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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 376829, 7 pages
http://dx.doi.org/10.1155/2013/376829
Research Article

Existence Results for Nonsmooth Vector Quasi-Variational-Like Inequalities

1Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
2Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India

Received 27 June 2013; Accepted 1 August 2013

Academic Editor: Jen-Chih Yao

Copyright © 2013 Mohammed Alshahrani et al. 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 introduce nonsmooth vector quasi-variational-like inequalities (NVQVLI) by means of a bifunction. We establish some existence results for solutions of these inequalities by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in Al-Homidan et al. (2012), one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

1. Introduction

The theory of quasi-variational inequalities (QVI) was started with a pioneerwork of A. Bensoussan and J. L. Lions in 1973, perhaps motivated by the stochastic control and impulse control problems. It was the paper of Bensoussan et al. [1] in which the term quasi-variational inequality was introduced. The quasi-variational inequality is an extension of a variational inequality [2] in which the underlying set depends on the solution itself. For further details on quasi-variational inequalities, we refer to [35] and the references therein. In 1980, Giannessi [6] initiated the theory of vector variational inequalities with applications to vector optimization. Since then, it has been growing up in different directions. One of such directions is the application to the theory of vector optimization. However, if the underlying objective function is not differentiable and not convex, then we need to define a nonsmooth vector variational-like inequality by means of Dini directional derivatives or Clarke directional derivatives. For studying such problems by using vector variational-like inequalities, Alshahrani et al. [7], Al-Homidan et al. [8], Ansari and Lee [9], Crespi et al. [10], and Lalitha and Mehta [11] considered a vector variational inequality, defined by means of Dini directional derivatives, called nonsmooth vector variational inequality. The nonsmooth vector optimization is studied in these references by using nonsmooth vector variational inequalities. Motivated by the extension of variational inequalities for vector-valued functions, several researchers started to study the QVI for vector-valued functions, known as vector quasi-variational inequalities (VQVI); see, for example, [1216] and the references therein. An optimization problem in which the feasible set depends on the solution itself is called quasi-optimization problem [14]. Such problems can be solved by using the vector quasi-variational inequality technique. To the best of our knowledge, no study has been done in the literature to study nonsmooth quasi-variational inequalities which are defined by means of a bifunction, in particular by means of Dini or Clarke directional derivatives. This paper can be treated as the beginning of the study of nonsmooth (vector) quasi-variational inequalities and nonsmooth vector quasi-optimization problem.

In this paper, we consider the vector quasi-variational-like inequality problems defined by means of a bifunction and present some existence results for solutions of these problems by using Fan-KKM theorem and a maximal element theorem. By using the technique and methodology adopted in [8], one can easily derive the relations among these inequalities and a vector quasi-optimization problem. Hence, the existence results for a solution of a vector quasi-optimization problem can be derived by using our results. The results of this paper extend several known results in the literature.

2. Formulations

We adopt the following ordering relations. We consider the cones and , where is the nonnegative orthant of , and is the origin of ; let be a set of . Then for all ,

Let be a real-valued function. The upper Dini directional derivative of at in the direction is defined as

For further details on Dini directional derivatives, we refer to the recent book [2].

Let be a nonempty subset of , a set-valued map, and a mapping. Let be a vector-valued function such that, for each fixed , is positively homogeneous in . In particular, we consider where a vector-valued function and

The nonsmooth (Stampacchia or Minty type) vector quasi-variational-like inequality problems are defined as follows.

Nonsmooth Stampacchia Vector Quasi-Variational-Like Inequality Problem (NSVQVLIP). Find such that and

Minty Vector Quasi-Variational-Like Inequality Problem (NMVQVLIP). Find such that and

When , for all , then these problems were studied in [8, 9, 11] with applications to vector optimization. Furthermore, if we consider the previous Dini directional derivative as a bifunction , with referring to a point in and referring to a direction from , that is, if , then the previously mentioned problems are studied in [7, 8, 10] and the references therein.

