#### Abstract

A new system of generalized mixed quasivariational inclusions (for short, SGMQVI) with relaxed cocoercive operators, which develop some preexisting variational inequalities, is introduced and investigated in Banach spaces. Next, the existence and uniqueness of solutions to the problem (SGMQVI) are established in real Banach spaces. From fixed point perspective, we propose some new iterative algorithms for solving the system of generalized mixed quasivariational inclusion problem (SGMQVI). Moreover, strong convergence theorems of these iterative sequences generated by the corresponding algorithms are proved under suitable conditions. As an application, the strong convergence theorem for a class of bilevel variational inequalities is derived in Hilbert space. The main results in this paper develop, improve, and unify some well-known results in the literature.

#### 1. Introduction

Generalized mixed quasivariational inclusion problems, which are extensions of variational inequalities introduced by Stampacchia [1] in the early sixties, are among the most interesting and extensively investigated classes of mathematics problems and have many applications in the fields of optimization and control, abstract economics, electrical networks, game theory, auction, engineering science, and transportation equilibria (see, e.g., [2–5] and the references therein). For the past few decades, existence results and iterative algorithms for variational inequality and variational inclusion problems have been obtained (see, e.g., [6–14] and the references cited therein). Recently, some new problems, which are called to be the system of variational inequality and equilibrium problems, received many attentions. Ansari et al. [2] considered a system of vector variational inequalities and obtained its existence results. In [3], Pang stated that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Verma [15] and J. K. Kim and D. S. Kim [16] investigated a system of nonlinear variational inequalities. Cho et al. [17] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces and obtained the existence and uniqueness properties of solutions for the system of nonlinear variational inequalities. In [18], Peng and Zhu introduced a new system of generalized mixed quasivariational inclusions involving -monotone operators. Very recently, Qin et al. [19] studied the approximation of solutions to a system of variational inclusions in Banach spaces and established a strong convergence theorem in uniformly convex and 2 uniformly smooth Banach spaces. In [20], Kamraksa and Wangkeeree gave a general iterative method for a general system of variational inclusions and proved a strong convergence theorem in strictly convex and 2 uniformly smooth Banach spaces. Further, Wangkeeree and Kamraksa [21] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities and then obtained the strong convergence of the iterative in Hilbert spaces. Petrot [22] applied the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. Zhao et al. [23] obtained some existence results for a system of variational inequalities by Brouwer’s fixed point theory and proved the convergence of an iterative algorithm in finite Euclidean spaces. Chen and Wan [24] also proved the existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces, proposed some iterative methods for finding the common element of the solutions set for the system of quasivariational inclusions and the fixed point set for Lipschitz mapping, and obtained the convergent rates of corresponding iterative sequences. On the other hand, various bilevel programming problems, bilevel decision problems, and mathematical program problems with equilibrium (variational inequalities) constraints have been wildly investigated (see, e.g., [25, 26]). To the best of our knowledge, there is few results concerning the algorithms and convergence analysis of solutions to bilevel variational inequalities in Hilbert spaces.

The aim of this paper is to introduce and study a new system of generalized mixed quasivariational inclusion problem (SGMQVI) in uniformly smooth Banach spaces which includes some previous variational inequalities as special cases. Furthermore, the existence and uniqueness theorems of solutions for the problem (SGMQVI) are established by using resolvent techniques. Thirdly, we also propose some new iterative algorithms for solving the problem (SGMQVI). Strong convergence of the iterative sequences generated by the corresponding iterative algorithms are proved under suitable conditions. As an application, we study the properties for the lower-level variational inequalities of a class of bilevel variational inequalities (for short, (BVI)) in Hilbert spaces and then suggest a reasonable iterative algorithm for (BVI). Finally, the strong convergence theorem for (BVI) are derived under appropriate assumptions. The results presented in this paper improve, develop, and extend the results of [8, 23, 24, 27].

#### 2. Preliminaries

Throughout this paper, let be a real -uniformly Banach space with its dual , denote the duality between and by and the norm of by , and let be a nonlinear mapping. If is a sequence in , we denote strong convergence of to by . A Banach space is called smooth if exists for all with . It is called uniformly smooth if the limit is attained uniformly for . The function is called the modulus of smoothness of . is called -uniformly smooth if there exists a constant such that .

*Example 2.1 (see [4]). *All Hilbert spaces, , and the Sobolev spaces , are 2-uniformly smooth, while and spaces are -uniformly smooth.

The generalized duality mapping defined as follows:
for all . As we know that is the usual normalized duality mapping, and for , , and for all and , and is single-valued if is smooth (see, e.g., [28]). If is a Hilbert space, then , where is the identity operator. Let , let be single-valued mappings, and let be set-valued mappings for all . We consider the system of generalized mixed quasivariational inclusions problem (for short, (SGMQVI)) as follows: find such that
where are positive constants. Denote the set of solutions to (SGMQVI) by .

