Research Article | Open Access

# Existence Results for Vector Mixed Quasi-Complementarity Problems

**Academic Editor:**Kaleem R. Kazmi

#### Abstract

We introduce strong vector mixed quasi-complementarity problems and the corresponding strong vector mixed quasi-variational inequality problems. We establish equivalence between strong mixed quasi-complementarity problems and strong mixed quasi-variational inequality problem in Banach spaces. Further, using KKM-Fan lemma, we prove the existence of solutions of these problems, under pseudomonotonicity assumption. The results presented in this paper are extensions and improvements of some earlier and recent results in the literature.

#### 1. Introduction

In 1980, Giannessi [1] introduced vector variational inequalities in a finite-dimensional Euclidean space. Motivated by Giannessi [1], Chen and Cheng [2] studied vector variational inequalities in infinite-dimensional Euclidean space and applied them to the vector optimization problems. Since then, vector variational inequalities and their generalizations have been studied and applied to vector optimization problems, vector complementarity problems, game theory, and so forth; see, for example, [1–20] and references therein. It is well known that the complementarity problems are closely related to variational inequality problems. Complementarity theory is introduced by Lemke [21] and Cottle and Dantzig [5]. It has emerged as an active and interesting field for researcher with wide range of applications in pure and applied sciences. Complementarity problems have been extended and generalized in various directions to study a large class of problems arising in industry, finance, optimization, physical, mathematical and engineering sciences; see, for example [4–12, 14, 15, 20]. Recently, vector complementarity problems and their relations with vector variational inequality problems have been investigated under pseudomonotone-type conditions and positiveness-type conditions; see, for example [6, 8–10, 20]. However, to the best of our knowledge, only a few existence results on the strong version of the vector variational inequality and vector complementarity problems were established.

Recently, Huang et al. [12] discussed equivalence results among a vector complementarity problem, a vector variational inequality problem, a vector optimization problem, and weak minimal element problem, under some monotonicity conditions and some inclusive-type conditions in ordered Banach spaces. In 2005, Huang and Fang [9] introduced several classes of strong vector F-complementarity problems and give some existence results for these problems in Banach spaces and discussed the least element problems of feasible sets and presented their relations with the strong vector F-complementarity problems.

Very recently, Khan [22] introduced and studied a generalized vector implicit Quasi-Complementarity problem and generalized vector implicit quasi variational inequality problem. He investigated the nonemptiness and closedness of solution sets of these problems and proved that solution sets of both the problems are equivalent to each other under some suitable conditions.

Inspired and motivated by the work going in this direction, in this paper we introduce and analyze a new class of strong vector Quasi-Complementarity problem and the corresponding strong vector mixed quasi variational inequality problem in the setting of Banach space and establish equivalence results between them. By using the KKM-Fan lemma, we derive the existence of solutions of strong vector mixed quasi variational inequalities under pseudomonotonicity assumption and show that the solution of the strong vector mixed quasi variational inequality is equivalent to the solution of strong vector mixed Quasi-Complementarity problems under suitable conditions. The results presented in this paper are the generalization and improvement of existing works of [6, 7, 9, 11, 15].

#### 2. Preliminaries

Throughout this paper unless otherwise stated let and be two real Banach spaces. Let be a nonempty, closed, convex subset of a real Banach space . A nonempty subset is called convex, pointed, connected, and reproduced cone, respectively, if it satisfies the following conditions: (i) , for all and ; (ii) ; (iii) ; (iv) .

Given in , we can define the relations “” and “" as follows: If “” is a partial order, then is called a Banach space ordered by . Let denote the space of all continuous linear mappings from into .

Now, we recall the following concepts and results needed in this paper.

