#### Abstract

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

#### 1. Introduction

Let be a real Hilbert space with inner product and norm , let be a nonempty closed convex subset of , and let be the metric projection of onto . Let be a nonlinear mapping on . We denote by the set of fixed points of and by the set of all real numbers. A mapping is called -Lipschitz continuous if there exists a constant such that In particular, if then is called a nonexpansive mapping; if then is called a contraction.

Let be a nonlinear mapping on . We consider the following variational inequality problem (VIP) [1]: find a point such that The solution set of VIP (2) is denoted by .

Let be a real-valued function, let be a nonlinear mapping, and let be a bifunction. The generalized mixed equilibrium problem (GMEP) [2] is to find such that We denote the set of solutions of GMEP (3) by .

In [2], it is assumed that is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous and convex function with restriction (A1) or (A2), where(H1), for all ;(H2) is monotone; that is, , for any ;(H3) is upper-hemicontinuous; that is, for each , (H4) is convex and lower semicontinuous, for each ;(A1)for each and , there exists a bounded subset and such that, for any , (A2) is a bounded set.

Given a positive number , let be the solution set of the auxiliary mixed equilibrium problem; that is, for each , where is a Fréchet differential and strongly convex function on . In particular, whenever , is rewritten as .

Let be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set : (assuming the existence of minimizers). We denote by the set of minimizers of CMP (7).

In 2011, combining the hybrid steepest-descent method in [3], the viscosity approximation method, and averaged mapping approach to the gradient-projection algorithm (GPA) in [4], Ceng et al. [5] introduced and analyzed the following iterative algorithm: where is -Lipschitzian mapping with constant is a -Lipschitzian and -strongly monotone operator with constants . Assume that , , with and , and . Under the control conditions that (i) , (ii) , (iii) either or , it was proven in [5] that the sequence generated by (8) converges strongly to some , which is a unique solution of the VIP

On the other hand, let be a single-valued mapping of into and let be a set-valued mapping with . Consider the following variational inclusion: find a point such that We denote by the solution set of the variational inclusion (10). Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with and as follows: where is a positive number.

Let and be two nonexpansive mappings. In 2009, Yao et al. [6] considered the following hierarchical VIP: find hierarchically a fixed point of , which is a solution to the VIP for monotone mapping ; namely, find such that The solution set of the hierarchical VIP (12) is denoted by . It is not hard to check that solving the hierarchical VIP (12) is equivalent to solving fixed point problem of the composite mapping ; that is, find such that . The authors [6] introduced and analyzed the following iterative algorithm for solving the hierarchical VIP (12):

We observed that Zeng et al. [7] introduced and considered the following triple hierarchical variational inequality (THVI).

Problem 1. Let be a positive integer. Assume that(i)each is a nonexpansive mapping with ;(ii) is -inverse strongly monotone;(iii) is -strongly monotone and -Lipschitz continuous;(iv).
Then, the objective is to

The authors [7] proposed the following algorithm for solving Problem 1.

Algorithm   (see [7, Algorithm ]). Let and satisfy assumptions (i)–(iv) in Problem 1. The following steps are presented for solving Problem 1.

Step  0. Take , choose arbitrarily, and let .

Step  1. Given , compute as where , for integer , with the mod function taking values in the set ; that is, if for some integers and , then if and if .

Update and go to Step  1.

In this paper, we introduce and study the following triple hierarchical variational inequality (THVI) with constraints of GMEP (3), CMP (7), and finitely many variational inclusions.

Problem 2. Let be two positive integers. Assume that(i) is a nonexpansive mapping, for , is -inverse strongly monotone, and is a convex functional with -Lipschitz continuous gradient ;(ii) is -inverse strongly monotone and is -strongly monotone and -Lipschitz continuous;(iii) is a bifunction from to satisfying (H1)–(H4) and is a lower semicontinuous and convex functional;(iv) is a maximal monotone mapping and is -inverse strongly monotone for ;(v), where .

