Hybrid Algorithms for Solving Variational Inequalities, Variational Inclusions, Mixed Equilibria, and Fixed Point Problems

Abstract

We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method.

1. Introduction

Throughout this paper, we assume that is a real Hilbert space with inner product and norm , is a nonempty closed convex subset of , and is the metric projection of onto . Let be a self-mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called -Lipschitzian if there exists a constant such that In particular, if , then is called a nonexpansive mapping [1]; if , then is called -contraction.

Recall that a mapping is called(i)-strongly monotone if there exists a constant such that (ii)-inverse strongly monotone if there exists a constant such that

It is obvious that if is -inverse strongly monotone, then is monotone and -Lipschitz continuous. In addition, a mapping is called strongly positive on if there exists a constant such that

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. In 2008, Peng and Yao [2] introduced the following generalized mixed equilibrium problem (GMEP) of finding such that We denote the set of solutions of GMEP (5) by . The GMEP (5) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied in [3–8].

We present some special cases of GMEP (5) as follows.

If , then GMEP (5) reduces to the generalized equilibrium problem (GEP) which is to find such that which was introduced and studied by S. Takahashi and W. Takahashi [9]. The set of solutions of GEP is denoted by .

If , then GMEP (5) reduces to the mixed equilibrium problem (MEP) which is to find such that which was considered and studied in [10]. The set of solutions of MEP is denoted by .

If , , then GMEP (5) reduces to the equilibrium problem (EP) which is to find such that The set of solutions of EP is denoted by . It is worth pointing out that the EP is a unified model of several problems, for instance, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.

Throughout this paper, it is assumed as in [2] that is a bifunction satisfying conditions (A1)–(A4) and is a lower semicontinuous and convex function with restriction (B1) or (B2), where(A1) for all ,(A2) is monotone, that is, for any ,(A3) is upper hemicontinuous, that is, for each , (A4) is convex and lower semicontinuous for each ,(B1)for each and , there exists a bounded subset and such that for any , (B2) is a bounded set.

Let be a single-valued mapping of into and a multivalued mapping with . Consider the following variational inclusion: find a point such that We denote by the solution set of the variational inclusion (11). In particular, if , then . If , then problem (11) becomes the inclusion problem introduced by Rockafellar [11]. It is known that problem (11) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria, and game theory.

In 1998, Huang [12] studied problem (11) in the case where is maximal monotone and is strongly monotone and Lipschitz continuous with . Subsequently, Zeng et al. [13] further studied problem (11) in the case which is more general than Huang’s one [12]. Moreover, the authors [13] obtained the same strong convergence conclusion as in Huang’s result [12]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [14–17] and the references therein.

On the other hand, consider the following variational inequality problem (): find a point such that The solution set of VIP (12) is denoted by .

In 1976, Korpelevič [18] proposed an iterative algorithm for solving the VIP (12) in Euclidean space : with being a given number, which is known as the extragradient method (see also [19]). The literature on the VIP is vast and Korpelevič’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [2, 5, 6, 8, 17, 20–28] and references therein, to name but a few.

VIP (12) was first discussed by Lions [29] and now is well known; there are a lot of different approaches towards solving VIP (12) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. VIP (12) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [30–33]. It is well known that, if is a strongly monotone and Lipschitz-continuous mapping on , then VIP (12) has a unique solution. Not only the existence and uniqueness of solutions are important topics in the study of VIP (12) but also how to actually find a solution of VIP (12) is important.

Let be a nonempty closed convex subset of a real Banach space . Let be an infinite family of nonexpansive self-mappings on and let be a sequence of nonnegative numbers in . For any , define a self-mapping on as follows: Such a mapping is called the -mapping generated by and ; see [34].

In 2008, Ceng and Yao [35] introduced and analyzed the following relaxed viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space.

Theorem 1 (see [35, Theorem 3.2]). Let be a strictly convex and reflexive Banach space with a uniformly Gateaux differentiable norm, let be a nonempty closed convex subset of , let be an infinite family of nonexpansive self-mappings on such that the common fixed point set , and let be a -contraction with the contraction coefficient . Let be a sequence of positive numbers in for some . For any given , let be the iterative sequence defined by where and are two sequences in with , is a sequence in , and is the -mapping defined by (14). Assume that(i), and ,(ii) and . Then there hold the following:(I);(II)the sequence converges strongly to some , provided and for some fixed , which is the unique solution of the :  where is the normalized duality mapping of .
Furthermore, let , . Given the nonexpansive mappings on , Atsushiba and Takahashi [36] defined, for each , mappings by
The is called the -mapping generated by and . Note that the nonexpansivity of implies the nonexpansivity of .
In 2008, Colao et al. [37] introduced and studied an iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space . Subsequently, combining Yamada’s hybrid steepest-descent method [38] and Colao et al.’s hybrid viscosity approximation method [37], Ceng et al. [7] proposed and analyzed the following hybrid iterative method for finding a common element of the set of solutions of (5) and the set of fixed points of a finite family of nonexpansive mappings .