*Definition 1. *A mapping is said to be *-convex* in first argument, if

*Definition 2. *Let and be the two nonlinear mappings. is said to be *monotone with respect to * if

*Definition 3. *Let and be the two nonlinear mappings. is said to be *pseudomonotone with respect to * if, for any given ,

*Remark 4. *Every monotone with respect to is pseudomonotone with respect to but converse does not hold in general. Definition 3 is vector version of -pseudomonotonicity studied by Kazmi et al. in [23, 24].

*Example 5. *Let , and

Now,
We have . It follows that
So, is pseudomonotone with respect to . However, for and , it follows that
This shows that is not a monotone with respect to .

*Definition 6. *A mapping is said to be *hemicontinuous* if, for any , the mapping is continuous at .

*Definition 7. *A mapping is said to be *positively homogeneous* in first argument, if for all and .

*Definition 8. *Let be a nonempty subset of a topological vector space . A set-valued map is said to be a *KKM mapping* if, for each nonempty finite subset , , where co denotes the convex hull.

Lemma 9 (KKM-Fan Lemma (see [25])). *Let be a nonempty subset of Hausdorff topological vector space . Let be a KKM-mapping such that for each is closed and for at least one is compact, then
*

#### 3. Strong Vector Mixed Quasi-Complementarity Problems

Throughout this section, let be a real Banach space and let be a nonempty, closed, and convex subset of . Let be an ordered Banach space induced by a pointed, closed, convex cone with nonempty interior. Let and be the two nonlinear mappings. In this paper, we consider the following strong vector mixed Quasi-Complementarity problems:(i)Strong vector mixed Quasi-Complementarity problem : Find such that , for all .(ii)Strong vector mixed Quasi-Complementarity problem : Find such that , for all .

Closely related to and problems, we consider the following strong vector mixed quasi variational inequality problem: Strong vector mixed quasi variational inequality problem (SVMQVIP): Find such that , for all

The strong vector mixed quasi variational inequality problem (SVMQVIP) is the generalization and extension of many previously known vectors as well as scalar mixed quasi variational inequalities. For the formulation, numerical results, existence results, sensitivity analysis, and dynamical aspects of the mixed quasi variational inequalities, see [3, 5, 7, 16, 17] and the references therein.

*Remark 10. *(1) If and , then and and (SVMQVIP) reduce, respectively, to the mixed Quasi-Complementarity problem (MQCP): (MQCP) Find such that , for all and mixed quasi variational inequality problem (MQVIP): (MQVIP) Find such that , for all ,which were introduced and studied by Farazjadeh et al. [7].

(2) If , then and reduce to the following strong vector complementarity problems (SVCP): Find such that , for all , Find such that , for all

and (SVMQVIP) reduces to the following strong vector variational inequality problem (SVVIP): (SVVIP) Find such that , for all

First, we will investigate the equivalences among and and (SVMQVIP), under some suitable assumptions.

Theorem 11. *
Suppose that , for all . If solves then solves (SVMQVIP). **
Let satisfy , for all and , for all . If solves (SVMQVIP) then also solves .*

*Proof. *(i) Let be the solution of . Then such that
Substituting in Inclusion (11), we get
Since , for all , we have
From Inclusions (10), (12), and (13), we have
From (11) and (14), we have
for all . Thus, is the solution of (SVMQVIP).

(ii) Now, let be the solution of (SVMQVIP), then
Since , for all , therefore it follows that , for all . By substituting and , respectively, in (16), we get
Since , for all , we have
From (17) and (18), we have
By using Inclusions (16) and (19), we have
for all , which implies that solves .

*Remark 12. *The condition , for all holds if is positively homogeneous; that is, for all . Hence, Theorem 11 generalizes and improves the theorems in [6, 9, 11, 14, 15].

Here we give an example of a function , which satisfies the condition , for all but not a positively homogeneous, which implies that previously known results in [6, 9, 11, 14, 15] cannot be applied.

*Example 13. *Let , defined by
Then satisfies but it is not positively homogeneous.