The main motivation of this paper is to establish some existence results for solutions of NMVQVLIP and NSVQVLIP by using Fan-KKM theorem or a maximal element theorem. Of course, by using the technique of [8], we can easily establish some results on the relations among NMVQVLIP, NSVQVLIP, and vector quasi-optimization problems [14]. Since the results are straightforward, we are not including them here.

3. Preliminaries

Let be a nonempty set. We denote by , int , and the closure of , the interior of , and the convex hull of , respectively.

Definition 1. Let be a nonempty set and a mapping. The set is said to be invex with respect to if, for all and all , we have .

We say that the map is skew if, for all ,

Condition C. Let be an invex set with respect to . Then, for all , , we have

We adopt the following definition of affineness.

A vector-valued function is called affine if, for all and for all with , we have

The following lemma can be easily proved.

Lemma 2 (see [17]). Let be a nonempty convex subset of a vector space and a mapping. If is affine in the first argument and skew, then it is also affine in the second argument.

Definition 3 (see [18, 19]). Let be a nonempty set. A vector-valued function is said to be -lower semicontinuous (resp., -upper semicontinuous) at if for any neighborhood of , there exists a neighborhood of such that for all (resp., for all ). is said to be -lower semicontinuous (resp., -upper semicontinuous) on if it is -lower semicontinuous (resp., -upper semicontinuous) at every point .

It is shown in [18] that a function is -lower semicontinuous if and only if, for all , the set is closed in .

Definition 4. Let be a nonempty convex set. A vector-valued function is said to be -convex if, for all and all ,

Definition 5 (see [18, 19]). Let be a nonempty convex set. A vector-valued function is said to be -quasiconvex if, for all , the set is convex.

It is shown in [18] that if is -quasiconvex, then the set is convex.

A vector-valued function is called positively homogeneous if for all and all , .

Definition 6. Let be a nonempty subset of a topological vector space . A set-valued map is said to be a KKM map provided and for each finite subset of , where denotes the convex hull of .

The following Fan-KKM theorem [20] will be used in the sequel.

Theorem 7 (see [20]). Let be a nonempty subset of a Hausdorff topological vector space . Assume that is a KKM map satisfying the following conditions: (i)for each , is closed; (ii)for at least one , is compact.
Then, .

We will use the following maximal element theorem to prove the existence of solutions of nonsmooth vector quasi-variational-like inequality problems.

Theorem 8 (see [21, Corollary 3.2]). Let be a nonempty convex subset of a Hausdorff topological vector space and two set-valued maps. Assume that the following conditions hold: (i)for all , ; (ii)for all , and is open in ; (iii)there exist a nonempty compact convex subset and a nonempty compact subset of such that for each , there exists such that .
Then, there exists such that .

4. Existence Results

Definition 9 (see [8]). Let be a nonempty set and a mapping. A vector-valued bifunction is said to be(a)-pseudomonotone with respect to on if, for all , (b)-properly subodd if for every , with and .

The definition of proper suboddness is considered in [11]. Of course, if , the definition of proper suboddness reduces to the definition of suboddness.

Definition 10. Let be a nonempty convex subset of . A function is said to be hemicontinuous if, for all , the mapping is continuous. The upper and lower hemicontinuity can be defined analogously.

Definition 11. Let be an invex set with respect to . A function is said to be -hemicontinuous if, for all , the mapping is continuous. The upper and lower -hemicontinuity can be defined analogously.

The following concept of -upper sign continuity for the bifunction is considered in [8].

Definition 12 (see [8]). Let be a nonempty invex set with respect to . A vector-valued bifunction is said to be -upper sign continuous if, for all and ,

Remark 13. It can be easily seen that if is skew and is -upper hemicontinuous in the first argument, then it is -upper sign continuous, but the converse is not true in general.

The following result provides the relations between NSVQVLIP and NMVQVLIP when the set-valued map is invex valued.

