Abstract
In this paper, by using a nonlinear scalarization technique, we obtain sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasi-equilibrium problem with set-valued mappings. The results are different from the recent ones in the literature.
Similar content being viewed by others
1 Introduction
The generalized vector quasi-equilibrium problem is a unified model of several problems, namely generalized vector quasi-variational inequalities, vector quasi-optimization problems, traffic network problems, fixed point and coincidence point problems, etc. (see, for example, [1, 2] and the references therein). It is well known that the stability analysis of a solution mapping for equilibrium problems is an important topic in optimization theory and applications. Stability may be understood as lower or upper semicontinuity, continuity, and Lipschitz or Hölder continuity. There have been many papers to discuss the stability of solution mapping for equilibrium problems when they are perturbed by parameters (also known the parametric (generalized) equilibrium problems). Last decade, many authors intensively studied the sufficient conditions of upper (lower) semicontinuity of various solution mappings for parametric (generalized) equilibrium problems, see [3–10]. Let us begin now, Yen [11] obtained the Hölder continuity of the unique solution of a classic perturbed variational inequality by the metric projection method. Mansour and Riahi [12] proved the Hölder continuity of the unique solution for a parametric equilibrium problem under the concepts of strong monotonicity and Hölder continuity. Bianchi and Pini [13] introduced the concept of strong pseudomonotonicity and got the Hölder continuity of the unique solution of a parametric equilibrium problem. Anh and Khanh [14] generalized the main results of [13] to two classes of perturbed generalized equilibrium problems with set-valued mappings. Anh and Khanh [15] further discussed the uniqueness and Hölder continuity of the solutions for perturbed equilibrium problems with set-valued mappings. Anh and Khanh [16] extended the results of [15] to the case of perturbed quasi-equilibrium problems with set-valued mappings and obtained the Hölder continuity of the unique solutions. Li et al. [17] introduced an assumption, which is weaker than the corresponding ones of [13, 14], and established the Hölder continuity of the set-valued solution mappings for two classes of parametric generalized vector quasi-equilibrium problems in general metric spaces. Li et al. [18] extended the results of [17] to perturbed generalized vector quasi-equilibrium problems.
Among many approaches for dealing with the lower semicontinuity, continuity and Hölder continuity of the solution mapping for a parametric vector equilibrium problem in general metric spaces, the scalarization method is of considerable interest. The classical scalarization method using linear functionals has been already used for studying the lower semicontinuity of the solution mapping [19–21] and the Hölder continuity [22] of the solution mapping to parametric vector equilibrium problems. Wang et al. [23] established the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result. Recently, by using this method, Peng [24] established the sufficient conditions for the Hölder continuity of the solution mapping to a parametric generalized vector quasi-equilibrium problem with set-valued mappings.
On the other hand, a useful approach for analyzing a vector optimization problem is to reduce it to a scalar optimization problem. Nonlinear scalarization functions play an important role in this reduction in the context of nonconvex vector optimization problems. The nonlinear scalarization function , commonly known as the Gerstewitz function in the theory of vector optimization [25, 26], has been also used to study the lower semicontinuity of the set-valued solution mapping to a parametric vector variational inequality [27]. Using this method, Bianchi and Pini [28] obtained the Hölder continuity of the single-valued solution mapping to a parametric vector equilibrium problem. Recently, Chen and Li [29] studied Hölder continuity of the solution mapping for both set-valued and single-valued cases to parametric vector equilibrium problems. The key role in their paper is a globally Lipschitz property of the Gerstewitz function. Very recently, by using the idea in [29], Chen [30] obtained Hölder continuity of the unique solution to a parametric vector quasi-equilibrium problem based on nonlinear scalarization approach under three different kinds of monotonicity hypotheses. It is natural to raise and give an answer to the following question.
Question Can one establish the Hölder continuity of a solution mapping to the parametric generalized vector quasi-equilibrium problem with set-valued mappings by using a nonlinear scalarization method?
Motivated and inspired by Peng [24] and Chen [30] and research going on in this direction, in this paper we aim to give positive answers to the above question. We first establish the sufficient conditions which guarantee the Hölder continuity of a solution mapping to the parametric generalized vector quasi-equilibrium problem with set-valued mappings by using a nonlinear scalarization method. We further study several kinds of the monotonicity conditions to obtain the Hölder continuity of the solution mapping. The main results of this paper are different from the corresponding results in Peng [24] and Chen [30]. These results improve the corresponding ones in recent literature.
