Abstract
We introduce the concept of Levitin-Polyak well-posedness for a system of generalized quasi-variational inclusion problems and show some characterizations of Levitin-Polyak well-posedness for the system of generalized quasi-variational inclusion problems under some suitable conditions. We also give some results concerned with the Hadamard well-posedness for the system of generalized quasi-variational inclusion problems.
MSC:49J40, 49K40, 90C31.
Similar content being viewed by others
1 Introduction
Well-posedness plays a crucial role in the theory and methodology of scalar optimization problems. In 1966, Tykhonov [1] first introduced the concept of well-posedness for a global minimizing problem, which has become known as Tykhonov well-posedness. Soon after, Levitin-Polyak [2] strengthened the concept of Tykhonov well-posedness, already known as the Levitin-Polyak (for short, LP) well-posedness. Subsequently, some authors studied the LP well-posedness for convex scalar optimization problems with functional constraints [3], vector optimization problems [4], variational inequality problems [5], generalized mixed variational inequality problems [6], generalized quasi-variational inequality problems [7], generalized vector variational inequality problems [8], equilibrium problems [9], vector equilibrium problems [10], generalized vector quasi-equilibrium problems [11] and generalized quasi-variational inclusion and disclusion problems [12]. Another important notion of the well-posedness for a minimizing problem is the well-posedness by perturbations or the extended well-posedness due to Zolezzi [13]. The notion of the well-posedness by perturbations establishes a form of continuous dependence of the solutions upon a parameter. Recently, Lemaire et al. [14] introduced the well-posedness by perturbations for variational inequalities and Fang et al. [15] considered the well-posedness by perturbations for mixed variational inequalities in Banach spaces. For more details about the well-posedness by perturbations, we refer readers to [16, 17] and the references therein.
On the other hand, for optimization problems, there is another concept of well-posedness, which has become known as the Hadamard well-posedness. The concept of Hadamard well-posedness was inspired by the classical idea of Hadamard, which goes back to the beginning of the last century. It requires the existence and uniqueness of the optimal solution together with continuous dependence on the problem data. Some results about the Hadamard well-posedness can be found in [18–20]. Recently, the concept of Hadamard well-posedness has been extended to vector optimization problems and vector equilibrium problems. Li and Zhang [21] investigated the Hadamard well-posedness for vector optimization problems. Zeng et al. [22] obtained a sufficient condition for the Hadamard well-posedness of a set-valued optimization problem. Salamon [23] investigated the generalized Hadamard well-posedness for parametric vector equilibrium problems with trifunctions.
Very recently, Lin and Chuang [24] studied the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems as constraints. Motivated by Lin, Wang et al. [25] investigated the well-posedness for generalized quasi-variational inclusion problems and for optimization problems with generalized quasi-variational inclusion problems as constraints. A system of generalized quasi-variational inclusion problems, which consists of a family of generalized quasi-variational inclusion problems defined on a product set, was first introduced by Lin [26]. It is well known that the system of generalized quasi-variational inclusion problems contains the system of variational inequalities, the system of equilibrium problems, the system of vector equilibrium problems, the system of vector quasi-equilibrium problems, the system of generalized vector quasi-equilibrium problems, the system of variational inclusions problems and variational disclusions problems as special cases. For more details, one can refer to [27–33] and the references therein. Nonetheless, to the best of our knowledge, there is no paper dealing with the Levitin-Polyak and Hadamard well-posedness for the system of generalized quasi-variational inclusion problems. Therefore, it is very interesting to generalize the concept of Levitin-Polyak and Hadamard well-posedness to the system of generalized quasi-variational inclusion problems.
Motivated and inspired by research work mentioned above, in this paper, we study the LP and Hadamard well-posedness for the system of generalized quasi-variational inclusion problems. This paper is organized as follows. In Section 2, we introduce the concept of LP well-posedness for the system of generalized quasi-variational inclusion problems. Some characterizations of the LP well-posedness for the system of generalized quasi-variational inclusion problems are shown in Section 3. Some results concerned with Hadamard well-posedness for the system of generalized quasi-variational inclusion problems are given in Section 4.
2 Preliminaries
Let I be an index set and be a metric space. For each , let be a metric space, and be Hausdorff topological vector spaces, be a nonempty closed and convex subset. Set , and . For each , let , and be set-valued mappings. Let be a continuous mapping. Throughout this paper, unless otherwise specified, we use these notations and assumptions.
Now, we consider the following system of generalized quasi-variational inclusion problems (for short, SQVIP).
Find such that, for each , and there exists satisfying
for all . We denote by S the solution set of (SQVIP).