Then, the objective is to

Motivated and inspired by the above facts, we introduce and analyze a hybrid iterative algorithm by combining Korpelevich’s extragradient method, the hybrid steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under mild conditions, the proposed algorithm converges strongly to a common element of the solution set of GMEP (3), the solution set of CMP (7), the solution set of finitely many variational inclusions, and the fixed point set of finitely many nonexpansive mappings , which is merely a unique solution of the THVI   (16). In addition, we also consider the application of the proposed algorithm to solving a hierarchical variational inequality problem with constraints of GMEP (3), CMP (7), and finitely many variational inclusions. That is, under appropriate conditions, it is proven that the proposed algorithm converges strongly to a unique solution of the VIP: ; equivalently, . The results obtained in this paper improve and extend the corresponding results announced by many others. We also observe that some recent and related results have been established in [814].

#### 2. Preliminaries

Throughout this paper, we assume that is 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,

Definition 3. A mapping is called(i)monotone if (ii)-strongly monotone if there exists a constant such that (iii)-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. Moreover, we also have that, for all and , So, if , then is a nonexpansive mapping from to .

Definition 4. A differentiable function is called(i)convex, if where is the Frechet derivative of at ;(ii)strongly convex, if there exists a constant such that
It is easy to see that if is a differentiable strongly convex function with a constant then is strongly monotone with constant .

The metric 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 5. For given and :(i);(ii);(iii) (this implies that is nonexpansive and monotone).

By using the technique of [15], we can readily obtain the following elementary result, where is the solution set of the mixed equilibrium problem [15].

Proposition 6 (see [16, Lemma 1 and Proposition 1]). Let be a nonempty closed convex subset of a real Hilbert space and let be a lower semicontinuous and convex function. Let be a bifunction satisfying the conditions (H1)–(H4). Assume that(i) is strongly convex with constant and the function is weakly upper semicontinuous, for each ;(ii)for each and there exists a bounded subset and such that, for any , Then, the following hold:(a)for each ;(b) is single-valued;(c) is nonexpansive if is Lipschitz continuous with constant and where , for ;(d) for all and , (e);(f) is closed and convex.
In particular, whenever is a bifunction satisfying the conditions (H1)–(H4) and , then we have that is firmly nonexpansive and In this case, is rewritten as . If, in addition, , then is rewritten as (see [17, Lemma 2.1] for more details).

Remark 7. Suppose that is strongly convex with constant and is Lipschitz continuous with constant . Then, is -strongly monotone and -Lipschitz continuous with positive constants . Utilizing Proposition 6 (d) we obtain that, for all and , which immediately implies that
The concept of -mapping was introduced in Atsushiba and Takahashi [18]. It is very useful in establishing the convergence of iterative methods for computing a common fixed point of nonlinear mappings (see, for instance, [19, 20]).
Let . Given the nonexpansive mappings on , Atsushiba and Takahashi define, for each , mappings , by
The is called the -mapping generated by and . Note that the nonexpansivity of implies the one of .

Proposition 8 (see [20]). Let be a nonempty closed convex subset of a Banach space . Let be a finite family of nonexpansive mappings of into itself such that , and let be real numbers such that for . For any , let be the -mapping of into itself generated by and . If is strictly convex, then .

Proposition 9 (see [21, Lemma 2.8]). Let be a nonempty convex subset of a Banach space . Let be a finite family of nonexpansive mappings of into itself and let be sequences in such that . Moreover, for every integer , let and be the -mapping generated by and and and , respectively. Then, for every , it follows that
In what follows, we recall some facts and tools in a real Hilbert space .

Lemma 10. Let be a real inner product space. Then, the following inequality holds:

Lemma 11. Let be a real Hilbert space. Then, the following hold:(a), for all ;(b), for all and with ;(c)If is a sequence in such that , it follows that

Definition 12. A mapping is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping; that is, where and is nonexpansive. More precisely, when the last equality holds, we say that is -averaged. Thus, firmly nonexpansive mappings (in particular, projections) are -averaged mappings.

Lemma 13 (see [22]). Let be a given mapping.(i) is nonexpansive if and only if the complement is -ism.(ii)If is -ism, then, for is -ism.(iii) is averaged if and only if the complement is -ism for some . Indeed, for is -averaged if and only if is -ism.

Lemma 14 (see [22]). Let be given operators.(i)If for some and if is averaged and is nonexpansive, then is averaged.(ii) is firmly nonexpansive if and only if the complement is firmly nonexpansive.(iii)If for some and if is firmly nonexpansive and is nonexpansive, then is averaged.(iv)The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where .(v)If the mappings are averaged and have a common fixed point, then The notation denotes the set of all fixed points of the mapping ; that is, .

