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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 701858, 8 pages
http://dx.doi.org/10.1155/2012/701858
Research Article

Existence Results for General Mixed Quasivariational Inequalities

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

Received 3 March 2012; Accepted 16 March 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 general mixed quasivariational inequality. We use the auxiliary principle technique to study the existence of a solution of the general mixed quasivariational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

1. Introduction

Variational inequalities theory, which was introduced and studied in the 1960’s has seen a dramatic increase in its application in various branches of pure and applied sciences. Variational inequalities have been extended and generalized in various directions using novel and innovative ideas. A useful and important generalization is called the mixed quasi variational inequality involving the bifunction. It has been shown that a wide class of problems which arise in the elasticity with nonlocal friction laws, fluid flow thorough porous media and structural analysis can be studied in the unified framework of the mixed quasi variational inequalities, see [112].

In recent years, Noor [13] has shown that the optimality conditions of the differentiable nonconvex functions involving one arbitrary functions can be characterized by a class of variational inequalities, which is called the general variational inequalities. We would like to mention that one can show that the minimum of the sum of differentiable nonconvex (𝑔-convex) function and a nondifferentiable 𝑔-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 general mixed quasi variational inequality involving the bifunction and three different operators. Due to the presence of the bifunction, projection and resolvent operator techniques, and their variation forms cannot be extended for solving the general mixed quasi variational inequalities. Thanks to the auxiliary principle technique, one can overcome this drawback. This technique is mainly due to Glowinski et al. [3]. This technique is more flexible and has been used to develop several numerical methods for solving the variational inequalities and the equilibrium problems. Noor [10, 14] has used this technique to study the existence of the general mixed quasi variational inequalities. This technique deals with considering an auxiliary problem and proving that the solution of the auxiliary problem is the solution of the original problem using the fixed point theory. This technique does not involve projection and resolvent of the operator. We again use the auxiliary principle technique to study the existence of a solution of the general mixed quasi variational inequalities, which is the main result (Theorem 3.1). We use this technique to suggest and analyze an iterative method for solving the general mixed quasi variational inequalities. Since the general mixed quasi 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 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 𝐻 be a real Hilbert space whose inner product and norm are denoted by , and , respectively. Let 𝐾 be a nonempty closed and convex set in 𝐻. Let 𝜑(,)𝐻×𝑅{} be a continuous bifunction.

For given nonlinear operators 𝑇,𝑔,𝐻𝐻, consider the problem of finding 𝑢𝐻 such that𝑇𝑢,𝑔(𝑣)𝑢+𝜑(𝑔(𝑣),𝑢)𝜑(𝑢,𝑢)0,𝑣𝐻𝑔(𝑣)𝐻.(2.1) Inequality of type (2.1) is called the general mixed quasi variational inequality.

One can show that the minimum of sum of differentiable nonconvex (𝑔-convex) function and a class of nondifferentiable nonconvex (𝑔-convex) function on the 𝑔-convex set 𝐾 in 𝐻 can be characterized by general mixed quasi variational inequality (2.1). For this purpose, we recall the following well-known concepts, see [10, 13, 1518].

Definition 2.1. Let 𝐾 be any set in 𝐻. The set 𝐾 is said to be 𝑔-convex if there exists a function 𝑔𝐻𝐻 such that [].𝑢+𝑡(𝑔(𝑣)𝑢)𝐾,𝑢,𝑣𝐻𝑢,𝑔(𝑣)𝐾,𝑡0,1(2.2)

Note that every convex set is 𝑔-convex, but the converse is not true, see [15].

Definition 2.2. The function 𝐹𝐾𝐻 is said to be 𝑔-convex on the 𝑔-convex set 𝐾 if there exists a function 𝑔 such that [].𝐹(𝑢+𝑡(𝑔(𝑣)𝑢))(1𝑡)𝐹(𝑢)+𝑡𝐹(𝑔(𝑣)),𝑢,𝑣𝐻𝑢,𝑔(𝑣)𝐾,𝑡0,1(2.3)

Clearly every convex function is 𝑔-convex, but the converse is not true 𝐼[𝑣]=𝐹(𝑣)+𝜑(𝑣,𝑣),𝑣𝐻.(2.4)

Using the technique of Noor [12, 18, 19], one can easily show the minimum of a differentiable 𝑔-convex function, and nondifferentiable nonconvex bifunction on a 𝑔-convex set 𝐾 in 𝐻 can be characterized by the general mixed quasi variational inequality (2.1).

