#### Abstract

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

#### 1. Introduction

In recent years, variational inequalities have appeared an interesting and dynamic field of pure and applied sciences. Variational techniques are being used to study a wide class of problem with applications in industry, structural engineering, mathematical finance, economics, optimization, transportation, and optimization problems. This has motivated to introduce and study several classes of variational inequalities. It is well known that the minimum of the differentiable convex functions on the convex set can be characterized by the variational inequalities. This result is due to Stampacchia [1]. However, we remark that if the convex function is directionally differentiable, then its minimum is characterized by a class of variational inequalities, which is called the bifunction variational inequality. For the formulation, applications, numerical results, and other aspects of bifunction variational inequalities, see [214]. Noor et al. [13] considered a new class of variational inequalities, which is called the trifunction variational inequality. It has been shown that the trifunction variational inequality includes the variational inequality and bifunction variational inequality as special cases.

Inspired and motivated by the ongoing research in this dynamic and fascinating field, we consider and analyze a new class of variational inequalities, called the mixed trifunction variational inequality. This new class of trifunction variational inequalities includes the trifunction(bifunction) variational inequality and the classical variational inequality as special cases.

There are a substantial number of numerical methods for solving the variational inequalities and trifunction equilibrium problems. Due to the nature of the trifunction variational inequality problem, projection methods and its variant form such as Wiener-Hopf equations cannot be used for solving the trifunction variational inequality. This fact motivated us to use the auxiliary principle technique of Glowinski et al. [15] as developed by Aslam Noor [16] and, Noor et al. [17]. This technique is quite flexible and general one. We again use this technique to suggest some explicit and proximal-point iterative methods for solving these problems. We also consider the convergence criteria of the proposed methods under suitable mild conditions, which is the main results (Theorems 3.4, 3.5, and 3.13) of this paper. Several special cases of our main results are also considered. Results obtained in this paper may be viewed as an improvement and refinement of the previously known results. The ideas and techniques of this paper stimulate further research in this area of pure and applied sciences.

#### 2. Preliminaries

Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a closed and convex set in .

For given trifunction and an operator , we consider the problem of finding such that which is called the trifunction variational inequality. We note that if , then problem (2.1) is studied in S. Takahashi and W. Takahashi [18], Yao et al. [19, 20], and Noor et al. [12, 14].

We now discuss some important special cases of the problem (2.1).

##### 2.1. Special Cases

(I) If , then problem (2.1) is equivalent to finding such that which is known as a variational inequality, introduced and studied by Stampacchia [1]. A wide class of problems arising in elasticity fluid flow through porous media, and optimization can be studied in the unified framework of problems (2.1). For the applications, formulation numerical results and other aspects of the variational inequalities and their generalizations, see [127].

(II) If , then problem (2.1) turns into the problem of finding such that which is known as trifunction variational inequalities, considered by Noor et al. [13].

For suitable and appropriate choice of the operator and spaces, one can obtain several new and known problems as special cases of the trifunction variational inequalities problems (2.1). For the applications, formulations, numerical methods, and other aspects of the variational inequalities, see [127].

Definition 2.1. An operator is said to be(i)monotone, if and only if, , for all ,(ii)partially relaxed strongly monotone, if there exists a constant such that Note that for , partially relaxed strong monotonicity reduces to monotonicity of the operator .

Definition 2.2. A trifunction with respect to an operator is said to be(a)jointly monotone if and only if (b)partially relaxed strongly jointly monotone if and only if there exists a constant such that It is clear that for , partially relaxed strongly jointly monotone trifunction is simply jointly monotone.

#### 3. Main Results

In this section, we suggest and analyze an iterative method for solving the trifunction variational-inequality problem (2.1) by using the auxiliary principle technique. This technique is mainly due to Glowinski et al. [15] as developed by Aslam Noor [16] and Noor et al. [814, 17, 25, 26].

For a given , consider the problem of finding such that where is a constant.

If , then is a solution of (2.1). This observation enables us to suggest and analyze the following iterative method for solving the mixed trifunction variational inequality (2.1).

Algorithm 3.1. For a given , compute from the iterative scheme We now discuss some special cases of Algorithm 3.1.
(III) If , then Algorithm 3.1 reduces to the following scheme for variational inequalities (2.2).

Algorithm 3.2. For a given , compute from the iterative scheme (IV) If , then Algorithm 3.1 reduces to the following.

Algorithm 3.3. For a given , compute from the iterative scheme which is used for finding the solution of trifunction variational inequality (2.3).

For suitable and appropriate choice of and spaces, one can define iterative algorithms for solving the different classes of trifunction variational inequalities and related optimization problems.

We now study the convergence analysis of Algorithm 3.1 using the technique of Noor et al. [814, 17, 25, 26], and this is the main motivation of our next result.