Proposition 14. Let be a nonempty invex set with respect to   such that Condition C holds. Let be a set-valued map such that, for each , is a nonempty and invex set with respect to . Let the vector-valued bifunction be -properly subodd, -pseudomonotone with respect to , and -upper sign continuous such that, for each fixed , is positively homogeneous. Then, is a solution of NSVQVLIP if and only if it is a solution of NMVQVLIP.

Proof. It is similar to the proof of Proposition 7.7 in [8]. However, we include it for the sake of completeness of the paper.
The -pseudomonotonicity of with respect to implies that every solution of NSVQVLIP is a solution of NMVQVLIP.
Conversely, let be a solution of NMVQVLIP. Then, , and
Since is invex, we have for all , and therefore, (13) becomes
By Condition C, , and thus,
By positive homogeneity and -proper suboddness of , we have
Thus, the -upper sign continuity of yields is a solution of NSVQVLIP.

The following result gives the equivalence between NSVQVLIP and NMVQVLIP when the set-valued map is convex valued.

Proposition 15. Let be a nonempty convex set, and let be affine in the first argument and skew. Let be a set-valued map such that, for each , is a nonempty and convex set. Let the vector-valued bifunction be -properly subodd, -pseudomonotone with respect to , and -upper sign continuous such that, for each fixed , is positively homogeneous. Then, is a solution of NSVQVLIP if and only if it is a solution of NMVQVLIP.

Proof. It is similar to the proof of Proposition 7.8 in [8]. However, we include it for the sake of completeness of the paper.
The -pseudomonotonicity of with respect to implies that every solution of NSVQVLIP is a solution of NMVQVLIP.
Conversely, let be a solution of NMVQVLIP. Then, , and
Since is convex, we have for all , and therefore, (17) becomes
Since is affine in the first argument and skew, by Lemma 2, is also affine in the second argument. Since by skewness of , we obtain
By positive homogeneity of in the second argument, we have
Since by skewness of , the -proper suboddness of implies that
The -upper sign continuity of yields is a solution of NSVQVLIP.

Throughout the rest of the paper, unless otherwise specified, we assume that is a set-valued map such that is nonempty convex for all , is open for all , and the set is closed.

We present some existence results for the solutions of NSVQVLIP and NMVQVLIP without boundedness assumption on the underlying set .

Theorem 16. Let be a nonempty convex set, and be skew, affine, and lower semicontinuous in the first argument. Let be -properly subodd, positively homogeneous in the second argument, and -pseudomonotone with respect to such that is continuous. Assume that there exist a nonempty compact convex subset of and such that, for all , and Then, there exists a solution of MVQVLIP.
Furthermore, if   is -upper sign continuous, then is a solution of SVQVLIP.

Proof. For all , we define two set-valued maps by
For all and for each , we also define other two set-valued maps and by
For each and for all , we have (see, e.g., [22]) and therefore,
The rest of the proof is divided into the following four steps.
(a) We claim that is a KKM map on .
Assume the contrary that is not a KKM map. Then, there exist a finite set in and with such that for all ; that is,
If , then , and therefore,
Hence,
Since is a convex cone and with , we have
Since is skew, we have . By the affineness of in the first argument, we have
Since is -proper subodd, we have
By positive homogeneity of , we obtain a contradiction of (29).
If , then . By the definition of , we have , and therefore, for all . Since is convex, we obtain , again a contradiction. Hence, is a KKM map.
(b) We show that , where and are the same as in the hypothesis.
Indeed, if , then ; that is, either or .
If , then and ; that is, and , a contradiction to our assumption that .
If , then if and only if , again a contradiction to our assumption that . Hence, .
(c) We show that .
Since is compact, is also compact. Moreover, since is a KKM map, for each finite subset of . Then by Theorem 7, we get .
(d) Next, we claim that .
Let ; then for each . For an arbitrary element , we have to show that .
Since , there exists a sequence such that converges to . Since , we have
Then, either or .
If , then and . It follows that and . Since is closed and , we have ; that is, . By -pseudomonotonicity of , we obtain
By the continuity of , we get . This implies that and ; that is, , and hence, . Therefore, , and .
Let . Since is open in , for all , is closed in . Since , we have . Hence, , which implies that
From (c), we get . Hence, there exists such that
This implies that .
If , then , a contradiction. Otherwise, ; then . Therefore, such that for all . From Proposition 15, is a solution of SVQVLIP.