Special cases are as follows:

(I) If and for all and , where are single-valued mappings, then the problem (SGMQVI) is equivalent to find such that
where are positive constants, which is called the system of generalized nonlinear mixed variational inclusions problem [8].

(II) If is a Hilbert space, and for all , where is a proper, convex, and lower semicontinuous functional, and denotes the subdifferential operator of , then the problem (SGMQVI) is equivalent to find such that
where are positive constants, which is called the generalized system of relaxed cocoercive mixed variational inequality problem [29].

(III) If is a Hilbert space, and is a closed convex subset of , and for all , where is the indicator function of defined by
then the problem (SGMQVI) is equivalent to find such that
where are positive constants, which is called the system of general variational inequalities problem [27].

(IV) If is a Hilbert space, and is a closed convex subset of , , and for all , where is the indicator function of defined by
then the problem (SGMQVI) is equivalent to find such that
where are positive constants, which is called the generalized system of relaxed cocoercive variational inequality problem [30].

(V) If for each , and for all , where , then the problem (SGMQVI) is equivalent to find such that
where are positive constants, which is called the system of quasivariational inclusion [19, 20].

(VI) If , for each , and for all , where and , then the problem (SGMQVI) is equivalent to find such that
where are positive constants, which is called the system of quasivariational inclusion [20].

We first recall some definitions and lemmas which are needed in our main results.

*Definition 2.2. *Let be a set-valued mapping, where is the effective domain of the mapping . is said to be (i)accretive if, for any and , there exists such that
(ii)-accretive (maximal-accretive) if is accretive and holds for every , where is the identity operator on .

*Remark 2.3. *If is a Hilbert space, then accretive operator and -accretive operator are reduced to monotone operator and maximal monotone operator, respectively.

*Definition 2.4 (see [24, 31]). *Let be a single-valued mapping. is said to be(i)-Lipschitz continuous mapping if there exists a constant such that
(ii)-relaxed cocoercive if there exist two constants and such that

*Remark 2.5 (see [24]). *(1) If , then a -Lipschitz continuous mapping reduces to a nonexpansive mapping.

(2) If , then a -Lipschitz continuous mapping reduces to a contractive mapping.

(3) It is easy to see that the identity operator is relaxed cocoercive, where for all .

*Definition 2.6 (see [24]). *Let be a mapping. is said to be (i)-Lipschitz continuous in the first variable if there exists a constant such that, for ,
(ii)-strongly accretive if there exists a constant such that
or equivalently,
(iii) relaxed cocoercive if there exist two constants and such that

*Remark 2.7. *(1) Every -strongly accretive mapping is a relaxed cocoercive for any positive constant . But the converse is not true in general.

(2) The conception of the cocoercivity is applied in several directions, especially for solving variational inequality problems by using the auxiliary problem principle and projection methods [14]. Several classes of relaxed cocoercive variational inequalities have been investigated in [5, 22, 28, 30].

*Definition 2.8 (see [9, 32]). *Let the set-valued mapping be -accretive. For any positive number , the mapping defined by
is called the resolvent operator associated with and , where is the identity operator on .

*Remark 2.9. *Let be a nonempty closed convex set. If is a Hilbert space and , the subdifferential of the indicator function , that is,
then , the metric projection operator from onto .

Lemma 2.10 (see [9, 32]). *Let the set-valued mapping be -accretive. Then the resolvent operator is single-valued and nonexpansive for all .*

Lemma 2.11 (see [33]). *Let , and be three nonnegative real sequences satisfying the following conditions:
**
for some with , and . Then .*

Lemma 2.12 (see [34]). *Let be a real -uniformly Banach space. Then there exists a constant such that
*

#### 3. Existence Theorems

In this section, we will investigate the existence and uniqueness of solutions for the problem (SGMQVI) in -uniformly smooth Banach space under some suitable conditions.

Theorem 3.1. *Let , be maximal accretive. Then is a solution of the problem (SGMQVI) if and only if
**
where are positive constants.*

*Proof. *Let be a solution of the problem (SGMQVI). Then, for any given positive constants , the problem (SGMQVI) is equivalent to
From Definition 2.8 and Lemma 2.10, it yields that (3.2) is equivalent to (3.1). This completes the proof.