Lemma 2.3. Let 𝐹𝐾𝐻 be a differentiable 𝑔-convex function on the 𝑔-convex set 𝐾. Then 𝑢𝐾 is the minimum of the functional 𝐼[𝑣] defined by (2.4) on the 𝑔-convex set 𝐾 if and only if 𝑢𝐾 satisfies the inequality 𝐹(𝑢),𝑔(𝑣)𝑢+𝜑(𝑔(𝑣),𝑢)𝜑(𝑢,𝑢)0,𝑣𝐻𝑔(𝑣)𝐾,(2.5) where 𝐹(𝑢) is the differential of 𝐹 at 𝑢𝐾.

Lemma 2.3 implies that 𝑔-convex programming problem can be studied via the general mixed variational inequality (2.1) with 𝑇𝑢=𝐹(𝑢).

We now list some special cases of the general mixed quasi variational inequality (2.1).(I)For 𝑔=𝐼, the identity operator, the general mixed quasi variational inequality (2.1) is equivalent to finding 𝑢𝐻 such that 𝑇𝑢,𝑣𝑢+𝜑(𝑢,𝑢)𝜑(𝑢,𝑢)0,𝑣𝐻𝑔(𝑣)𝐻,(2.6) which is also called the mixed quasi variational inequality, see [13, 8, 9, 11, 12].(II)If the bifunction 𝜑(,) is the indicator function of a closed convex-valued set 𝐾(𝑢) in 𝐻, that is, 𝜑(𝑢,𝑢)=𝐾(𝑢)(𝑢)=0,𝑢𝐾(𝑢),+,otherwise,(2.7) then problem (2.1) is equivalent to finding 𝑢𝐾(𝑢) such that 𝑇𝑢,𝑔(𝑣)𝑢0,𝑣𝐻𝑔(𝑣)𝐾(𝑢).(2.8) Problems of type (2.8) are called general quasi variational inequalities.(III)If 𝐾(𝑢)𝐾, the convex set, then problem (2.8) is equivalent to finding 𝑢𝐾 such that 𝑇𝑢,𝑔(𝑣)𝑢0,𝑣𝐻𝑔(𝑣)𝐾,(2.9) which is called the general variational inequality, introduced and studied by Noor [13].(IV)If 𝑔=𝐼, the identity operator, then, problem (2.5) is equivalent to finding 𝑢𝐾(𝑢) such that 𝑇𝑢,𝑣𝑢0,𝑣𝐾(𝑢),(2.10) which is called the quasi variational inequality.(V) If 𝐾(𝑢)=𝐾, the convex set, then problem (2.10) is equivalent to finding 𝑢𝐾 such that 𝑇𝑢,𝑣𝑢0,𝑣𝐾,(2.11) which is called the classical variational inequality, introduced and studied by Stampacchia [20]. For the applications, formulations, generalizations, numerical method and other aspects of the variational inequalities, see [130] and the references therein.

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 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.

We also need the following concepts and results.

Definition 2.4. For all 𝑢,𝑣𝐻, an operator 𝑇𝐻𝐻 is said to be(i)strongly monotone, if there exists a constant 𝛼>0 such that 𝑇𝑢𝑇𝑣,𝑢𝑣𝛼𝑢𝑣2,(2.12)(ii)Lipschitz continuous, if there exists a constant 𝛽>0 such that𝑇𝑢𝑇𝑣𝛽𝑢𝑣.(2.13)

From (i) and (ii), it follows that 𝛼𝛽.

Definition 2.5. The bifunction 𝜑(,) is said to be skew symmetric, if 𝜑(𝑢,𝑢)𝜑(𝑢,𝑣)𝜑(𝑣,𝑢)+𝜑(𝑣,𝑣)0,𝑢,𝑣𝐻.(2.14)

Clearly, if the bifunction 𝜑(,) is linear in both arguments, then𝜑(𝑢,𝑢)𝜑(𝑢,𝑣)𝜑(𝑣,𝑢)+𝜑(𝑣,𝑣)=𝜑(𝑢𝑣,𝑢𝑣)0,𝑢,𝑣𝐻,(2.15)

which shows that the bifunction 𝜑(,) is nonnegative.

Remark 2.6. It is worth mentioning that the points (𝑢,𝑢),(𝑢,𝑣),(𝑣,𝑢),and(𝑣,𝑣) make up a set of the four vertices of the square. In fact, the skew-symmetric bifunction 𝜑(,) can be written in the form 121𝜑(𝑢,𝑢)+21𝜑(𝑣,𝑣)21𝜑(𝑢,𝑣)+2𝜑(𝑣,𝑢),𝑢,𝑣𝐻.(2.16) 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 has 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. [3] to study the existence of a solution of the general mixed quasi variational inequality (2.1).