If the mapping is perturbed by a parameter , that is, such that, for some , for all , then, for any , we define a parametric system of generalized quasi-variational inclusion problem (for short, PSQVIP): Find such that, for each , and there exists satisfying
for all .
Some special cases of (SQVIP) are as follows:
-
(I)
If, for each , is a mapping, is a pointed, closed and convex cone with for every , reduces to a single-valued mapping and for all , then (SQVIP) reduces to the system of vector equilibrium problems: Find such that, for each , and there exists satisfying
for all , which has been studied by Peng and Wu [34] and the references therein.
-
(II)
If, for each , and are set-valued mappings, is a pointed, closed and convex cone with for all , for all , then (SQVIP) reduces to the system of set-valued vector quasi-equilibrium problems of Chen et al. [35]: Find such that, for each , and there exists satisfying
for all .
-
(III)
If the index set I is a single set, then (SQVIP) reduces to the generalized quasi-variational inclusion problem studied in Wang et al. [12, 25] and the references therein.
Definition 2.1 Let and be a sequence such that . A sequence is called a LP approximating solution sequence for (SQVIP) corresponding to if, for each and , there exists a sequence of nonnegative real numbers with and such that
and
for all , where denote the closed interval .
Definition 2.2 (1) (SQVIP) is said to be LP well-posed by perturbations if it has a unique solution and, for all with , every LP approximating solution sequence for (SQVIP) corresponding to converges strongly to the unique solution.
-
(2)
(SQVIP) is said to be generalized LP well-posed by perturbations if the solution set S for (SQVIP) is nonempty and, for all sequences with , every LP approximating solution sequence for (SQVIP) corresponding to has some subsequence which converges strongly to some point of S.
Definition 2.3 [36]
Let , be two topological spaces. A set-valued mapping is said to be:
-
(1)
upper semicontinuous (for short, u.s.c.) at if, for any neighborhood V of , there exists a neighborhood U of x such that for all ;
-
(2)
lower semicontinuous (for short, l.s.c.) at if, for each open set V in with , there exists an open neighborhood of x such that for all ;
-
(3)
u.s.c. (resp., l.s.c.) on if it is u.s.c. (resp., l.s.c.) on every point ;
-
(4)
continuous on if it is both u.s.c. and l.s.c. on ;
-
(5)
closed if the graph of F is closed, i.e., the set is closed in .
Definition 2.4 [37]
Let and be two metric spaces. A set-valued mapping is said to be -subcontinuous if, for any sequence converging strongly in , the sequence with has a strongly convergent subsequence.
Definition 2.5 [38]
Let A be a nonempty subset of X, the measure of noncompactness μ of the set A is defined by
Definition 2.6 [38]
Let A and B be two nonempty subsets of a Banach space X. The Hausdorff metric between A and B is defined by
where with .
3 The Levitin-Polyak well-posedness for (SQVIP)
In this section, we discuss some metric characterizations of the LP well-posedness for (SQVIP). First, we introduce the following LP approximating solution set for (SQVIP):
for all , where denotes the closed ball centered at with radius δ.
Clearly, we have the following:
-
(1)
for all ;
-
(2)
if and , then .
Next, we present some properties of .
Proposition 3.1 For each , let be compact-valued, be closed-valued and be closed for all . Then .
Proof Clearly, . Hence we only need to show that . If not, then there exists such that . Thus, for any and , we have . For each and , it follows that and there exist and such that
and
for all . Clearly, . Since and is a compact set, there exist a subsequence of and such that and, for each ,
for all . For all , there exists such that
for all . Clearly, . Since is closed for all , this together with (2) implies that
for all . Since is closed-valued, it follows from (1) that and so , which is a contradiction. This completes the proof. □
Example 3.1 Let I be a single set, , and . For any , let
Then it is easy to see that all the conditions of Proposition 3.1 are satisfied. By Proposition 3.1, . Indeed, for all ,
and
Therefore, .
Proposition 3.2 For each , assume that
-
(i)
P is a finite-dimensional space;
-
(ii)
is u.s.c. and compact-valued;
-
(iii)
is -subcontinuous, l.s.c. and closed;
-
(iv)
is closed.
Then is closed for any .