Theorem 14. *
If solves Problem then solves (SVMQVIP). **
Let satisfy , for all and , for all . If solves (SVMQVIP) then solves . *

*Proof. *(a) Let be the solution of . Then such that
Now,
for all . Thus, is the solution of (SVMQVIP).

(b) Now, let be the solution of (SVMQVIP), then
Since , for all , therefore it follows that . By substituting and , respectively, in (24), we get
Since , for all , we have
From (25) and (26), we have
By using (27), we have
for all . Then (27) and (28) imply that solves .

#### 4. Existence Results

First, we prove following Minty-type lemma with the help of pseudomonotone mapping with respect to .

Lemma 15. *Let be -convex in first argument and let be a hemicontinuous mapping and pseudomonotone with respect to . Then the following two problems are equivalent: *

*Proof. *(29) (30). The result directly follows from pseudomonotonicity with respect to .

Now, (30) (29). For any given , we know that , for all , as is convex. Since is a solution of problem (30), so for each , it follows that

Now, we have
For , we get
Since is hemicontinuous and is closed, letting in inclusion (33), we get
Hence,
Therefore, is solution of problem (29). This completes the proof.

Now, with the help of Lemma 15, we have following existence theorem for (SVMQVIP).

Theorem 16. * Let be real reflexive Banach space and let be a Banach space. Let be a nonempty, bounded, closed, and convex subset of . Let be -convex and upper semicontinuous in first and second arguments, respectively. Let be hemicontinuous and pseudomonotone with respect to . Then (SVMQVIP) has solution.*

*Proof. * Define two set-valued mappings as follows:
and are nonempty, since . We claim that is a KKM mapping. If this is not true, then there exists a finite set and with such that . Now, by the definition of , we have
Now, we have
which is not possible. Thus, our claim is verified. So is a KKM mapping.

Now, since is pseudomonotone with respect to , therefore for every and so is also a KKM mapping. Now we claim that for each is closed in the weak topology of .

Indeed, suppose , the weak closure of . Since is reflexive, there is a sequence in such that converges weakly to . Then
Since is upper semicontinuous and is closed, therefore,
and so . This shows that is weakly closed, for each . Our claim is then verified. Since is reflexive and is nonempty, bounded, closed and convex, is a weakly compact subset of and so is also weakly compact. According to Lemma 9 (KKM-Fan Lemma),
This implies that there exists such that
Therefore by Lemma 15, we conclude that there exists such that
This completes the proof.

Theorem 17. *Let satisfy , for all . If all the assumptions of Theorem 16 hold, then is solvable. In addition, if , for all , then is solvable.*

*Proof. *The conclusion follows directly from Theorems 11, 14, and 16.