The structure of the paper is as follows. Section 2 presents the parametric generalized vector quasi-equilibrium problem and materials used in the rest of this paper. We establish, in Section 3, a sufficient condition for the Hölder continuity of the solution mapping to a parametric generalized vector quasi-equilibrium problem.
2 Preliminaries
Throughout the paper, unless otherwise specified, we denote by and the norm and the metric on a normed space and a metric space, respectively. A closed ball with center and radius is denoted by . We always consider X, Λ, M as metric spaces, and Y as a linear normed space with its topological dual space . For any , we define , where denotes the value of at y. Let be a pointed, closed and convex cone with , where intC stands for the interior of C. Let
be the dual cone of C. Since , the dual cone of C has a weak* compact base. Let . Then
is a weak*-compact base of . Clearly, is a weak∗-compact base of , that is, is convex and weak∗-compact such that and .
Let , the nonlinear scalarization function [25, 26] is defined by
It is well known that is a continuous, positively homogeneous, subadditive and convex function on Y, and it is monotone (that is, ) and strictly monotone (that is, ) (see [25, 26]). In case, , and , the nonlinear scalarization function can be expressed in the following equivalent form [[25], Corollary 1.46]:
Lemma 2.1 [[25], Proposition 1.43]
For any fixed , and ,
-
(i)
(that is, );
-
(ii)
;
-
(iii)
, where ∂C denotes the boundary of C;
-
(iv)
.
The property (i) of Lemma 2.1 plays an essential role in scalarization. From the definition of , property (iv) in Lemma 2.1 could be strengthened as
For any , the set defined by
is a weak∗-compact set of (see [[19], Lemma 5.1]). The following equivalent form of can be deduced from [[31], Corollary 2.1] or [[32], Proposition 2.2] ([[25], Proposition 1.53]).
Proposition 2.2 [[30], Proposition 2.2]
Let . Then, for ,
Proposition 2.3 [[30], Proposition 2.3]
is Lipschitz on Y, and its Lipschitz constant is
The following example can be found in [[30], Example 2.1].
Example 2.4
-
(i)
If and , then the Lipschitz constant of is (). Indeed, for all .
-
(ii)
If and . Take , then
and the Lipschitz constant is . Hence,
Now we recall some basic definitions and their properties which will be used in the sequel.
Definition 2.5 (Classical notion)
Let and . A set-valued mapping is said to be -Hölder continuous at on a neighborhood of if and only if
When X is a normed space, we say that the vector-valued mapping is -Hölder continuous at on a neighborhood of iff
Definition 2.6 Let and . A set-valued mapping is said to be -Hölder continuous at , on neighborhoods and of and if and only if
for all , .
3 Main results
By using a nonlinear scalarization technique, we present the sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasi-equilibrium problem.
Let and be neighborhoods of and , respectively, and let and be set-valued mappings. For each and , we consider the following parametric generalized vector quasi-equilibrium problem (PGVQEP):
Find such that
For each and , let
The weak solution set of (6) is denoted by
For each , and fixed , the -solution set of (6) is denoted by
We first establish the following lemmas which will be used in the sequel.
Lemma 3.1 For each , and fixed ,
Proof Let , and fixed . For any , we have
Therefore, for each and each , we have
By Lemma 2.1(i), we conclude that . Since z is arbitrary, we have
which gives that .
On the other hand, for each , we have that
Thus, for each and each , we have that . By Lemma 2.1(i), we can obtain . Therefore, we have , which implies that
Hence, . The proof is completed. □
Lemma 3.2 Suppose that and are the given neighborhoods of and , respectively.
-
(a)
If for each , is -Hölder continuous at , then for any fixed , the function
is -Hölder continuous at .
-
(b)
If for each and , is -Hölder continuous on , then for any fixed , the function
is -Hölder continuous on .
Proof (a) Let . The -Hölder continuity of implies that there exists a neighborhood of such that for all ,
So, for any , there exist and such that
By using Proposition 2.3, we obtain
which gives that
Since is arbitrary and , we have
Applying the symmetry between and , we arrive at
It follows from the last two inequalities that
Therefore, we conclude that is -Hölder continuous at .
-
(b)
It follows by a similar argument as in part (a). The proof is completed. □
Now, by using the nonlinear scalarization technique, we propose some sufficient conditions for Hölder continuity of the solution mapping for (PGVQEP).
Theorem 3.3 For each fixed , let be nonempty in a neighborhood of . Assume that the following conditions hold.
-
(i)
is -Hölder continuous on ;
-
(ii)
For each , is -Hölder continuous at ;
-
(iii)
For each and , is -Hölder continuous on ;
-
(iv)
is -Hölder strongly monotone with respect to , that is, there exist constants , such that for every , ,
-
(v)
, , where is the Lipschitz constant of on Y.
Then, for every , the solution of (PVQGEP) is unique, and as a function of λ and μ satisfies the Hölder condition: for all ,
where , .
Proof Let . The proof is divided into the following three steps based on the fact that
where , .
Step 1: We prove that
for all and .
If , then we are done. So, we assume that . Since and , by the -Hölder continuity of , there exist and such that
and
Since and , by Lemma 3.1, we obtain
and
By virtue of (iv), we have
By combining (12) and (13) with the last inequality, we have
Whence, assumption (iv) implies that
Step 2: We prove that
for all and .
If , then we are done. So, we assume that . Since and , by the -Hölder continuity of and , there exist and such that
and
Again, by the Hölder continuity of , there exist and such that
and
Since and , by Lemma 3.1, we obtain the following:
and
By virtue of (iv), we have
By combining (20) and (21) with the last inequality, we have
By virtue of (16), (17), (18) and (19), we get
Whence, condition (v) implies that
Step 3: Let and . It follows from (9) and (15) that
Thus,
Taking and , we see that the diameter of is 0, that is, this set is a singleton . This implies that the (PGVQEP) has a unique solution in a neighborhood of . The proof is completed. □
Definition 3.4 Let be a set-valued mapping. A set-valued mapping is said to be
-
(A)
-Hölder strongly monotone with respect to if there exist and , such that for every with ,
-
(B)
-Hölder strongly pseudomonotone with respect to and , such that for every with ,
-
(C)
quasi-monotone on if with ,
The following proposition provides the relation among monotonicity conditions defined above.
Proposition 3.5
-
(i)
(A) ⇒ (iv).
-
(ii)
(B) and (C) ⇒ (iv).
Proof (i) From the definition of (A), we have
(ii) Assume that F satisfies definitions (B) and (C). We consider two cases.
Case 1. , , then there exists such that . From Lemma 2.1, we have
which implies that . Hence,
Case 2. , , then there exists such that . By a similar argument as in the previous case, we have the desired result. □
Remark 3.6 The converse of Proposition 3.5 does not hold in general, even in the special case and . See, for example, Examples 1.1 and 1.2 in [15]. Therefore, Theorem 3.3 still holds when condition (iv) is replaced by condition (A) or conditions (B) and (C). We can immediately obtain the following two theorems.
Theorem 3.7 Theorem 3.3 still holds when condition (iv) is replaced by condition (A).
Theorem 3.8 Theorem 3.3 still holds when condition (iv) is replaced by conditions (B) and (C).
Let be a vector-valued mapping. Then (PGVQEP) becomes the following parametric vector quasi-equilibrium problem (PVQEP):
Find such that
Remark 3.9 In the case of a vector-valued mapping, condition (iv) in Theorem 3.3 and condition (ii′′) coincide. Also, condition (A) and conditions (B) and (C) are the same as conditions (ii) and (ii′) in [30], respectively. It is obvious that Theorems 3.3, 3.7 and 3.8 extend Theorems 3.3, 3.1 and 3.2 in [30], respectively, in the case that the vector-valued mapping is extended to a set-valued one.
4 Applications
Since the parametric generalized vector quasi-equilibrium problem (PGVQEP) contains as special cases many optimization-related problems, including quasi-variational inequalities, traffic equilibrium problems, quasi-optimization problems, fixed point and coincidence point problems, complementarity problems, vector optimization, Nash equilibria, etc., we can derive from Theorem 3.3 a direct consequence for such special cases. We discuss now only some applications of our results.
4.1 Quasi-variational inequalities
In this section, we assume that X is a normed space. Let and be set-valued mappings, where denotes the space of all bounded linear mappings of X into Y. Setting in (6), we obtain parametric generalized vector quasi-variational inequalities (PGVQVI) in the case of set-valued mappings as follows:
For each and , let
The solution set of (25) is denoted by
For each , and fixed , the -solution set of (25) is
Theorem 4.1 Assume that for each fixed , is nonempty in a neighborhood of the considered point . Assume further that the following conditions hold.
-
(i')
is -Hölder continuous on ;
-
(ii')
For each , is -Hölder continuous at ;
-
(iii')
is bounded in , and is bounded;
-
(iv')
is -Hölder strongly monotone with respect to , i.e., there exist constants , such that for every : ,
-
(v')
, , where is the Lipschitz constant of on Y.
Then, for every , the solution of (PGVQVI) is unique, , and this function satisfies the Hölder condition: for all ,
where , .
Proof We verify that all the assumptions of Theorem 3.3 are fulfilled. First, (i′), (iv′) and (v′) are the same as (i), (iv) and (v) in Theorem 3.3. We need only to verify conditions (ii) and (iii). Taking such that
and
We put and . For any fixed , by assumption (ii′), we have
Then
Hence
Also, we put and . We need to show that
For each fixed and ,
Hence, condition (iii) is verified, and so we obtain the result. □
For (PGVQVI), if we put , , then (25) becomes the following parametric generalized quasi-variational inequality problem in the case of scalar-valued one:
For each and , let
The solution set of (26) is denoted by
For each , and fixed , the -solution set of (25) is
It follows from Lemma 2.1 that coincides with .
Corollary 4.2 Assume that is nonempty in a neighborhood of the considered point . Assume further that conditions (i′)-(iii′) and (v′) in Corollary 4.1 hold. Replace (iv′) by (iv′′).
(iv′′) is -Hölder strongly monotone, i.e., there exist constants , , such that for every : ,
Then, for every , the solution of (PGVQVI) is unique, , and this function satisfies the Hölder condition: for all ,
where , .
Proof It is not hard to show that (iv′′) implies (iv′). Indeed, for any with ,
Therefore, (iv′) is satisfied. □
Remark 4.3 Corollary 4.2 extends Corollary 3.1 in [33] since the mapping T is a multivalued mapping.
4.2 Traffic equilibrium problems
The foundation of the study of traffic network problems goes back to Wardrop [34], who stated the basic equilibrium principle in 1952. Over the past decades, a large number of efforts have been devoted to the study of traffic assignment models, with emphasis on efficiency and optimality, in order to improve practicability, reduce gas emissions and contribute to the welfare of the community. The variational inequality approach to such problems begins with the seminal work of Smith [35] who proved that the user-optimized equilibrium can be expressed in terms of a variational inequality. Thus, the possibility of exploiting the powerful tools of variational analysis has led to dealing with a large variety of models, reaching valuable theoretical results and providing applications in practical situations. In this paper, we are concerned with a class of equilibrium problems which can be studied in the framework of quasi-variational inequalities, see [36, 37].
Let a set N of nodes, a set L of links, a set of origin-destination pairs (O/D pairs for short) be given. Assume that there are paths connecting the pairs , , whose set is denoted by . Set ; i.e., there are in whole m paths in the traffic network. Let stand for the path flow vector. Assume that the travel cost of the path , , is a set . So, we have a multifunction with . Let the capacity restriction be
where are given real numbers. Extending the Wardrop definition to the case of multivalued costs, we propose the following definition.
A path flow vector H is said to be a weak equilibrium flow vector if
where and are among indices corresponding to .
A path flow vector H is said to be a strong equilibrium flow vector if
Suppose that the travel demand of the O/D pair , , depends on the weak (or strong) equilibrium problem flow H. So, considering all the O/D pairs, we have a mapping . We use the Kronecker notation
Then the matrix
is called an O/D pair/path incidence matrix. The path flow vectors meeting the travel demands are called the feasible path flow vectors and form the constraint set, for a given weak (or strong) equilibrium flow H,
Assume further that the path costs are also perturbed, i.e., depend on a perturbation parameter μ of a metric space M: , .
Our traffic equilibrium problem is equivalent to a quasi-variational inequality as follows (see [38]).
