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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 962978, 9 pages
http://dx.doi.org/10.1155/2012/962978
Research Article

Some New Classes of Extended General Mixed Quasi-Variational Inequalities

Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

Received 30 May 2012; Accepted 10 June 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Muhammad Aslam Noor and Khalida Inayat Noor. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider and study a new class of variational inequality, which is called the extended general mixed quas-variational inequality. We use the auxiliary principle technique to study the existence of a solution of the extended general mixed quasi-variational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

1. Introduction

Mixed quasi-variational inequalities are very important and significant generalization of variational inequalities involving the nonlinear bifunction. It is well known that a large class of problems arising in various branches of pure and applied sciences can be studied in the general framework of mixed quasi-variational inequalities. In recent years, Noor [14] has shown that the optimality conditions of the differentiable nonconvex functions involving two arbitrary functions can be characterized by a class of variational inequalities, which is called the extended general variational inequalities. For the numerical results, existence theory, and other aspects of the extended general variational inequalities, see [113] and the references therein. We would like to mention that one can show that the minimum of a sum of differentiable nonconvex (-convex) function and a nondifferentiable nonconvex (-convex) bifunction can be characterized by a class of variational inequality. Motivated by this result, we introduce a new class of mixed variational inequalities, which is called extended general mixed quasi-variational inequality involving four different operators. Due to the nature of the problem, one cannot use the projection and resolvent operator techniques to study the existence of such type of the variational inequalities. To overcome this drawback, one uses the auxiliary principle technique. This technique is mainly due to Glowinski et al. [14]. This technique is more flexible and has been used to develop several numerical methods for solving the variational inequalities and the equilibrium problems. Noor [1, 9] has used this technique to study the existence of the extended general mixed variational inequality and its variant forms. We again use the auxiliary principle technique to study the existence of a solution of the extended general mixed quasi-variational inequalities. Since the extended general variational inequalities include various classes of variational inequalities and complementarity problems as special cases, results proved in this paper continue to hold for these problems. Results proved in this paper may be viewed as important and significant improvement of the previously known results. It is interesting to explore the applications of these extended general variational inequalities in mathematical and engineering sciences with new and novel aspects. This may lead to new research in this field.

2. Preliminaries

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

For given nonlinear operators , consider the problem of finding such that Inequality of type (2.1) is called the extended general mixed quasi-variational inequality.

One can show that the minimum of a sum of differentiable nonconvex (-convex) function and a nondifferentiable nonconvex (-convex) function on the -convex set in can be characterized by the extended general mixed quasi-variational inequality (2.1). For this purpose, we recall the following well known concepts, see [1, 5, 1518].

Definition 2.1. Let be any set in . The set is said to be -convex, if there exist functions such that
Note that every convex set is -convex, but the converse is not true, see [15]. If , then the -convex set is called the -convex set, which was introduced by Youness [19]. See also Cristescu and Lupşa [15] for its various extensions and generalization.

Definition 2.2. The function is said to be -convex on the -convex set , if there exist two functions such that
Clearly every convex function is -convex, but the converse is not true. For , Definition 2.2 is due to Youness [19].
We now consider the function , defined by
Using the technique of Noor [2, 4, 5], one can easily show that the minimum of a differentiable -convex function and a nondifferentiable -convex bifunction on a -convex set in can be characterized by the extended general mixed quasi-variational inequality (2.1).

Lemma 2.3. Let be a differentiable -convex function, and let be a nondifferentiable bifunction on the -convex set . Then is the minimum of the functional , defind by (2.4) on the -convex set , if and only if, satisfies the inequality where is the differential of at .

Lemma 2.3 implies that -convex programming problem can be studied via the extended general mixed variational inequality (2.1) with .

We now list some special cases of the extended general variational inequalities.(i)If , then Problem(2.1) is equivalent to finding such that which is known as general mixed quasi-variational inequality, introduced and studied by Noor [20, 21]. It turned out that odd order and nonsymmetric obstacle, free, moving, unilateral, and equilibrium problems arising in various branches of pure and applied sciences can be studied via general variational inequalities.

We would like to emphasize the fact that general variational inequalities (2.6), (2.5), and (2.6) are quite different from each other and have different applications. For , the identity operator, the extended general variational inequality (2.1) is equivalent to finding such that which is known as the classical mixed quasi-variational inequality. For the applications, formulation and numerical methods for solving the mixed quasi-variational inequalities (2.7), see [114, 1618, 2034]. If the bifunction is the indicator function of a closed convex-valued set in , that is, then problem (2.1) is equivalent to finding such that Problems of type (2.9) are called extended general quasi-variational inequalities, studied by Noor [10]. If , the convex set in , Then, problem (2.9) is equivalent to finding such that which is called the extended general variational inequality, introduced and studied by Noor [14]. For the formulation, applications, numerical method and other aspects of the extended general variational inequality (2.10), see [14, 612, 35, 36].