Proof For any , let and . Then there exists such that, for each ,
and there exists such that
for all . Since P is a finite-dimensional space, we can suppose that . In order to prove that , we first prove that, for each ,
Assume that the left inequality does not hold. Then there exists such that
Thus there exist an increasing sequence and a sequence with such that
Since, for each , is closed and -subcontinuous, the sequence has a subsequence, which is still denoted by , converging strongly to a point . It follows that, for each ,
which is a contradiction. Thus, for each , . Since, for each , is u.s.c. and compact-valued, there exist a subsequence of and such that . For any , since is l.s.c., there exists a sequence with such that and, for each ,
Since is closed and is continuous, we obtain
for all . Thus and so is closed. This completes the proof. □
Remark 3.1 If I is a single set and for all , then Propositions 3.1 and 3.2 can be considered as a generalization of Properties 3.1 and 3.2 of [25], respectively.
In this paper, let for all . It is clear that is a metric space.
Theorem 3.1 For each , let be complete. We assume that
-
(i)
is u.s.c. and compact-valued;
-
(ii)
is -subcontinuous, l.s.c. and closed;
-
(iii)
is closed.
Then (SQVIP) is LP well-posed by perturbations if and only if, for any ,
as .
Proof Suppose that (SQVIP) is LP well-posed by perturbations. Then (SQVIP) has a unique solution for any . This implies that for any .
Now, we show that
as . If not, then there exist , sequences and of nonnegative real numbers with , and the sequences and with satisfying
for all . Since , there exist and and such that
for all and
for all . Clearly, and . Thus and are both the LP approximating solution sequences for (SQVIP) corresponding to and , respectively. Since (SQVIP) is LP well-posed by perturbations, and have to converge strongly to the unique solution of (SQVIP), which is a contradiction to (5).
Conversely, suppose that (4) holds. Let be any sequence with and be the LP approximating solution sequence for (SQVIP) corresponding to . Then there exist a sequence of nonnegative real numbers with and such that
and
for all . Set . Then and and . It follows from (4) that is a Cauchy sequence and so it converges strongly to a point . By the similar arguments as in the proof of Proposition 3.2, we can show that and there exists such that
for all . Thus is a solution of (SQVIP).
Finally, to complete the proof, it is sufficient to prove that (SQVIP) has a unique solution. If (SQVIP) has two distinct solutions x and , then it is easy to see that for any . It follows that
for all , which contradicts (4). Thus (SQVIP) has a unique solution. This completes the proof. □
Remark 3.2 If I is a single set, for all , then Theorem 3.1 can be seen as a generalization of Theorem 3.1 of [25].
Example 3.2 Let I be a single set, , and . For all , let
Then A is -subcontinuous, l.s.c. and closed, T is u.s.c. and compact-valued and G is closed. For any , we have
and
for sufficiently small . Therefore, as .
Theorem 3.2 For each , let be complete and P be a finite-dimensional space. We assume that
-
(i)
is u.s.c. and compact-valued;
-
(ii)
is -subcontinuous, l.s.c. and closed;
-
(iii)
is closed.
Then (SQVIP) is generalized LP well-posed by perturbations if and only if, for any ,
as .
Proof Suppose that (SQVIP) is generalized LP well-posed by perturbations. Then S is nonempty.
Now, we prove that S is compact. Indeed, let be a sequence in S. Then is the LP approximating solution sequence for (SQVIP) corresponding to . Since (SQVIP) is generalized LP well-posed by perturbations, has a subsequence which converges strongly to a point of S. This implies that S is compact. For any , since , we have and
Since S is compact,
In order to prove , we need to prove that
as . Assume that this is not true. Then there exist , and the sequences and of nonnegative real numbers with and with such that, for n sufficiently large,
Since , there exists such that, for each , and there exists satisfying
for all , it follows that and is the LP approximating solution sequence for (SQVIP) corresponding to . By the generalized LP well-posedness by perturbations of (SQVIP), there exists a subsequence of which converges strongly to a point of S, which contradicts (10).
Conversely, suppose that (9) holds. From Propositions 3.1 and 3.2, is closed for any and . Since as , theorem on p.412 in [38] can be applied and one concludes that the set S is nonempty compact and
as . Let be any sequence in P with . If is the LP approximating solution sequence for (SQVIP) corresponding to , then there exists a sequence of nonnegative numbers with and with such that
and
for all . For each , let . Then and . Thus it follows from (11) that
as . The compactness of S implies that (SQVIP) is generalized LP well-posed by perturbations. This completes the proof. □
Remark 3.3 If I is a single set, for every , then Theorem 3.2 can be considered as a generalization of Theorem 3.2 of [25].
Example 3.3 Let I be a single set, , and . For all , let
Then all the conditions of Theorem 3.2 are satisfied. It follows that, for any ,
and
for sufficiently small . Therefore, as . From Theorem 3.2, (SQVIP) is generalized LP well-posedness by perturbations.
