#### Abstract

We introduce a new general system of variational inclusions in Banach spaces and propose a new iterative scheme for finding common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of nonexpansive semigroups in a uniformly convex and 2-uniformly smooth Banach space. Furthermore, strong convergence theorems are established under some certain control conditions. As applications, finding a common solution for a system of variational inequality problems and minimization problems is given.

#### 1. Introduction

In the theory of variational inequalities and variational inclusions, the development of an efficient and implementable iterative algorithm is interesting and important. The important generalization of variational inequalities called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in optimization, nonlinear programming, finance, economics, and applied sciences.

Variational inequalities are being used as a mathematical programming tool in modeling a wide class of problems arising in several branches of pure and applied mathematics. Several numerical techniques for solving variational inequalities and the related optimization problem have been considered by many authors.

Throughout this paper, we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space and be its dual space. Let denote the unit sphere of . is said to be uniformly convex if for each , there exists a constant such that for all , The norm on is said to be Gâteaux differentiable if the limit exists for each , and in this case is smooth. Moreover, we say that the norm is said to have a uniformly Gâteaux differentiable if the above limit is attained uniformly for all and in this case is said to be uniformly smooth. We define a function , called the modulus of smoothness of , as follows: It is know that is uniformly smooth if and only if . Let be a fixed real number . A Banach space is said to be -uniformly smooth if there exists a constant such that for all . From [1], we know the following property.

Let be a real number with and let be a Banach space. Then, is -uniformly smooth if and only if there exists a constant such that The best constant in the above inequality is called the -uniformly smoothness constant of (see [1] for more details).

Let be a real Banach space and the dual space of . Let denote the pairing between and . For , the generalized duality mapping In particular, if , the mapping is called the normalized duality mapping and usually, we write . If is a Hilbert space, then is the identity. Further, we have the following properties of the generalized duality mapping : (1) for all with , (2) for all and , (3) for all .

It is know that if is smooth, then is single-valued, which is denoted by .

Definition 1.1. Let be a nonempty closed convex subset of . A mapping is said to be (i)nonexpansive if (ii)Lipschitzian if there exists a constant such that (iii)contraction if there exists a constant such that

Remark 1.2. We denote as the set of fixed points of . We know that is nonempty if is bounded; for more detail see [2].

Definition 1.3. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ,(ii) for all ,(iii)for each the mapping is continuous,(iv) for all and .

Remark 1.4. We denote by the set of all common fixed points of , that is . We know that is nonempty if is bounded, see [3].

Let be a nonempty closed convex subset of a smooth Banach space . Recall the following definitions of a nonlinear mapping , the following are mentioned.

Definition 1.5. Given a mapping , (i) is said to be accretive(ii) is said to be -strongly accretive if there exists a constant such that (iii) is said to be -inverse-strongly accretive or -cocoercive if there exists a constant such that (iv) is said to be -relaxed cocoercive if there exists a constant such that (v) is said to be -relaxed cocoercive if there exist positive constants and such that

Remark 1.6. (1) Every -strongly accretive mapping is an accretive mapping.
(2) Every -strongly accretive mapping is a -relaxed cocoercive mapping for any positive constant but the converse is not true in general. Then, the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.
(3) Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, e.g., [4]).
(4) The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods [5]. Several classes of relaxed cocoercive variational inequalities have been studied in [6, 7].

The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities, inclusions, and related optimization problems.

Definition 1.7. Let be a multivalued maximal accretive mapping. The single-valued mapping , defined by is called resolvent operator associated with , where is any positive number and is the identity mapping.

In 2010, Qin et al. [8] introduced a system of quasivariational inclusions as follows. Find such that where and are nonlinear mappings for all . As special cases of problem (1.15), we have the following. (1)If and , then problem (1.15) is reduced to the following. Find such that (2)Further, if in problem (1.16), then problem (1.16) is reduced to the following. Find such that

The problem (1.17) is called variational inclusion problem denoted by .

Here we have examples of the variational inclusion (1.17). If , where is a nonempty closed convex subset of , and then the variational inclusion problem (1.17) is equivalent (see [9]) to finding such that This problem is called Hartman-Stampacchia variational inequality problem denoted by .

Let be a subset of , and let be a mapping of into . Then, is said to be sunny if whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for all , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto .

In 2006, Aoyama et al. [10] considered the following problem: find such that They proved that the variational inequality (1.21) is equivalent to a fixed point problem. The element is a solution of the variational inequality (1.21) if and only if satisfies the following equation: where is a constant and is a sunny nonexpansive retraction from onto .

The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 1.8 (see [11]). Let be a smooth Banach space and a nonempty subset of . Let be a retraction and the normalized duality mapping on . Then the following are equivalent: (1) is sunny and nonexpansive,(2), for all , .

Proposition 1.9 (see [12]). Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space and a nonexpansive mapping of into itself with . Then the set is a sunny nonexpansive retract of .

