than one constraint. The aim of this paper is to investigate a common solution of variational inclusion, equilibrium and fixed point problems. The organization of this paper is as follows. In Section 1, we provide some necessary preliminaries. In Section 2, a hybrid method is introduced and analyzed. Strong convergence theorems are established in the framework of Hilbert spaces. In Section 3, applications of the main results are discussed.
In what follows, we always assume that H is a real Hilbert space with the inner product and the norm . Let C be a nonempty, closed, and convex subset of H and let be the metric projection from H onto C. Let be a mapping. stands for the fixed point set of S; that is, .
If , then S is said to be nonexpansive. Let be a mapping. If C is nonempty closed and convex, then the fixed point set of S is nonempty.
For such a case, A is also said to be α-inverse-strongly monotone.
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 denote the domain of M. It is well known that is firmly nonexpansive, and , where , and .
In this paper, the set of such an is denoted by .
To study the equilibrium problems (1.1), 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 semicontinuous.
In order to prove our main results, we also need the following lemmas.
Lemma 1.1 
Lemma 1.2 
- (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 .
Lemma 1.3 
is nonexpansive and , for each ;
for each and for each positive integer k, the limit exists;
- (3)the mapping defined by(1.3)
is a nonexpansive mapping satisfying and it is called the W-mapping generated by and .
Lemma 1.4 
Throughout this paper, we always assume that , .
Lemma 1.5 
Let be a mapping and let be a maximal monotone operator. Then .
Lemma 1.6 
Lemma 1.7 
Then D is maximal monotone and if and only if .
2 Main results
, , ;
Then the sequence converges strongly to , where .
Since is bounded, there exists a subsequence of which converges weakly to w. Without loss of generality, we may assume that . Therefore, we see that . We also have .
In view of Lemma 1.4, we obtain that . This implies from (2.22) that . This is a contradiction. Thus, we have .
This implies that . This implies that , that is, .
This implies that . This implies that , that is, .
Using Lemma 1.1, we find that . This completes the proof. □
In this section, we consider some applications of the main results.
In this paper, we use to denote the solution set of the inequality. It is well known that is a solution of the inequality iff x is a fixed point of the mapping , where is a constant, I stands for the identity mapping. If A is α-inverse-strongly monotone and , then the mapping is nonexpansive. It follows that is closed and convex.
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 semicontinuous convex function on H, we see that the subdifferential of is a maximal monotone operator. It is clearly that , , and .
, , ;
Then the sequence converges strongly to .
Putting , where is a k-strict pseudo-contraction, we find that A is -inverse-strongly monotone.
Next, we consider fixed points of strict pseudo-contractions.
, , ;
Then the sequence converges strongly to .
Proof Taking , wee see that is a -strict pseudo-contraction with and for . In view of Theorem 3.1, we find the desired conclusion immediately. □
The author is very grateful to the reviewers for useful suggestions which improved the contents of this paper.
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