4 The Hadamard well-posedness for (SQVIP)
In this section, for each , we assume that , and are finite-dimensional spaces, is a nonempty closed and convex subset.
For each , let be the collection of all such that
-
(i)
is continuous and bounded compact-convex-valued;
-
(ii)
is u.s.c. and bounded compact-convex-valued;
-
(iii)
is u.s.c. and bounded compact-convex-valued.
Definition 4.1 [39]
A sequence of nonempty subsets of is said to be convergent to D in the sense of Painlevé-Kuratowski (for short, ) if
where , the inner limit, consists of all possible limit points of the sequences with for all and , the outer limit, consists of all possible cluster points of such sequences.
Definition 4.2 [22]
A sequence of nonempty set-valued mappings is said to be convergent to a set-valued mapping in the sense of Painlevé-Kuratowski (for short, ) if , where and .
We say that, for each , a sequence converges to in the sense of Painlevé-Kuratowski (for short, ) if , and .
Next, we give the definition of the Hadamard well-posedness for (SQVIP). As mentioned above, we denote by S the solution set of (SQVIP) determined by for each . Similarly, we denote by the solution set of (SQVIP) n determined by for each and , where (SQVIP) n is formulated as follows:
Find such that, for each , there exists satisfying
for all .
Definition 4.3 (SQVIP) is said to be Hadamard well-posed if its solution set and, when, for each , every sequence of pairs converges to in the sense of Painlevé-Kuratowski, any sequence satisfying has a subsequence which converges strongly to a point in S.
Theorem 4.1 For each , let for all . Then the solution set for (SQVIP) n is closed.
Proof Without loss of generality, we suppose that . Take any sequence satisfying . For each , since is closed, it follows that K is closed and . Now, implies that, for each , there exists such that
for all . For each , since , is u.s.c. and compact-valued, this implies that there exist and a subsequence of such that . Since is continuous and compact-valued, it follows that is closed, this implies that . For each , since is continuous, there exists a sequence with such that
Since is u.s.c. and compact-valued, we know that is closed, which implies that . Therefore, is closed. This completes the proof. □
Theorem 4.2 For each , let be a nonempty compact subset of , for all , and . Then
Proof Suppose that (12) does not hold. Then there exists satisfying
From (13), it follows that there exists such that the sequence has a subsequence, which is still denoted by , converging strongly to . For each , since is compact, we know that K is compact. Again, from , and Theorem 4.1, it follows that and S are both compact. Thus, for n sufficiently large, there exists satisfying
where and denotes the ball with the center y and the radius ϵ. It follows from that, for each , there exists such that
for all . For each , since , we have
Again, since is bounded and is bounded, there exists a subsequence of with converging strongly to a point . This together with (14) implies that . By similar arguments, we also know that . Since , we have
By Theorem 5.37 of [39], for all , there exist a sequence converging strongly to and such that for all and . It follows from (14) that there exists such that . Since , we have
Again, since , , , and are bounded, it follows that there exists a subsequence of converging strongly to a point and so, from (16), . Since for all , we get . This implies that , which is a contradiction. This completes the proof. □
Theorem 4.3 For each , let be a nonempty compact subset of , and . Then (SQVIP) is Hadamard well-posed.
Proof For each , let , and be a sequence satisfying . For each , by the compactness of , we know that K is compact. Again, from and the compactness of K, it follows that and so, from Theorem 4.2,
Thus has a subsequence which converges strongly to an element in S and so (SQVIP) is Hadamard well-posed. This completes the proof. □
References
Tykhonov AN: On the stability of the functional optimization problem. Ž. Vyčisl. Mat. Mat. Fiz. 1966, 6: 631–634.
Levitin ES, Polyak BT: Convergence of minimizing sequences in conditional extremum problem. Sov. Math. Dokl. 1996, 7: 764–767.