Remark 17. (a) Theorem 16 extends and generalizes [8, Theorem 7.34], [9, Theorem 5.1], and [11, Theorem 2.2].
(b) If is a closed map, then the set is closed.

By using maximal element Theorem 8, we present the following existence result for solutions of NSVQVLIP and NMVQVLIP without boundedness assumption on the underlying set.

Theorem 18. Let be a nonempty convex set, and let be skew. Let be -pseudomonotone with respect to such that for all ; the set is convex, and the set is closed in . Assume that there exist a nonempty compact convex subset and a nonempty compact subset of such that, for each , there exists such that and . Then, there exists a solution of NMVQVLIP.
Furthermore, if is -properly subodd, -upper sign continuous, and for each fixed , is positively homogeneous, then is a solution of NSVQVLIP.

Proof. For each , define two set-valued maps by
Then, for all . Indeed, by skewness of , for all . By assumption, . Thus, .
By hypothesis, the complement of in , is closed in for each . Therefore, is open in for all .
Define other two set-valued maps by
Since, for all , , we have .
By -pseudomonotonicity of , we have for all . Since and are convex, for all , we have
Since, for each , and are open in , is open in . Also, since, for each , (see, e.g., the proof of [23, Lemma 2.3]) and is open in , we have that is open in . Therefore, by Theorem 8, there exists such that . If, then , a contradiction to our assumption. So, , and thus, . Therefore,
Thus, is a solution of NMVQVLIP.

By Proposition 15, is a solution of NSVQVLIP.

Remark 19. If, for each fixed , the vector-valued function is -quasiconvex, then the set is convex.

Remark 20. For all , the set is convex, if is affine in the first argument and is -convex in the second argument.

Indeed, let . Since is a convex cone, for all , we have

By adding these relations, we get

Since is -convex in the second argument, we have

By combining relations (45) and (46), we obtain

Since is affine in the first argument, we get and hence, . Thus, for all , is a convex set.

Remark 21. The set is closed in if the vector-valued function is -lower semicontinuous for each fixed .

5. Conclusions

In this paper, we defined vector quasi-variational-like inequality problems by means of a bifunction and established two existence results for solutions of these problems. One can treat upper Dini directional derivative as a bifunction, and hence, the bifunction can be replaced by upper Dini directional derivative. Then, we get the so-called nonsmooth vector quasi-variational-like inequality problem. By using the technique and methodology given in [8], one can easily derive some relations between vector quasi-variational-like inequality problems and a vector quasi-optimization problem. Since the results are straightforward, we have not included them in this paper. Of course, the results of this paper extend several known results in the literature, namely, [9, Theorem 5.1], [11, Theorem 3.1], and [8, Theorem 7.34] from vector variational (-like) inequality problems to vector quasi-variational (-like) inequality problems. This paper can be treated as the first in this direction.

Acknowledgments

This research was done during the visit of the second author to King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, and it was supported by KFUPM Funded Research Project no. IN121035.