For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [13, 14]. More precisely, take , and define a contraction by where is a fixed point. Banachs contraction mapping principle guarantees that has a unique fixed point , that is, It is unclear, in general, what the behavior of is as , even if has a fixed point. However, in the case of having a fixed point, Ceng et al. [15] proved that, if is a Hilbert space, then converges strongly to a fixed point of . Reich [14] extended Browders result to the setting of Banach spaces and proved that, if is a uniformly smooth Banach space, then converges strongly to a fixed point of , and the limit defines the (unique) sunny nonexpansive retraction from onto .

Reich [14] showed that, if is uniformly smooth and is the fixed point set of a nonexpansive mapping from into itself, then there is a unique sunny nonexpansive retraction from onto and it can be constructed as follows.

Proposition 1.10 (see [14]). Let be a uniformly smooth Banach space and a nonexpansive mapping such that . For each fixed and every , the unique fixed point of the contraction converges strongly as to a fixed point of . Define by . Then is the unique sunny nonexpansive retract from onto ; that is, satisfies the property,

Many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inclusion and variational inequalities (1.15)–(1.17) and (1.21) by using different iterative methods (see, e.g., [10, 1627]).

Recently, Qin et al. [8] considered the problem of finding the solutions of a general system of variational inclusion (1.15) with -inverse strongly accretive mappings. To be more precise, they obtained the following results.

Lemma 1.11. For any , where , is a solution of the problem (1.15) if and only if is a fixed point of the mapping defined by

Theorem QCCK (see [8]). Let be a uniformly convex and 2-uniformly smooth Banach space with the smoothness constant . Let be a maximal monotone mapping and be a -inverse-strongly accretive mapping, respectively, for all . Let be a -strict pseudocontraction such that . Define a mapping by , for all . Assume that , where is defined as in Lemma 1.11. Let and let be a sequence defined by where , , , and , are sequences in . If the control consequences and satisfy the following restrictions: (C1),(C2) and , then converges strongly to , where is the sunny nonexpansive retraction from onto and is a solution of the problem (1.15), where .

Iterative methods for nonexpansive mappings have recently been applied to solve minimization problems; see, for example, [2832]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is 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 the fixed point set of a nonexpansive mapping on and is a given point in .

In [33], Moudafi introduced the viscosity approximation method and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with a constant and satisfies certain conditions, converges strongly to a fixed point of in which is the unique solution of the following variational inequality: In 2006, Marino and Xu [34] introduced the following general iterative method: where is a strongly positive bounded linear operator on a Hilbert space . They proved that, if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.32) converges strongly to the unique solution of the variational inequality: which is the optimality condition for the minimization problem: where is the fixed point set of a nonexpansive mapping and is a potential function for (i.e., for all ).

In a smooth Banach space, we always assume that is strongly positive (see [35]), that is, a constant with the property where is the identity mapping and is the normalized duality mapping.

Recently, Sunthrayuth and Kumam [36] introduced the following iterative method for nonexpansive semigroup in Banach spaces, They proved strong convergence theorem of the iterative scheme defined by (1.36) converges strongly to the common fixed point of solving the variational inequality In 2010, Kamraksa and Wangkeeree [37] introduced a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. They proved the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in a strictly convex and 2-uniformly smooth Banach space.

In this paper, we are motivated and inspire by idea of Qin et al. [8] and Sunthrayuth and Kumam [36].

First, we introduce a new general system of variational inclusions in Banach spaces as follows.

Let be Banach spaces. We consider a system of quasivariational inclusions as follows. Finding such that which is called a new general system of variational inclusions in Banach spaces, and are nonlinear mappings for all . As special cases of problem (1.38), we have the following. (1)If and , then problem (1.38) is reduced to the following. Finding such that (2)Further, if , , then problem (1.38) is reduced to problem (1.15).

Second, we study a general iterative approximation method (3.1) below, for finding common elements of the set of solutions of a new general system of variational inclusions (1.38) with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings and the set common fixed points of nonexpansive semigroup in the framework of Banach spaces. Moreover, we prove the strong convergence of the proposed iterative method under some certain control conditions. The results presented in this paper extend and improve the results of Qin et al. [8] and Sunthrayuth and Kumam [36], and many authors.

#### 2. Preliminaries

This section collects some results that will be used in the proofs of our main results.

Lemma 2.1 (see [38]). The resolvent operator associated with is single valued and nonexpansive for all .

Lemma 2.2 (see [39]). Let be a real 2-uniformly smooth Banach space with the best smoothness constant . Then, the following inequality holds:

Lemma 2.3 (see [40]). In a real Banach space , the following inequality holds:

Now, we present the concept of a uniformly asymptotically regular semigroup (see [4143]).

Definition 2.4. Let be a nonempty closed convex subset of a Banach space , be a continuous operator semigroup on . Then is said to be uniformly asymptotically regular (in short, u.a.r.) on if for all and any bounded subset of such that

Lemma 2.5 (see [44]). Let be a nonempty closed convex subset of a uniformly Banach space , be a bounded closed convex subset of . If we denote a nonexpansive semigroup on such that . For all , the set , then

Remark 2.6. It is easy to check that the set defined by Lemma 2.5 is a u.a.r. nonexpansive semigroup on (see [45] for more detail).