#### References

- F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in
*Variational Inequalities and Complementarity Problems*, R. W. Cottle, F. Gianneesi, and J. L. Lions, Eds., pp. 151–186, Wiley, Chichester, UK, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G. Y. Chen and G. M. Cheng, “Vector variational inequalities and vector optimization,” in
*Lecture Notes in Economics and Mathematical Systems*, vol. 285, pp. 408–416, Springer, Berlin, Germany, 1987. View at: Google Scholar - A. Bnouhachem and M. A. Noor, “Numerical method for general mixed quasi-variational inequalities,”
*Applied Mathematics and Computation*, vol. 204, no. 1, pp. 27–36, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Y. Chen and X. Q. Yang, “The vector complementary problem and its equivalences with the weak minimal element in ordered spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 153, no. 1, pp. 136–158, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. W. Cottle and G. B. Dantzig, “Complementary pivot theory of mathematical programming,”
*Linear Algebra and its Applications*, vol. 1, pp. 163–185, 1968. View at: Google Scholar - Y.-P. Fang and N.-J. Huang, “The vector
*F*-complementary problems with demipseudomonotone mappings in Banach spaces,”*Applied Mathematics Letters*, vol. 16, no. 7, pp. 1019–1024, 2003. View at: Publisher Site | Google Scholar | MathSciNet - A. P. Farajzadeh, M. Aslam Noor, and S. Zainab, “Mixed quasi complementarity problems in topological vector spaces,”
*Journal of Global Optimization*, vol. 45, no. 2, pp. 229–235, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Isac, “A special variational inequality and the implicit complementarity problem,”
*Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics*, vol. 37, no. 1, pp. 109–127, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet - N.-j. Huang and Y.-p. Fang, “Strong vector
*F*-complementary problem and least element problem of feasible set,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 61, no. 6, pp. 901–918, 2005. View at: Publisher Site | Google Scholar | MathSciNet - N.-J. Huang and C.-J. Gao, “Some generalized vector variational inequalities and complementarity problems for multivalued mappings,”
*Applied Mathematics Letters*, vol. 16, no. 7, pp. 1003–1010, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N.-j. Huang and J. Li, “
*F*-implicit complementarity problems in Banach spaces,”*Zeitschrift für Analysis und ihre Anwendungen*, vol. 23, no. 2, pp. 293–302, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. J. Huang, X. Q. Yang, and W. K. Chan, “Vector complementarity problems with a variable ordering relation,”
*European Journal of Operational Research*, vol. 176, no. 1, pp. 15–26, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. R. Kazmi and S. A. Khan, “Existence of solutions to a generalized system,”
*Journal of Optimization Theory and Applications*, vol. 142, no. 2, pp. 355–361, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. S. Lee, M. F. Khan, and Salahuddin, “Vector
*F*-implicit complementarity problems with corresponding variational inequality problems,”*Applied Mathematics Letters*, vol. 20, no. 4, pp. 433–438, 2007. View at: Publisher Site | Google Scholar | MathSciNet - J. Li and N.-j. Huang, “Vector
*F*-implicit complementarity problems in Banach spaces,”*Applied Mathematics Letters*, vol. 19, no. 5, pp. 464–471, 2006. View at: Publisher Site | Google Scholar | MathSciNet - U. Mosco, “Implicit variational problems and quasi variational inequalities,” in
*Nonlinear Operators and the Calculus of Variations*, vol. 543 of*Lecture Notes in Mathematics*, pp. 83–156, Springer, Berlin, Germany, 1976. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. A. Noor, “Mixed quasi variational inequalities,”
*Applied Mathematics and Computation*, vol. 146, no. 2-3, pp. 553–578, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Usman and S. A. Khan, “A generalized mixed vector variational-like inequality problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 11, pp. 5354–5362, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. U. Verma, “On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators,”
*Journal of Mathematical Analysis and Applications*, vol. 213, no. 1, pp. 387–392, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Q. Yang, “Vector complementarity and minimal element problems,”
*Journal of Optimization Theory and Applications*, vol. 77, no. 3, pp. 483–495, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. E. Lemke, “Bimatrix equilibrium points and mathematical programming,”
*Management Science*, vol. 11, pp. 681–689, 1965. View at: Google Scholar | MathSciNet - S. A. Khan, “Generalized vector implicit quasi complementarity problems,”
*Journal of Global Optimization*, vol. 49, no. 4, pp. 695–705, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. R. Kazmi, A. Khaliq, and A. Raouf, “Iterative approximation of solution of generalized mixed set-valued variational inequality problem,”
*Mathematical Inequalities & Applications*, vol. 10, no. 3, pp. 677–691, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. R. Kazmi, F. A. Khan, and M. Shahza, “Two step algorithm for solving regularized generalized mixed variational inequality problem,”
*Bulletin of the Korean Mathematical Society*, vol. 47, no. 4, pp. 675–685, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Fan, “A generalization of Tychonoff's fixed point theorem,”
*Mathematische Annalen*, vol. 142, pp. 305–310, 1961. View at: Google Scholar | MathSciNet

#### Copyright

Copyright © 2013 Suhel Ahmad Khan and Naeem Ahmad. 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.