Lemma 4.4 A path vector flow is a weak equilibrium flow if and only if it is a solution of the following quasi-variational inequality:
Lemma 4.5 A path vector flow is a strong equilibrium flow if and only if it is a solution of the following quasi-variational inequality:
Corollary 4.6 Assume that solutions of the traffic network equilibrium problem exist and all the assumptions of Corollary 4.2 are satisfied. Then, in a neighborhood of , the solution is unique and satisfies the same Hölder condition as in Corollary 4.2.
4.3 Quasi-optimization problem
For the normed linear space Y and pointed, closed and convex cone C with nonempty interior, we denote the ordering induced by C as follows:
The orderings ≥ and > are defined similarly. Let be a vector-valued mapping. For each , consider the problem of parametric quasi-optimization problem (PQOP) finding such that
Since the constraint set depends on the minimizer , this is a quasi-optimization problem. Setting , (PVQEP) becomes a special case of (PQOP).
The following results are derived from Theorem 3.8 (Theorem 3.3 cannot be applied since , and ).
Theorem 4.7 For (PQOP), assume that the solution exists in a neighborhood of the considered point . Assume further that the following conditions hold.
-
(i)
is -Hölder continuous on ;
-
(ii)
For each , is -Hölder continuous at ;
-
(iii)
For each and , is -Hölder continuous on ;
-
(iv)
is -Hölder strongly monotone with respect to , i.e., there exist constants , such that for every : ,
-
(v)
, , where is the Lipschitz constant of on Y.
Then, for every , the solution of (PVQGEP) is unique, , and this function satisfies the Hölder condition:
for all ,
where , .
5 Conclusions
In this paper, by using a nonlinear scalarization technique, we obtain sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasi-equilibrium problem in the case where the mapping F is a general set-valued one. As applications, we derived this Hölder continuity for some quasi-variational inequalities, traffic network problems and quasi-optimization problems.
References
Ansari QH: Vector equilibrium problems and vector variational inequalities. In Vector Variational Inequalities and Vector Equilibria Mathematical Theories. Edited by: Giannessi F. Kluwer Academic, Dordrecht; 2000:1–16.
Ansari QH, Yao JC: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett. 1999, 12: 53–56.
Chuong TD, Yao JC: Isolated and proper efficiencies in semi-infinite vector optimization problems. J. Optim. Theory Appl. 2014, 162: 447–462. 10.1007/s10957-013-0425-2
Xue XW, Li SJ, Liao CM, Yao JC: Sensitivity analysis of parametric vector set-valued optimization problems via coderivatives. Taiwan. J. Math. 2011, 15: 2533–2554.
Chuong TD, Yao JC, Yen ND: Further results on the lower semicontinuity of efficient point multifunctions. Pac. J. Optim. 2010, 6: 405–422.
Gong XH, Kimura K, Yao JC: Sensitivity analysis of strong vector equilibrium problems. J. Nonlinear Convex Anal. 2008, 9: 83–94.
Kimura K, Yao JC: Semicontinuity of solution mappings of parametric generalized vector equilibrium problems. J. Optim. Theory Appl. 2008, 138: 429–443. 10.1007/s10957-008-9386-2
Kimura K, Yao JC: Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. J. Ind. Manag. Optim. 2008, 4: 167–181.
Kimura K, Yao JC: Sensitivity analysis of solution mappings of parametric vector quasiequilibrium problems. J. Glob. Optim. 2008, 41: 187–202. 10.1007/s10898-007-9210-9
Wangkeeree R, Wangkeeree R, Preechasilp P: Continuity of the solution mappings of parametric generalized vector equilibrium problems. Appl. Math. Lett. 2014, 29: 42–45.
Yen ND: Hölder continuity of solutions to parametric variational inequalities. Appl. Math. Optim. 1995, 31: 245–255. 10.1007/BF01215992
Mansour MA, Riahi H: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 2005, 306: 684–691. 10.1016/j.jmaa.2004.10.011
Bianchi M, Pini R: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 2003, 31: 445–450. 10.1016/S0167-6377(03)00051-8
Anh LQ, Khanh PQ: On the Hölder continuity of solutions to multivalued vector equilibrium problems. J. Math. Anal. Appl. 2006, 321: 308–315. 10.1016/j.jmaa.2005.08.018
Anh LQ, Khanh PQ: Uniqueness and Hölder continuity of solutions to multivalued vector equilibrium problems in metric spaces. J. Glob. Optim. 2007, 37: 349–465.
