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Iterative Algorithms for Mixed Equilibrium Problems, System of Quasi-Variational Inclusion, and Fixed Point Problem in Hilbert Spaces
We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of Peng and Yao, 2009, Qin et al. 2010 and many authors.
Throughout this paper, we assume that is a real Hilbert space with inner product and norm denoted by and , respectively. Let be a nonempty closed convex subset of . A mapping is called nonexpansive if , . They use to denote the set of fixed points of ; that is, . It is assumed throughout the paper that is a nonexpansive mapping such that . Recall that a self-mapping is a contraction on if there exists a constant , and such that .
Let be a proper extended real-valued function and let be a bifunction of into , where is the set of real numbers. Ceng and Yao  considered the following mixed equilibrium problem for finding such that The set of solutions of (1) is denoted by . We see that is a solution of problem (1) which implies that . If , then the mixed equilibrium problem (1) becomes the following equilibrium problem for finding such that The set of solutions of (2) is denoted by . The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (2). Some methods have been proposed to solve the equilibrium problem (see [2–14]).
Let be a mapping. The variational inequality problem, denoted by , is for finding such that for all . The variational inequality problem has been extensively studied in the literature. See, for example, [15, 16] and the references therein. A mapping of into is called monotone if for all . is called -inverse-strongly monotone if there exists a positive real number such that for all We consider a system of quasi-variational inclusion for finding such that where and are nonlinear mappings for each . The set of solutions of problem (6) is denoted by . As special cases of problem (6), we have the following.(1)If and , then problem (6) is reduced to (7) for finding such that (2)Further, if , then problem (7) is reduced to (8) for finding such that where is the zero vector in . The set of solutions of problem (8) is denoted by . A set-valued mapping is called monotone if for all and imply . A monotone mapping is maximal if its graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for for all imply . Let be a monotone mapping of into and let be the normal cone to at ; that is, , and define Then, is the maximal monotone and if and only if ; see .
Let be a set-valued maximal monotone mapping; then, the single-valued mapping defined by is called the resolvent operator associated with , where is any positive number and is the identity mapping. The following characterizes the resolvent operator.(R1)The resolvent operator is single-valued and nonexpansive for all ; that is, (R2)The resolvent operator is 1-inverse-strongly monotone; see ; that is, (R3)The solution of problem (8) is a fixed point of the operator for all ; see also ; that is, (R4)If , then the mapping is nonexpansive.(R5) is closed and convex.
Let be a strongly positive linear bounded operator on ; that is, there exists a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is a strongly positive linear bounded operator and is a potential function for (i.e., for ).
In 2007, Plubtieng and Punpaeng  proposed the following iterative algorithm: They proved that if the sequences and of parameters satisfy appropriate conditions, then the sequences and both converge to the unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ).
In 2009, Peng and Yao  introduced an iterative algorithm based on extragradient method which solves the problem for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings, and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. The sequences generated by are for all , where is -mapping. They proved the strong convergence theorems under some mild conditions.
In 2010, Qin et al.  introduced an iterative method for finding solutions of a generalized equilibrium problem, the set of fixed points of a family of nonexpansive mappings, and the common variational inclusions. The sequences generated by and are a sequence generated by where is a contraction and is inverse-strongly monotone mappings for and is called a -mapping generated by and . They proved the strong convergence theorems under some mild conditions. Liou  introduced an algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of variational inclusion in a real Hilbert space. The sequences generated by are for all , where is a strongly positive bounded linear operator and are inverse-strongly monotone. They proved the strong convergence theorems under some suitable conditions.
Next, Petrot et al.  introduced the new following iterative process for finding the set of solutions of quasi-variational inclusion problem and the set of fixed point of a nonexpansive mapping. The sequence is generated by for all , where are three sequences in and . They proved that generated by (22) converges strongly to which is the unique solution in .
In 2011, Jitpeera and Kumam  introduced a shrinking projection method for finding the common element of the common fixed points of nonexpansive semigroups, the set of common fixed point for an infinite family, the set of solutions of a system of mixed equilibrium problems, and the set of solution of the variational inclusion problem. Let , , , , and be sequences generated by , , , , and where , . We proved the strong convergence theorem under certain appropriate conditions.
In this paper, motivated by the above results, we introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusions, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Then, we prove strong convergence theorems which are connected with [5, 26–29]. Our results extend and improve the corresponding results of Jitpeera and Kumam , Liou , Plubtieng and Punpaeng , Petrot et al. , Peng and Yao , Qin et al. , and some authors.
