Existence Results for Vector Mixed Quasi-Complementarity Problems
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.
In 1980, Giannessi  introduced vector variational inequalities in a finite-dimensional Euclidean space. Motivated by Giannessi , Chen and Cheng  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  and Cottle and Dantzig . 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.  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  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  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].
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 )). 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. .
(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.
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 .
Example 13. Let , defined by Then satisfies but it is not positively homogeneous.
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
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.
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