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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.

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 [

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. [

Let

For given trifunction

We now discuss some important special cases of the problem (

(I) If

(II) If

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 (

An operator

monotone, if and only if,

partially relaxed strongly monotone, if there exists a constant

A trifunction

jointly monotone if and only if

partially relaxed strongly jointly monotone if and only if there exists a constant

In this section, we suggest and analyze an iterative method for solving the trifunction variational-inequality problem (

For a given

If

For a given

(III) If

For a given

For a given

For suitable and appropriate choice of

We now study the convergence analysis of Algorithm

Let

Let

Using the relation

Let

Let

We again use the auxiliary principle technique to suggest and analyze several proximal point algorithms for solving the trifunction variational inequalities (

(V) For a given

Note that if

For a given

Note that if

For a given

For a given

For a given

If

For a given

For a given

We now again use the auxiliary mixed trifunction variational inequality (

For a given

For suitable and appropriate choice of

We would like to mention that one can study the convergence analysis of Algorithm

Let

Let

Using the relation

Let

Its proof is similar to the Proof of Theorem

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.

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.