Theorem 3.1. Let 𝑇 be a strongly monotone with constant 𝛼>0 and Lipschitz continuous with constant 𝛽>0. Let 𝑔 be a strongly monotone and Lipschitz continuous operator with constants 𝜎>0 and 𝛿>0, respectively. Let the bifunction 𝜑(,) be skew symmetric. If there exists a constant 𝜌>0 such that ||||𝛼𝜌𝛽2||||<𝛼2𝛽2𝑘(2𝑘)𝛽2,𝛼>𝛽𝑘(2𝑘),𝑘<1,(3.1) where 𝜃=𝑘+12𝜌𝛼+𝜌2𝛽2,(3.2)𝑘=12𝜎+𝛿2.(3.3) then the 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 𝑢𝐾 satisfying the general mixed quasi variational inequality (2.1), we consider the problem of finding a solution 𝑤𝐾 such that 𝜌𝑇𝑢+𝑤𝑔(𝑢),𝑔(𝑣)𝑤+𝜌𝜑(𝑔(𝑣),𝑤)𝜌𝜑(𝑤,𝑤)0,𝑣𝐻𝑔(𝑣)𝐾,(3.4) where 𝜌>0 is a constant. The inequality of type (3.4) is called the auxiliary general mixed quasi variational inequality associated with the problem (2.1). It is clear that the relation (3.4) defines a mapping 𝑢𝑤. It is enough to show that the mapping 𝑢𝑤 defined by the relation (3.4) has a unique fixed point belonging to 𝐻 satisfying the general mixed quasi variational inequality (2.1). Let 𝑤1𝑤2 be two solutions of (3.4) related to 𝑢1,𝑢2𝐻, respectively. It is sufficient to show that for a well chosen 𝜌>0, 𝑤1𝑤2𝑢𝜃1𝑢2,(3.5) with 0<𝜃<1, where 𝜃 is independent of 𝑢1 and 𝑢2. Taking 𝑔(𝑣)=𝑤2 (respectively 𝑤1) in (3.4) related to 𝑢1 (respectively 𝑢2), adding the resultant and using the skew-symmetry of the bifunction 𝜑(,), we have 𝑤1𝑤2,𝑤1𝑤2𝑢𝑔1𝑢𝑔2𝜌𝑇𝑢1𝑇𝑢2,𝑤1𝑤2,(3.6) from which we have 𝑤1𝑤2𝑔𝑢1𝑢𝑔2𝜌𝑇𝑢1𝑇𝑢2𝑢1𝑢2𝑔𝑢1𝑢𝑔2+𝑢1𝑢2𝜌𝑇𝑢1𝑇𝑢2.(3.7) Since 𝑇 is both strongly monotone and Lipschitz continuous operator with constants 𝛼>0 and 𝛽>0 respectively, it follows that 𝑢1𝑢2𝜌𝑇𝑢1𝑇𝑢22𝑢2𝑢222𝜌𝑢1𝑢2,𝑇𝑢1𝑇𝑢2+𝜌2𝑇𝑢1𝑇𝑢2212𝜌𝛼+𝜌2𝛽2𝑢1𝑢22.(3.8) In a similar way, using the strongly monotonicity with constant 𝜎>0 and Lipschitz continuity with constant 𝛿>0, we have 𝑢1𝑢2𝑔𝑢1𝑢𝑔212𝜎+𝛿2𝑢1𝑢2.(3.9) From (3.7), (3.8), and (3.9) and using the fact that the operator is firmly expanding, we have 𝑤1𝑤2𝑘+12𝜌𝛼+𝜌2𝛽2𝑢1𝑢2𝑢=𝜃1𝑢2.(3.10) From (3.1) and (3.2), it follows that 𝜃<1 showing that the mapping defined by (3.4) has a fixed point belonging to 𝐾, which is the solution of (2.1), the required result.

We note that if 𝑤=𝑢, then 𝑤 is a solution of the general mixed quasi variational inequality (2.1). This observation enables us to suggest and analyze the following iterative method for solving the general mixed quasi variational inequality (2.1), and this is one of the main motivation of this paper.

Algorithm 3.2. For a given 𝑢0𝐻, find the approximate solution 𝑢𝑛+1 by the iterative scheme 𝜌𝑇𝑢𝑛+𝑢𝑛+1𝑢𝑔𝑛,𝑔(𝑣)𝑢𝑛+1𝑔+𝜌𝜑(𝑣),𝑢𝑛+1𝑢𝜌𝜑𝑛+1,𝑢𝑛+10,𝑣𝐻,(3.11) where 𝜌>0 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. One can consider the convergence analysis of Algorithm 3.2 using the technique of Noor [24]. 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 general mixed quasi variational inequality. We have shown that this class is related 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 general mixed quasi variational inequalities under some suitable conditions. Our technique does not involve the projection or resolvent operator. We have also suggested an iterative method for solving the general mixed quasi variational inequality. 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 general mixed quasi variational inequalities in various branches of pure and applied sciences.

Acknowledgment

The authors would like to thank Dr. S. M. Junaid Zaidi, Rector, CIIT, for providing excellent research facilities.