Theorem 3.4. Let be a solution of (2.1) and an approximate solution obtained from Algorithm 3.1. If the trifunction and the operator are partially relaxed strongly monotone operators with constants and , respectively, then

Proof. Let is a solution of (2.1), then, replacing by in (2.1), we have Let be the approximate solution obtained from Algorithm 3.1. Taking in (3.2), we have Adding (3.6) and (3.7), we have which implies that where we have used the partially relaxed strong monotonicity of the trifunction and operator .
Using the relation , for all , and from (3.9), one can have which is the required result (3.5).

Theorem 3.5. Let be a finite dimensional space. If is the approximate solution obtained from Algorithm 3.1 and be a solution of problem (2.1). Then .

Proof. Let be a solution of (2.1). For , we see that the sequence is nonincreasing and consequently is bounded. Also from (3.5), we have which implies that Let be the cluster point of and the subsequence of this sequence converges to . Replacing by in (3.2) and taking the limit as and using (3.12), we have which shows solves the trifunction variational inequality (2.1) and Thus, it follows from the above inequality that the sequence has exactly one cluster point and , the required result.

We again use the auxiliary principle technique to suggest and analyze several proximal point algorithms for solving the trifunction variational inequalities (2.1), and this is another motivation of this paper. Here, we show that the one can suggest several form of the auxiliary mixed trifunction variational inequalities type. Each type gives rise to different type of the inertial proximal point method for solving the mixed trifunction variational inequality (2.1). This is the beauty of the auxiliary principle technique. To convey an idea of the technique, we only consider some special cases. The interested readers are invited to explore the novel applications of this technique in other fields of the mathematical and engineering sciences.

(V) For a given , consider the problem of finding such that where and are constants.

Note that if , then is a solution of (2.1). This observation enables us to suggest and analyze the following iterative method for solving the mixed trifunction variational inequality (2.1).

Algorithm 3.6. For a given , compute from the iterative scheme Algorithm 3.6 is called the inertial proximal point method for solving the mixed trifunction variational inequality (2.1).
Note that if , then Algorithm 3.6 reduces to the following inertial proximal point method for solving (2.1).

Algorithm 3.7. For a given , compute from the iterative scheme (VI) For a given , consider the problem of finding a such that Note that if , then is a solution of (2.1). This observation enables us to suggest and analyze the following proximal iterative method for solving the mixed trifunction variational inequalities (2.1).

Algorithm 3.8. For a given , compute from the iterative scheme (VII) For a given , consider the problem of finding a such that Note that if , then is a solution of (2.1). This observation enables us to suggest and analyze the following iterative method for solving the mixed trifunction variational inequality (2.1).

Algorithm 3.9. For a given , compute from the iterative scheme Some special cases of these algorithms are as below.

If , then Algorithm 3.6 reduces to the following scheme for variational inequalities given as (2.2).

Algorithm 3.10. For a given , compute from the iterative scheme If , then Algorithm 3.6 reduces to the following.

Algorithm 3.11. For a given , compute from the iterative scheme which is used for solving the trifunction variational inequality (2.3).

We now again use the auxiliary mixed trifunction variational inequality (3.15) to suggest the following the implicit iterative method for solving the mixed trifunction variational inequality (2.1).

Algorithm 3.12. For a given , compute from the iterative scheme We remark that Algorithm 3.12 is quite different from Algorithms 3.63.11. One can easily show that the convergence of Algorithm 3.12 also requires the montonicity of the trifunction and the operator .

For suitable and appropriate choice of and spaces, one can define iterative algorithms as special cases of Algorithms 3.7 and 3.8 for solving the different classes variational inequalities and related optimization problems.

We would like to mention that one can study the convergence analysis of Algorithm 3.7 using the technique of Theorems 3.4 and 3.5. However, for the sake of completeness and to convey the main ideas, we include the main steps of the proof.

Theorem 3.13. Let be a solution of (2.1) and an approximate solution obtained from Algorithm 3.7. If trifunction and operator are monotone, then

Proof. Let be a solution of (2.1). Then, replacing by in (2.1), we have Let be the approximate solution obtained from Algorithm 3.6. Taking in (3.16), we have Adding (3.26) and (3.27), we have which implies that where we have used the monotonicity of the operator and the trifunction .
Using the relation , for all and from (3.29), we have which is the required result (3.25).

Theorem 3.14. Let be a finite dimensional space. If is the approximate solution obtained from Algorithm 3.6 and is a solution of problem (2.1), then .

Proof. Its proof is similar to the Proof of Theorem 3.5.

#### 4. Conclusion

In this paper, we have used the auxiliary principle technique to suggest and analyze several explicit and inertial proximal point algorithms for solving the trifunction equvariational inequality problem. We have also discussed the convergence criteria of the proposed new iterative methods under some suitable weaker conditions. In this sense, our results can be viewed as refinement and improvement of the previously known results. Note this technique does not involve the projection and the resolvent technique. We have also shown that this technique can be used to suggest several iterative methods for solving various classes of equilibrium and variational inequalities problems. Results proved in this paper may inspire further research in variational inequalities and related optimization problems.

#### Acknowledgments

This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia, and Research Grant no: KSU.VPP.108. The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.