References

  1. A. Bensoussan, M. Goursat, and J. L. Lions, “Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires,” Comptes Rendus Mathematique, vol. 276, pp. A1279–A1284, 1973. View at Zentralblatt MATH · View at MathSciNet
  2. Q. H. Ansari, C. S. Lalitha, and M. Mehta, Generalized Convexity, Nonsmooth Variational Inequalities and Nonsmooth Optimization, CRC Press, Taylor and Francis Group, London, UK, 2014.
  3. C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, John Wiley & Sons, New York, NY, USA, 1984. View at MathSciNet
  4. A. Bensoussan and J. L. Lions, Applications des Inéquations Variationnelles en Contrôle Stochastique, Dunod, Paris, France, 1978. View at MathSciNet
  5. A. S. Kravchuk and P. J. Neittaanmäki, Variational and Quasi Variational Inequalities in Mechanics, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  6. F. Giannessi, “Theorems of alternative, quadratic programs and complementarity problems,” in Variational Inequalities and Complementarity Problems, R. W. Cottle, F. Giannessi, and J. L. Lions, Eds., pp. 151–186, Wiley, Chichester, UK, 1980. View at Zentralblatt MATH · View at MathSciNet
  7. M. Alshahrani, Q. H. Ansari, and S. Al-Homidan, “Nonsmooth variational-like inequalities and nonsmooth vector optimization,” Optimization Letters, 2013. View at Publisher · View at Google Scholar
  8. S. Al-Homidan, Q. H. Ansari, and J. C. Yao, “Nonsmooth invexities, invariant monotonicities and nonsmooth vector variational inequalities with applications to vector optimization,” in Adances in Vector Optimization, Q. H. Ansari and J. C. Yao, Eds., Springer, Berlin, Germany, 2012.
  9. Q. H. Ansari and G. M. Lee, “Nonsmooth vector optimization problems and Minty vector variational inequalities,” Journal of Optimization Theory and Applications, vol. 145, no. 1, pp. 1–16, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. G. P. Crespi, I. Ginchev, and M. Rocca, “Some remarks on the Minty vector variational principle,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 165–175, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. S. Lalitha and M. Mehta, “Vector variational inequalities with cone-pseudomonotone bifunctions,” Optimization, vol. 54, no. 3, pp. 327–338, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. H. Ansari, W. K. Chan, and X. Q. Yang, “The system of vector quasi-equilibrium problems with applications,” Journal of Global Optimization, vol. 29, no. 1, pp. 45–57, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Q. H. Ansari, W. K. Chan, and X. Q. Yang, “Weighted quasi-variational inequalities and constrained Nash equilibrium problems,” Taiwanese Journal of Mathematics, vol. 10, no. 2, pp. 361–380, 2006. View at Zentralblatt MATH · View at MathSciNet
  14. Q. H. Ansari and J. C. Yao, “On vector quasi-equilibrium problems,” in Equilibrium Problems and Variational Models, P. Daniele, F. Giannessi, and A. Maugeri, Eds., pp. 1–18, Kluwer Academic Publishers, London, UK, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Mathematical Theories, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  16. G. X. Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, NY, USA, 1999. View at MathSciNet
  17. J. W. Peng, “Criteria for generalized invex monotonicities without Condition C,” European Journal of Operational Research, vol. 170, no. 2, pp. 667–671, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. Bianchi, N. Hadjisavvas, and S. Schaible, “Vector equilibrium problems with generalized monotone bifunctions,” Journal of Optimization Theory and Applications, vol. 92, no. 3, pp. 527–542, 1997. View at Publisher · View at Google Scholar
  19. D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989. View at MathSciNet
  20. K. Fan, “A generalization of Tychonoff's fixed point theorem,” Mathematische Annalen, vol. 142, pp. 305–310, 1961. View at MathSciNet
  21. L. J. Lin, Z. T. Yu, Q. H. Ansari, and L. P. Lai, “Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities,” Journal of Mathematical Analysis and Applications, vol. 284, no. 2, pp. 656–671, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. Q. Khanh and L. M. Luu, “Some existence results for vector quasivariational inequalities involving multifunctions and applications to traffic equilibrium problems,” Journal of Global Optimization, vol. 32, no. 4, pp. 551–568, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. X. P. Ding, “Existence of solutions for quasi-equilibrium problems in noncompact topological spaces,” Computers & Mathematics with Applications, vol. 39, no. 3-4, pp. 13–21, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet