A hybrid approximation algorithm for finding common solutions of equilibrium problems, a finite family of variational inclusions, and fixed point problems in Hilbert spaces
© Deng and Chen; licensee Springer. 2013
Received: 14 April 2012
Accepted: 13 March 2013
Published: 10 April 2013
In this paper, we introduce an iterative method for finding a common element of the set of fixed points of nonexpansive mappings, the set of solutions of a finite family of variational inclusions with set-valued maximal monotone mappings and inverse strongly monotone mappings, and the set of solutions of an equilibrium problem in Hilbert spaces. Furthermore, using our new iterative scheme, under suitable conditions, we prove some strong convergence theorems for approximating these common elements. The results presented in the paper improve and extend many recent important results.
Keywordsvariational inclusions inverse strongly monotone maximal monotone mapping fixed point equilibrium problem
We denote by the set of fixed points of T.
where h is a potential function for γf.
where θ is the zero vector in H. The set of solutions of problem (1.5) is denoted by . The formulation (1.5) extends this formalism to a finite family of variational inclusions covering, in particular, various forms of feasibility problems (see, e.g., ).
for all , where , , and ; B is a strongly positive bounded linear operator on H and is a sequence of nonexpansive mappings on H. They proved that under certain appropriate conditions imposed on and , the sequences , , and generated by (1.6) converge strongly to , where .
for all , where , , , and , f is an L-Lipschitz mapping on H, B is a k-Lipschitzian and η-strongly monotone operator on H with coefficients and , and is a sequence of nonexpansive mappings on H. Under suitable conditions, some strong convergence theorems for approximating to these common elements are proved. Our results extend and improve some corresponding results in [10, 14] and the references therein.
This section collects some lemmas which are used in the proofs of the main results in the next section.
For solving the equilibrium problem for a bifunction , let us assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, that is, for all ;
(A4) for each , is convex and lower semicontinuous.
Lemma 2.1 
- (1)is single-valued;
- (2)is firmly nonexpansive, that is, for any ,
- (4)is closed and convex.
By the proof of Lemma 5 in , we have the following lemma.
Lemma 2.3 
That is, is strongly monotone with coefficient .
Lemma 2.4 
- (ii)or .
- (i)Monotone if
- (ii)Strongly monotone if there exists a constant such that
For such a case, A is said to be α-strongly-monotone.
- (iii)Inverse-strongly monotone if there exists a constant such that
For such a case, A is said to be α-inverse-strongly-monotone.
- (iv)k-Lipschitz continuous if there exists a constant such that
Let I be the identity mapping on H. It is well known that if is α-inverse-strongly monotone, then A is a -Lipschitz continuous and monotone mapping. In addition, if , then is a nonexpansive mapping.
A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for , for every implies .
where λ is a positive number. It is worth mentioning that the resolvent operator is single-valued, nonexpansive, and 1-inverse-strongly monotone (see, for example, ) and that a solution of problem (1.3) is a fixed point of the operator for all ; see, for instance, . Furthermore, a solution of a finite family of variational inclusion problems (1.5) is a common fixed point of , , .
Lemma 2.6 
Let be a maximal monotone mapping, and let be a Lipschitz-continuous mapping. Then the mapping is a maximal monotone mapping.
Lemma 2.8 (The resolvent identity)
Lemma 2.9 
Let H be a Hilbert space. Let , be -inverse-strongly monotone mappings, let , be maximal monotone mappings, and let be a bounded sequence in H. Assume that , , satisfy the following:
Lemma 2.10 
Let H be a real Hilbert space and B be a k-Lipschitzian and η-strongly monotone operator with , . Let and . Then for , is a contraction with a constant .
3 Main results
Let H be a real Hilbert space and T be a nonexpansive mapping on H. Assume that the set is nonempty, that is, . Since is closed convex, the nearest point projection from H onto is well defined. Recall also that f is an L-Lipschitz mapping on H with coefficient . Let B is a k-Lipschitzian and η-strongly monotone operator on H with coefficients and .
for all , where , , satisfy (H1)-(H2), and satisfy
Proof Using the definition of in Lemma 2.9, we have . We divide the proof into several steps.
Step 1. The sequence is bounded.
Hence is bounded and therefore , , , and are also bounded.
Step 2. We show that .
Since is bounded, it follows that . Hence, using conditions (C1)-(C5), (3.9) and Lemma 2.4, we have as .
Since and , we have (3.10).
Step 4. We prove .
Step 5. We show .
and hence . From (A3), we have for all . Therefore, .
This is a contradiction. Hence .
Since , is maximal monotone, this implies that , , i.e., . So, we obtain the result.
where is a unique solution of the variation (3.3).
Step 7. We prove that .
where and . It is easily verified that , , and . Hence, by Lemma 2.4, the sequence converges strongly to ω. Furthermore, from (3.17) and (3.19), we obtain that the sequences and converge strongly to ω. □
Let and in Theorem 3.1; we obtain the following corollary.
for all , where , , satisfy (H1)-(H2), and satisfy:
This work is supported in part by National Natural Science Foundation of China (71272148), the Ph.D. Programs Foundation of Ministry of Education of China (20120032110039) and China Postdoctoral Science Foundation (Grant No. 20100470783).
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