Let be a convex functional with -Lipschitz continuous gradient . It is well known that the gradient-projection algorithm (GPA) generates a sequence determined by the gradient and the metric projection : or, more generally, where, in both (37) and (38), the initial guess is taken from arbitrarily and the parameters or are positive real numbers. The convergence of algorithms (37) and (38) depends on the behavior of the gradient .

Lemma 15 (see [23, demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . 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 .

Lemma 16. Let be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 5 (i)) implies
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 17 (see [3, Lemma 3.1]). is a contraction provided ; that is, where .

Lemma 18 (see [3]). 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, .

Recall that a Banach space is said to satisfy Opial’s property [23] if for any given sequence which converges weakly to an element , there holds the inequality It is well known that every Hilbert space satisfies Opial’s property in [23].

Finally, recall that 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, -Lipschitz-continuous mapping, and let be the normal cone to at ; that is, Define Then, is maximal monotone and

Let be a maximal monotone mapping. Let be two positive numbers.

Lemma 19 (see [24]). There holds the resolvent identity For , we observe that there holds the following relation:

Lemma 20 (see [25]). is single-valued and firmly nonexpansive; that is,
Consequently, is nonexpansive and monotone.

Lemma 21 (see [26]). Let be a maximal monotone mapping with . Then, for any given is a solution of problem (11) if and only if satisfies

Lemma 22 (see [27]). 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 23 (see [26]). Let be a maximal monotone mapping with and let be a monotone, continuous, and single-valued mapping. Then, for each . In this case, is maximal monotone.

#### 3. Main Results

In this section, we will introduce and analyze a hybrid iterative algorithm for finding a solution of the THVI (16) with constraints of several problems: the GMEP (3), the CMP (7), and finitely many variational inclusions in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. We prove the strong convergence of the proposed algorithm to a unique solution of THVI (16) under suitable conditions. In addition, we also consider the application of the proposed algorithm to solving a hierarchical VIP with the same constraints.

We are now in a position to state and prove the first main result in this paper.

Theorem 24. Let be a nonempty closed convex subset of a real Hilbert space . Let be a convex functional with -Lipschitz continuous gradient . Let be two integers. Let be a bifunction from to satisfying (H1)–(H4), let be a lower semicontinuous and convex functional, and let be -inverse strongly monotone. Let be a maximal monotone mapping and let be -inverse strongly monotone for . Let be a finite family of nonexpansive mappings on . Let be -inverse strongly monotone and let be -strongly monotone and -Lipschitz continuous. Assume that , where . Let , where and . For every , let be the -mapping generated by and . Assume that(i) is strongly convex with a constant and its derivative is Lipschitz continuous with a constant such that the function is weakly upper semicontinuous for each ;(ii)for each , there exist a bounded subset and such that, for any , (iii) and ;(iv), and ;(v) and , for ;(vi) and , for .For arbitrarily given , let be a sequence generated by where (here is nonexpansive, for each ), (), and . Then, whenever is firmly nonexpansive, the following hold:(i);(ii);(iii) provided additionally.

Proof. Let . Taking into account that , we may assume, without loss of generality, that for all . Since is -Lipschitzian, it follows that is -ism. By Lemma 13 (ii) we know that for is -ism. So by Lemma 13 (iii) we deduce that is -averaged. Now since the projection is -averaged, it is easy to see from Lemma 14 (iv) that the composite is -averaged for . Hence, we obtain that, for each , is -averaged for each . Therefore, we can write where is nonexpansive and for each . It is clear that Since is -Lipschitz continuous, we get Put for and , where is the identity mapping on . Then, we have .
We divide the rest of the proof into several steps.
Step  1. We prove that is bounded.
Indeed, take arbitrarily. Since , is -inverse strongly monotone and ; utilizing the nonexpansivity of , we have, for any , Utilizing (21) and Lemma 20 we have Combining (60) and (61), we have Since where , it is clear that for each . Since is -inverse strongly monotone and , utilizing the nonexpansivity of , we obtain from (21), (55), and (62) that Utilizing Lemma 17, we obtain from (55) and that where . By induction, we find that Thus, is bounded and so are the sequences and .
Step  2. We prove that .
Indeed, put for all . Utilizing (21) and (51), we obtain that