Anh LQ, Khanh PQ: Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions. J. Glob. Optim. 2008, 42: 515–531. 10.1007/s10898-007-9268-4
Li SJ, Li XB, Teo KL: The Hölder continuity of solutions to generalized vector equilibrium problems. Eur. J. Oper. Res. 2011, 199: 334–338.
Li SJ, Chen CR, Li XB, Teo KL: Hölder continuity and upper estimates of solution to vector quasiequilibrium problems. Eur. J. Oper. Res. 2011, 210: 148–157. 10.1016/j.ejor.2010.10.005
Chen CR, Li SJ, Teo KL: Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim. 2009, 45: 309–318. 10.1007/s10898-008-9376-9
Gong XH: Continuity of the solution set to parametric weak vector equilibrium problems. J. Glob. Optim. 2008, 139: 3–46.
Gong XH, Yao JC: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 2008, 138: 197–205. 10.1007/s10957-008-9379-1
Li SJ, Li XB: Hölder continuity of solutions to parametric weak generalized Ky Fan inequality. J. Optim. Theory Appl. 2011, 149: 540–553. 10.1007/s10957-011-9803-9
Wang QL, Lin Z, Li XX: Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem. Positivity 2014. 10.1007/s11117-014-0273-9
Peng ZY: Hölder continuity of solutions to parametric generalized vector quasiequilibrium problems. Abstr. Appl. Anal. 2012. 10.1155/2012/236413
Chen GY, Huang XX, Yang XQ: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin; 2005.
Luc DT: Theory of Vector Optimization. Springer, Berlin; 1989.
Chen CR, Li SJ: Semicontinuity of the solution set map to a set-valued weak vector variational inequality. J. Ind. Manag. Optim. 2007, 3: 519–528.
Bianchi M, Pini R: Sensitivity of parametric vector equilibria. Optimization 2006, 55: 221–230. 10.1080/02331930600662732
Chen CR, Li MH: Hölder continuity of solutions to parametric vector equilibrium problem with nonlinear scalarization. Numer. Funct. Anal. Optim. 2014, 35: 685–707. 10.1080/01630563.2013.818549
Chen CR: Hölder continuity of the unique solution to parametric vector quasiequilibrium problems via nonlinear scalarization. Positivity 2013, 17: 133–150. 10.1007/s11117-011-0153-5
Chen GY, Yang XQ, Yu H: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Glob. Optim. 2005, 32: 451–466. 10.1007/s10898-003-2683-2
Li SJ, Yang XQ, Chen GY: Nonconvex vector optimization of set-valued mappings. J. Math. Anal. Appl. 2003, 283: 337–350. 10.1016/S0022-247X(02)00410-9
Anh LQ, Khanh PQ: Hölder continnuity of the unique solution to quasequilibrium problems in metric spaces. J. Optim. Theory Appl. 2009, 141: 37–54. 10.1007/s10957-008-9508-x
Wardrop JG: Some theoretical aspects of road traffic research. Proc. Inst. Civil Eng., Part II 1952.
Smith MJ: The existence, uniqueness and stability of traffic equilibrium. Transp. Res., Part B, Methodol. 1979, 13: 295–304.
De Luca M, Maugeri A: Quasi-variational inequality and applications to equilibrium problems with elastic demands. In Nonsmooth Optimization and Related Topic. Edited by: Clarke FM, Demyanov VF, Giannessi F. Plenum, New York; 1989:61–77.
De Luca M: Existence of solutions for a time-dependent quasi-variational inequality. Rend. Circ. Mat. Palermo Suppl. 1997, 48: 101–106.
Khanh PQ, Luu LM: On the existence of solution to vector quasivariational inequalities and quasicomplementarity with applications to traffic equilibria. J. Optim. Theory Appl. 2004, 123: 1533–1548.
Acknowledgements
The authors were partially supported by the Thailand Research Fund and Naresuan University, Grant No. RSA5780003. The authors would like to thank the referees for their remarks and suggestions, which helped to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Wangkeeree, R., Preechasilp, P. On the Hölder continuity of solution maps to parametric generalized vector quasi-equilibrium problems via nonlinear scalarization. J Inequal Appl 2014, 425 (2014). https://doi.org/10.1186/1029-242X-2014-425
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-425