We would like to mention that one can obtain several known and new classes of variational inequalities as special cases of the problem (2.1). From the above discussion, it is clear that the extended general mixed quasi-variational inequalities (2.1) is most general and includes several previously known classes of variational inequalities and related optimization problems as special cases. These variational inequalities have important applications in mathematical programming and engineering sciences, see [140].

We also need the following concepts and results.

Definition 2.4. For all , an operator is said to be:(i)strongly monotone, if there exists a constant such that (ii)Lipschitz continuous, if there exists a constant such that
From (i) and (ii), it follows that .

Remark 2.5. It follows from the strong monotonicity of the operator , that which implies that
This observation enables us to define the following concept.

Definition 2.6. The operator is said to firmly expanding if

Definition 2.7. The bifunction is said to be skew-symmetric, if,
Clearly, if the bifunction is linear in both arguments, then, which shows that the bifunction is nonnegative.

Remark 2.8. It is worth mentioning that the points , and make up a set of the four vertices of the square. In fact, the skew-symmetric bifunction can be written in the form This shows that the arithmetic average value of the skew-symmetric bifunction calculated at the north-east and south-west vertices of the square is greater than or equal to the arithmetic average value of the skew-symmetric bifunction computed at the north-west and south-west vertices of the same square. The skew-symmetric bifunction have the properties which can be considered an analogs of monotonicity of gradient and nonnegativity of a second derivative for the convex functions.

3. Main Results

In this section, we use the auxiliary principle technique of Glowinski et al. [14] as developed by Noor [1, 8, 20, 21] and M. A. Noor and K. I. Noor [13] to study the existence of a solution of the extended general mixed quasi-variational inequality (2.1).

Theorem 3.1. Let be a strongly monotone with constant and Lipschitz continuous with constant . Let be a strongly monotone and Lipschitz continuous operator with constants and , respectively. Let the bifunction be skew-symmetric. If the operator is firmly expanding and there exists a constant such that where then the extended general mixed quasi-variational inequality (2.1) has a unique solution.

Proof. We use the auxiliary principle technique to prove the existence of a solution of (2.1). For a given satisfying the extended general mixed quasi-variational inequality (2.1), we consider the problem of finding a solution such that where is a constant.
The inequality of type (3.4) is called the auxiliary extended general mixed variational inequality associated with the problem (2.1). It is clear that the relation (3.4) defines a mapping . It is enough to show that the mapping defined by the relation (3.4) has a unique fixed point belonging to satisfying the general variational inequality (2.1). Let be two solutions of (3.4) related to , respectively. It is sufficient to show that for a well chosen , with , where is independent of and . Taking (resp., ) in (3.4) related to (resp., ), adding the resultant, and using the skew-symmetry of the bifunction , we have from which we have Since is both strongly monotone and Lipschitz continuous operator with constants and , respectively, it follows that In a similar way, using the strong monotonicity with constant and Lipschitz continuity with constant , we have From (3.2), (3.3), (3.7), (3.8), and (3.9), and using the fact that the operator is firmly expanding, we have From (3.1), it follows that showing that the mapping defined by (3.4) has a fixed point belonging to , which is solution of (2.1), the required result.

We note that if , then is a solution of the extended general mixed quasi variational inequality (2.1). This observation enables us to suggest the following iterative method for solving the extended general mixed quasi-variational inequality (2.1).

Algorithm 3.2. For a given , find the approximate solution by the iterative scheme: where is a constant. Algorithm 3.2 is called the explicit iterative method. For different and suitable choice of the operators and spaces, one can obtain various iterative methods for solving the quasi-variational inequalities and its variant forms. We leave this to the interested readers.

4. Conclusion

In this paper, we have introduced and studied a new class of variational inequalities, which is called the extended general mixed variational inequality. We have shown that this class is related to the optimality conditions of the nonconvex differentiable functions. One can easily obtain various classes of variational inequalities as special cases of this new class. We have used the auxiliary principle technique to study the existence of a solution of the extended general mixed quasi-variational inequalities under some suitable conditions. Our technique does not involve the projection or resolvent operator. We expect that the results proved in this paper may stimulate further research in this field. The interested readers are encouraged to find the novel and new applications of the extended general mixed quasi-variational inequalities in various branches of pure and applied sciences.

Acknowledgments

The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, and COMSATS Institute of Information Technology, Pakistan, for providing excellent research facilities.

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