Theorem 3.2. *Let be a real -uniformly smooth Banach space. Let , be -accretive mapping, Let be -relaxed cocoercive and Lipschitz continuous in the first variable with constant , and Let be -relaxed cocoercive and Lipschitz continuous with constant . Then, for each , the mapping has at most one fixed point. If
**
then the implicit function determined by
**
is continuous on .*

*Proof. *Let . We show by contradiction that has at most one fixed point. Suppose to the contrary that and such that
Since is Lipschitz continuous in the first variable with constant , then
which is a contradiction. Therefore, the mapping has at most one fixed point.

Next, we show that the implicit function is continuous on . For any sequence, as . Since is -relaxed cocoercive and Lipschitz continuous in the first variable with constant and is -relaxed cocoercive and Lipschitz continuous with constant , one has
Therefore, from Lemma 2.10, we get
From (3.3), it follows that the implicit function is continuous on . This completes the proof.

If and for all , then Theorem 3.2 is reduced to the following result.

Corollary 3.3 (see [24]). *Let be a real -uniformly smooth Banach space. Let be -accretive mapping; Let be -relaxed cocoercive and Lipschitz continuous in the first variable with constant . Then, for each , the mapping has at most one fixed point. If
**
then the implicit function determined by
**
is continuous on .*

Theorem 3.4. *Let be a real -uniformly smooth Banach space. Let be -accretive mapping, Let be -relaxed cocoercive and Lipschitz continuous in the first variable with constant , and let be -relaxed cocoercive and Lipschitz continuous with constant for . Assume that
**
Then the problem (SGMQVI) has a solution. Moreover, the solutions set of (SGMQVI) is a singleton.*

*Proof. *By Theorem 3.1, the problem (SGMQVI) has a solution if and only if (3.1) holds. For the convenience, we define a mapping by
Since are -relaxed cocoercive and Lipschitz continuous in the first variable with constant and are -relaxed cocoercive and Lipschitz continuous with constant for , by Theorem 3.2, we know that and are continuous on . For any ,
From Lemma 2.10, it yields that
Note that, for each ,
Therefore, we obtain
It follows from (3.12) that the mapping is contractive. By banach’s contraction principle, there exists a unique such that . Therefore, by Theorem 3.2, there exists an unique such that is a solution of the problem (SGMQVI), where for , that is, . This completes the proof.

#### 4. Convergence Analysis

In this section, we introduce several implicit algorithms with errors and explicit algorithms without errors for solving the system of generalized mixed quasivariational inclusions problem (SGMQVI) and then explore the convergence analysis of the iterative sequences generated by the corresponding algorithms.

From Section 3, we know that the system of generalized mixed quasivariational inclusions problem (SGMQVI) is equivalent to the fixed point problem (3.1). This equivalent formulation is crucial from the numerical analysis point of view. As we know, this fixed point formulation has been used to suggest and analyze some iterative methods for solving variational inequalities and related optimization problems. By using the relations between the problem (SGMQVI) and the fixed point problem (3.1), we construct the following iterative algorithms for solving the system of generalized mixed quasivariational inclusions problem (2.3).

*Algorithm 4.1. *Let be positive constants for all . For any given points , define sequences in by the following implicit algorithm:
where and is a real sequence in .

If for all , then Algorithm 4.1 is reduced to the following result.

*Algorithm 4.2. *Let be positive constants for all . For any given points , define sequences in by the following implicit algorithm
where is a real sequence in .

Now we construct an explicit algorithms for solving the system of generalized mixed quasivariational inclusions problem (SGMQVI).

*Algorithm 4.3. *Let be positive constants for all . For any given points , define sequences in by the following explicit algorithm
where is a real sequence in .

*Remark 4.4. *If is a Hilbert space, and is a closed convex subset of , for all , and for all , where is the indicator function of , and denotes the subdifferential operator of , then, from Remark 2.9, Algorithms 4.1 and 4.3 are reduced to the Algorithms 4.5 and 4.6 for solving the system of general variational inequalities problem (2.7).

*Algorithm 4.5. *Let be positive constants for all . For any given points , define sequences in by the following implicit algorithm:
where and is a real sequence in .

*Algorithm 4.6. *Let be positive constants for all . For any given points , define sequences in by the following explicit algorithm:
where is a real sequence in .

Theorem 4.7. *Let be a real -uniformly smooth Banach space, and let , and be the same as in Theorem 3.4. Assume that is a real sequence in and satisfy the following conditions:*(i)*;*(ii)*;*(iii)*;*(iv)*.**Then the sequences generated by Algorithm 4.1 converge strongly to , respectively, such that .*

*Proof. *By Theorem 3.4, we know that there exist an unique point such that . Then, from Theorem 3.1, one has
Therefore, from both (4.1) and (4.6), we have
Since are -relaxed cocoercive and Lipschitz continuous in the first variable with constant and are -relaxed cocoercive and Lipschitz continuous with constant for , we can conclude