Theorem 2 (see [7, Theorem 3.1]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction satisfying assumptions (A1)–(A4) and let be a lower semicontinuous and convex function with restriction (B1) or (B2). Let the mapping be -inverse strongly monotone, and let be a finite family of nonexpansive mappings on such that . Let be a -Lipschitzian and -strongly monotone operator with constants and a -Lipschitzian mapping with constant . Let and , where . Suppose that and are two sequences in , is a sequence in , and is a sequence in with . For every , let be the -mapping generated by and . Given arbitrarily, suppose that the sequences and are generated iteratively by where the sequences , , and the finite family of sequences satisfy the following conditions:(i) and ;(ii);(iii) and ;(iv) for all . Then, both and converge strongly to , where is a unique solution of the variational inequality:
On the other hand, whenever a real Hilbert space, Yao et al. [4] very recently introduced and analyzed an iterative algorithm for finding a common element of the set of solutions of (5), the set of solutions of the variational inclusion (11), and the set of fixed points of an infinite family of nonexpansive mappings.

Theorem 3 (see [4, Theorem 3.2]). Let be a lower semicontinuous and convex function and let be a bifunction satisfying conditions (A1)–(A4) and (B1). Let be a strongly positive bounded linear operator with coefficient and let be a maximal monotone mapping. Let the mappings be -inverse strongly monotone and -inverse strongly monotone, respectively. Let be a -contraction. Let , , and be three constants such that , , and . Let be a sequence of positive numbers in for some and an infinite family of nonexpansive self-mappings on such that . For arbitrarily given , let the sequence be generated by where , are two real sequences in and is the -mapping defined by (14) (with and ). Assume that the following conditions are satisfied:(C1) and ;(C2). Then, the sequence converges strongly to , where is a unique solution of the :

Motivated and inspired by the above facts, in this paper, we introduce and analyze a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inclusion (11) in a real Hilbert space. Furthermore, it is proven that the proposed hybrid iterative algorithm is strongly convergent under some mild conditions imposed on algorithm parameters. Here our hybrid algorithm is based on Korpelevič’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. The results obtained in this paper improve and extend the corresponding results announced by many others.

2. Preliminaries

Let be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let be a nonempty closed convex subset of . We write to indicate that the sequence converges weakly to and to indicate that the sequence converges strongly to . Moreover, we use to denote the weak -limit set of the sequence ; that is,

The metric (or nearest point) projection from onto is the mapping which assigns to each point the unique point satisfying the property

Some important properties of projections are gathered in the following proposition.

Proposition 4 (see [31, 39]). For given and :(i), ; (ii), ; (iii), .
Consequently, is nonexpansive and monotone.
If is an -inverse strongly monotone mapping (-ism) of into , then it is obvious that is -Lipschitzian. We also have that, for all and , So, if , then is a nonexpansive mapping from to .
It is also easy to see that a projection is -ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.
A set-valued mapping is called monotone if, for all , and imply A set-valued mapping is called maximal monotone if is monotone and for each , where is the identity mapping of . We denote by the graph of . It is known that a monotone mapping is maximal if and only if, for , for every implies .
Let be a monotone, -Lipschitzian mapping and let be the normal cone to at ; that is, Defining then, is maximal monotone and if and only if ; see [11].
Assume that is a maximal monotone mapping. Then, for , associated with , the resolvent operator can be defined as In terms of Huang [12] (see also [13]), there holds the following property for the resolvent operator .

Lemma 5. is single-valued and firmly nonexpansive; that is, Consequently, is nonexpansive and monotone.

Lemma 6. Let be a nonempty closed convex subset of and a monotone mapping. In the context of the (12), there holds the following relation:

Lemma 7 (see [17]). Let be a maximal monotone mapping with . Then, for any given , is a solution of problem (11) if and only if satisfies

Lemma 8 (see [13]). Let be a maximal monotone mapping with and let be a strongly monotone, continuous, and single-valued mapping. Then, for each , the equation has a unique solution for .