Let be a real Hilbert space with inner product and norm and let be a nonempty closed convex subset of . Then, For every point , there exists a unique nearest point in , denoted by , such that is called the metric projection of onto . It is well known that is a nonexpansive mapping of onto and satisfies Moreover, is characterized by the following properties: and Let be a monotone mapping of into . In the context of the variational inequality problem, the characterization of projection (27) implies the following: It is also known that satisfies the Opial condition ; that is, for any sequence with , the inequality holds for every with .
For the infinite family of nonexpansive mappings of , and sequence in , see ; we define the mapping of into itself as follows:
Lemma 1 (Shimoji and Takahashi ). Let be a nonempty closed convex subset of a real Hilbert space . Let be a family of infinitely nonexpanxive mappings with and let be a real sequence such that for every . Then (1) is nonexpansive and for each ;(2)for each and for each positive integer , the limit exists;(3)the mapping defined by is a nonexpansive mapping satisfying and it is called the -mapping generated by , and ;(4)if is any bounded subset of , then .
For solving the mixed equilibrium problem, let us give the following assumptions for a bifunction and a proper extended real-valued function satisfies the following conditions:(A1) for all ;(A2) is monotone; that is, for all ;(A3)for each , ;(A4)for each , is convex and lower semicontinuous;(A5)for each , is weakly upper semicontinuous;(B1)for each and , there exist a bounded subset and such that for any , (B2) is a bounded set.
We need the following lemmas for proving our main results.
Lemma 2 (Peng and Yao ). Let be a nonempty closed convex subset of . Let be a bifunction that satisfies (A1)–(A5) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows: for all . Then, the following hold: (1)for each ;(2) is single-valued;(3) is firmly nonexpansive; that is, for any , ;(4);(5) is closed and convex.
Lemma 3 (Xu ). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (1),(2) or .Then, .
Lemma 4 (Suzuki ). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .
Lemma 5 (Marino and Xu ). Assume is a strongly positive linear bounded operator on a Hilbert space with coefficient and . Then, .
Lemma 6. For given is a solution of problem (6) if and only if is a fixed point of the mapping defined by where , are positive constants, and are two mappings.
Proof. This completes the proof.
Now, we prove the following lemmas which will be applied in the main theorem.
Lemma 7. Let be defined as in Lemma 6. If is -inverse-strongly monotone and , and , respectively, then is nonexpansive.
Proof. For any and , , we have This shows that is nonexpansive on .
3. Main Results
In this section, we show a strong convergence theorem for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of a infinite family of nonexpansive mappings in a real Hilbert space.
Theorem 8. Let be a nonempty closed convex subset of a real Hilbert Space . Let be a bifunction of into real numbers satisfying (A1)–(A5) and let be a proper lower semicontinuos and convex function. Let be nonexpansive mappings for all , such that . Let be a contraction of into itself with coefficient and let be -inverse-strongly monotone mapping of into . Let be a strongly positive bounded linear self-adjoint on with coefficient and , let be a maximal monotone mapping. Assume that either or holds and let be the -mapping defined by (31). Let , , , and be sequences generated by , , and where and , , , and satisfy the following conditions:(C1) and ,(C2), (C3), .Then, converges strongly to , where , is the metric projection of onto and , where is solution to the problem (6).
Proof. Let ; that is , . Putting , one can see that .
We divide our proofs into the following steps: (1)sequences , , , and are bounded;(2);(3), and ;(4);(5), where ;(6).
Step 1. From conditions (C1) and (C2), we may assume that . By the same argument as that in , we can deduce that is positive and . For all and . since is a -inverse-strongly monotone and are -inverse-strongly monotone, we have It follows that ; hence is nonexpansive.
In the same way, we conclude that and ; hence are nonexpansive. Let . It follows that By Lemma 2, we have for all , . Then, for , we obtain Hence, we have From (40) and (44), we deduce that It follows by mathematical induction that Hence, is bounded and also , , , , , and are all bounded.
Step 2. We show that .
Putting , we get , . We note that It follows that By the definition of , where is an approximate constant such that . Since for all and , we compute It follows that Substituting (51) into (49), We note that Applying (52) and (53) in (48), we get By conditions (C1)–(C3), imply that Hence, by Lemma 4, we obtain It follows that We obtain that
Step 3. We can rewrite (40) as . We observe that it follows that By conditions (C1), (C2), and (58), imply that From (42) and (43), we get By (40), we obtain