Lemma 2.7 (see [46]). Let be a nonempty closed convex subset of and let be a u.a.r. nonexpansive semigroup on such and at least there exists a which is demicompact. Then, for each , there exists a sequence , such that converges strongly to some point in , where .

Remark 2.8. By Lemma 2.7, we can see that, for each , there is a corresponding unique point , thus we can define a mapping such that and it is easy to see that .

Remark 2.9. From the definition of , we can see that is a nonexpansive mapping. Actually, by Lemma 2.7, let ; then there exists a sequence of such that . Further, for any other point , by the definition of a u.a.r., we can get a subsequence of such that , then

Lemma 2.10 (see [35]). Let be a nonempty closed convex subset of a reflexive, smooth Banach space which admits a weakly sequentially continuous duality mapping from into , be a nonexpansive mapping such that , be a contraction mapping with a coefficient and be a strongly positive bounded linear operator with a coefficient . Let such that and which satisfies . Then the sequence defined by converges strongly to the common fixed point as , where is a unique solution in of the variational inequality

Lemma 2.11 (see [47]). Let be a closed convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on defined by for all is well defined, nonexpansive and holds.

Lemma 2.12 (see [48]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.13 (see [35]). Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and . Then .

If a Banach space admits a sequentially continuous duality mapping from weak topology to weak star topology, then by Lemma 1 of [49], we have that duality mapping is a single value. In this case, the duality mapping is said to be a weakly sequentially continuous duality mapping, that is, for each with , we have (see [4951] for more details).

A Banach space is said to be satisfying Opial's condition if for any sequence for all implies

By Theorem 1 in [49], it is well known that if admits a weakly sequentially continuous duality mapping, then satisfies Opial's condition, and is smooth.

Lemma 2.14 (see [50], Demiclosed principle). Let be a nonempty closed convex subset of a reflexive Banach space which satisfies Opial's condition and suppose is nonexpansive. Then the mapping is demiclosed at zero, that is, and implies .

Lemma 2.15 (see [30]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i), (ii) or . Then, .

Let be a real 2-uniformly smooth Banach space with the smoothness constant . Let be an -Lipschitzian and relaxed -cocoercive mapping, we defined a function by Consequence, we put

In order to prove our main result, the following lemmas are needed.

Lemma 2.16. Let be a real 2-uniformly smooth Banach space with the smoothness constant . Let be an -Lipschitzian and relaxed -cocoercive mapping. Then In particular, if , then is nonexpansive.

Proof. For all , from Lemma 2.2 and by the cocoercivity of the mapping , we have It follows that It is clear that, if . Thus, we have is nonexpansive.

Lemma 2.17. Let be a 2-uniformly smooth Banach space . Let be the a maximal monotone mapping and be an -Lipschitzian and relaxed -cocoercive mapping for . Let be a mapping defined by If and for all , then the mapping is nonexpansive.

Proof. For all , we have From Lemma 2.16 and by the nonexpansiveness of for all , we have is a nonexpansive mapping, which implies that the mapping is nonexpansive.

Lemma 2.18. For all , where and , is a solution of the problem (1.38) if and only if is a fixed point of the mapping defined as in Lemma 2.17.

Proof. Let be a solution of the problem (1.38). Then, we have which implies that We can deduce that (2.18) is equivalent to This completes the proof.

#### 3. Main Results

In this section, we prove that the iterative scheme (3.1) below converges strongly to common element of the set of solutions of the variational inclusion with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mapping and the set of fixed point of a family of nonexpansive semigroup in a uniformly convex and 2-uniformly smooth Banach space under some certain control conditions.

Now, we prove the strong convergence theorem of the sequence (3.1) for solving the problem (1.38).

Theorem 3.1. Let be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly continuous duality mapping and has the smoothness constant . Let be a maximal monotone mapping and be a -Lipschitzian and relaxed -cocoercive mapping with and for all . Let be a nonexpansive semigroup from into itself and at least there exists a which is demicompact. Assume that , where is defined as in Lemma 2.17. Let be a contraction mapping with a coefficient and be a strongly positive linear bounded self adjoint operator with a coefficient such that and . Let be a sequence defined by where , , are the sequences in which satisfies and is a positive real divergent sequence satisfy the following restrictions: (C1) and , (C2), (C3). Then the sequence defined by (3.1) converges strongly to , which solves the variational inequality and is a solution of general system of variational inequality problem (1.38), where and .

Proof. First, we show is bounded. By the condition (C1), we may assume, with no loss of generality, that . Since is a linear bounded operator on , by (1.35), we have , . Observe that that is, is positive. It follows that Taking , it follows from Lemma 2.17 that Putting and . Then . It follows from Lemmas 2.1 and 2.16 that and setting . From (3.6), we obtain It follows from (3.7) that By induction, we have Hence, is bounded, so are , , and . On the other hand, by nonexpansiveness of and for all , we have Now, we estimate for , it follows that It follows from (3.10) and (3.12) that where is an appropriate constant such that .
Setting , for all . Then , for all , we have