Lemma 9 (see [17]). Let be a maximal monotone mapping with and a monotone, continuous, and single-valued mapping. Then for each . In this case, is maximal monotone.

Lemma 10 ([40], see also [10]). Assume that satisfies (A1)–(A4) 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:(i)for each , ;(ii) is single-valued;(iii) is firmly nonexpansive, that is, for any , (iv);(v) is closed and convex.

Proposition 11 (see [5, Proposition 2.1]). Let , , , , and be as in Lemma 10. Then the following inequality holds: for all and .

Remark 12. From the conclusion of Proposition 11, it immediately follows that for all and .

Lemma 13 (see [41]). Let and be bounded sequences in a Banach space and a sequence in with Suppose that for each and Then .

We have the following crucial lemmas concerning the -mapping defined by (14).

Lemma 14 (see [42, Lemma 3.2]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence of positive numbers in for some . Then, for every and , the limit exists.

Lemma 15 (see [42, Lemma 3.3]). Let be a nonempty closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive self-mappings on such that and let be a sequence of positive numbers in for some . Then, .

Remark 16. Using Lemma 14, we can define the mapping as follows: Such a is called the -mapping generated by the sequences and . As pointed out in [43], if is a bounded sequence in , then we have Throughout this paper, we always assume that is a sequence of positive numbers in for some .

Lemma 17 (see [44]). Let be a sequence of nonnegative numbers satisfying the conditions where and are sequences of real numbers such that(i) and , or equivalently, (ii), or . Then .

Lemma 18 (see [39, demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on with . Then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .

The following lemma is an immediate consequence of an inner product.

Lemma 19. In a real Hilbert space , there holds the inequality
Let be a nonempty closed convex subset of a real Hilbert space . We introduce some notations. Let be a number in and let . Associating with a nonexpansive mapping , we define the mapping by where is an operator such that, for some positive constants , is -Lipschitzian and -strongly monotone on ; that is, satisfies the conditions for all .

Lemma 20 (see [44, Lemma 3.1]). is a contraction provided ; that is, where .

Remark 21. (i) Since is -Lipschitzian and -strongly monotone on , we get . Hence, whenever , we have which implies So, .
(ii) In Lemma 20, put and . Then we know that , and

3. A Strong Convergence Theorem

In this section, we will prove a strong convergence theorem for a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inclusion (11) in a real Hilbert space.

Theorem 22. Let be a nonempty closed convex subset of a real Hilbert space . Let , be two integers. Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function, where . Let and be -inverse strongly monotone and -inverse strongly monotone, respectively, where , . Let be a -Lipschitzian and -strongly monotone operator with positive constants and let be a -Lipschitzian mapping with constant . Let be a maximal monotone mapping and let the mapping be -inverse strongly monotone. Let , , and , where . Let be a sequence of positive numbers in for some and an infinite family of nonexpansive self-mappings on such that . For arbitrarily given , let the sequence be generated by where , are two real sequences in and is the -mapping defined by (14). Assume that the following conditions are satisfied:(i) and ;(ii);(iii) and for all ;(iv) and for all . Assume that either (B1) or (B2) holds. Then the sequence converges strongly to , where is a unique solution of the :

Proof. Let . Note that is a -Lipschitzian and -strongly monotone operator with positive constants and is a -Lipschitzian mapping with constant . Then, we have where , and hence Since , it is known that . Therefore, is a contraction of into itself, which implies that there exists a unique element such that .
We divide the remainder of the proof into several steps.
Step  1. Let us show that is bounded.
Indeed, taking into account the control conditions (i) and (ii), we may assume, without loss of generality, that for all . Put for all and , for all and , and , where is the identity mapping on . Then we have that and . Take arbitrarily. Then from (24) and Lemma 10 we have Similarly, we have Combining (55) and (56), we have Since the mapping is -inverse strongly monotone with , we have It is clear that, if , then is nonexpansive. Set for each . It follows that which, together with (57), yields Utilizing Lemma 20, from (49) we obtain By induction, we get Therefore, is bounded and hence , , , , , and are also bounded.
Step  2. Let us show that as .
Indeed, define for each . Then from the definition of we obtain It follows that Utilizing (14) and the nonexpansivity of and , we obtain that for each where for some . Note that Note that