References

  1. C. Baiocchi and A. Capelo, Variational and Quasi Variational Inequalities, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1984.
  2. F. Giannessi and A. Maugeri, Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, NY, USA, 1995.
  3. R. Glowinski, J. L. Lions, and R. Trémolières, Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and its Applications, North-Holland Publishing, Amsterdam, The Netherlands, 1981.
  4. M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988. View at Publisher · View at Google Scholar
  5. M. A. Noor, “Quasi variational inequalities,” Applied Mathematics Letters, vol. 1, pp. 367–370, 1988.
  6. M. A. Noor, “Wiener-Hopf equations and variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, pp. 197–206, 1988.
  7. M. A. Noor, “Some algorithms for general monotone mixed variational inequalities,” Mathematical and Computer Modelling, vol. 29, no. 7, pp. 1–9, 1999. View at Publisher · View at Google Scholar
  8. M. A. Noor, “Fundamentals of mixed quasi variational inequalities,” International Journal of Pure and Applied Mathematics, vol. 15, no. 2, pp. 137–258, 2004.
  9. M. A. Noor, “Mixed quasi variational inequalities,” Applied Mathematics and Computation, vol. 146, no. 2-3, pp. 553–578, 2003. View at Publisher · View at Google Scholar
  10. M. A. Noor, “Auxiliary principle technique for solving general mixed variational inequalities,” Journal of Advanced Mathematical Studies, vol. 3, no. 2, pp. 89–96, 2010.
  11. M. A. Noor, K. I. Noor, and T. M. Rassias, “Some aspects of variational inequalities,” Journal of Computational and Applied Mathematics, vol. 47, no. 3, pp. 285–312, 1993. View at Publisher · View at Google Scholar
  12. M. A. Noor, K. Inayat Noor, and Th. M. Rassias, “Set-valued resolvent equations and mixed variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 220, no. 2, pp. 741–759, 1998. View at Publisher · View at Google Scholar
  13. M. A. Noor, “Differentiable non-convex functions and general variational inequalities,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 623–630, 2008. View at Publisher · View at Google Scholar
  14. M. A. Noor, K. Inayat Noor, and E. Al-Said, “On new proximal point methods for solving the variational inequalities,” Journal on Applied Mathematics, vol. 2012, 7 pages, 2012.
  15. G. Cristescu and L. Lupsa, Non-Connected Convexities and Applications, vol. 68 of Applied Optimization, Kluwer Academic, Dodrecht, The Netherlands, 2002.
  16. M. A. Noor, “Extended general variational inequalities,” Applied Mathematics Letters, vol. 22, no. 2, pp. 182–186, 2009. View at Publisher · View at Google Scholar
  17. M. A. Noor, “On a class of general variational inequalities,” Journal of Advanced Mathematical Studies, vol. 1, pp. 31–42, 2008.
  18. M. A. Noor, “Projection iterative methods for extended general variational inequalities,” Journal of Applied Mathematics and Computing, vol. 32, no. 1, pp. 83–95, 2010. View at Publisher · View at Google Scholar
  19. M. A. Noor, “Nonconvex Wiener-Hopf equations and variational inequalities,” Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 73–82, 2011.
  20. G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” vol. 258, pp. 4413–4416, 1964.
  21. M. A. Noor, “Some recent advances in variational inequalities. I. Basic concepts,” New Zealand Journal of Mathematics, vol. 26, no. 1, pp. 53–80, 1997.
  22. M. A. Noor, “Some recent advances in variational inequalities. II. Other concepts,” New Zealand Journal of Mathematics, vol. 26, no. 2, pp. 229–255, 1997.
  23. M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar
  24. M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar
  25. M. A. Noor, “General variational inequalities and nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 810–822, 2007. View at Publisher · View at Google Scholar
  26. M. A. Noor, “Strongly nonlinear nonconvex variational inequalities,” Journal of Advanced Mathematical Studies, vol. 4, no. 1, pp. 77–84, 2011.
  27. M. A. Noor, “Some aspects of extended general variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 303569, 16 pages, 2012. View at Publisher · View at Google Scholar
  28. M. A. Noor, K. Inayat Noor, and H. Yaqoob, “On general mixed variational inequalities,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 227–246, 2010. View at Publisher · View at Google Scholar
  29. M. A. Noor, S. Ullah, K. Inayat Noor, and E. Al-Said, “Iterative methods for solving extended general mixed variational inequalities,” Computers & Mathematics with Applications, vol. 62, no. 2, pp. 804–813, 2011. View at Publisher · View at Google Scholar
  30. M. A. Noor, K. Inayat Noor, Y. Z. Huang, and E. Al-Said, “Implicit schemes for solving extended general nonconvex variational inequalities,” Journal on Applied Mathematics, vol. 2012, 2012.