Konsulova AS, Revalski JP: Constrained convex optimization problems-well-posednesss and stability. Numer. Funct. Anal. Optim. 1994, 15: 889–907. 10.1080/01630569408816598
Huang XX, Huang XQ: Levitin-Polyak well-posedness of constrained vector optimization problems. J. Glob. Optim. 2007, 37: 287–304. 10.1007/s10898-006-9050-z
Hu R, Fang YP: Levitin-Polyak well-posedness of variational inequalities. Nonlinear Anal. 2010, 72: 373–381. 10.1016/j.na.2009.06.071
Li XB, Xia FQ: Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces. Nonlinear Anal. 2012, 75: 2139–2153. 10.1016/j.na.2011.10.013
Jiang B, Zhang J, Huang XX: Levitin-Polyak well-posedness of generalized quasivariational inequalities problems with functional constraints. Nonlinear Anal. 2009, 70: 1492–1530. 10.1016/j.na.2008.02.029
Xu Z, Zhu DL, Huang XX: Levitin-Polyak well-posedness in generalized vector variational inequality problem with functional constraints. Math. Methods Oper. Res. 2008, 67: 505–524. 10.1007/s00186-007-0200-y
Long XJ, Huang NJ, Teo KL: Levitin-Polyak well-posedness for equilibrium problems with functional constraints. J. Inequal. Appl. 2008., 2008: Article ID 657329
Li SJ, Li MH: Levitin-Polyak well-posedness of vector equilibrium problems. Math. Methods Oper. Res. 2008, 69: 125–140.
Li MH, Li SJ, Zhang WY: Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. J. Ind. Manag. Optim. 2009, 5: 683–696.
Wang SH, Huang NJ: Levitin-Polyak well-posedness for generalized quasi-variational inclusion and disclusion problems and optimization problems with constraints. Taiwan. J. Math. 2012, 16: 237–257.
Zolezzi T: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 1996, 91: 257–266. 10.1007/BF02192292
Lemaire B, Ould Ahmed Salem C, Revalski JP: Well-posedness by perturbations of variational problems. J. Optim. Theory Appl. 2002, 115: 345–368. 10.1023/A:1020840322436
Fang YP, Huang NJ, Yao JC: Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur. J. Oper. Res. 2010, 201: 682–692. 10.1016/j.ejor.2009.04.001
Huang XX: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 2001, 53: 101–116. 10.1007/s001860000100
Zolezzi T: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. 1995, 25: 437–453. 10.1016/0362-546X(94)00142-5
Lucchetti R, Patrone F: Hadamard and Tykhonov well-posedness of certain class of convex functions. J. Math. Anal. Appl. 1982, 88: 204–215. 10.1016/0022-247X(82)90187-1
Yang H, Yu J: Unified approaches to well-posedness with some applications. J. Glob. Optim. 2005, 31: 371–383. 10.1007/s10898-004-4275-1
Yu J, Yang H, Yu C: Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems. Nonlinear Anal. 2007, 66: 777–790. 10.1016/j.na.2005.10.018
Li SJ, Zhang WY: Hadamard well-posedness vector optimization problems. J. Glob. Optim. 2010, 46: 383–393. 10.1007/s10898-009-9431-1
Zeng J, Li SJ, Zhang WY, Xue XW: Hadamard well-posedness for a set-valued optimization problems. Optim. Lett. 2013. 10.1007/s11590-011-0439-3
Salamon J: Closedness of the solution map for parametric vector equilibrium problems with trifunctions. J. Glob. Optim. 2010, 47: 173–183. 10.1007/s10898-009-9464-5
Lin LJ, Chuang CS: Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraints. Nonlinear Anal. 2009, 70: 3607–3617.
Wang SH, Huang NJ, O’Regan D: Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints. J. Glob. Optim. 2013, 55: 189–208. 10.1007/s10898-012-9980-6
Lin LJ: Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems. J. Glob. Optim. 2007, 38: 21–39. 10.1007/s10898-006-9081-5
Ansari QH, Yao JC: System of generalized variational inequalities and their applications. Appl. Anal. 2000, 76: 203–217. 10.1080/00036810008840877
Lin LJ, Du WS: Systems of equilibrium problems with applications to generalized Ekeland’s variational principle and systems of semi-infinite problems. J. Glob. Optim. 2008, 40: 663–677. 10.1007/s10898-007-9146-0
Huang NJ, Li J, Yao JC: Gap functions and existence of solutions for a system of vector equilibrium problems. J. Optim. Theory Appl. 2007, 133: 201–212. 10.1007/s10957-007-9202-4
Ansari QH, Chan WK, Yang XQ: The system of vector quasi-equilibrium problems with applications. J. Glob. Optim. 2004, 29: 45–57.
Ansari QH, Schaible S, Yao JC: The system of generalized vector equilibrium problems with applications. J. Glob. Optim. 2002, 23: 3–16.
Huang NJ, Li J, Wu SY: Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings. J. Glob. Optim. 2008, 41: 401–415. 10.1007/s10898-007-9248-8
Lin LJ, Tu CI: The studies of systems of variational inclusions problems and variational disclusions problems with applications. Nonlinear Anal. 2008, 69: 1981–1998. 10.1016/j.na.2007.07.041
Peng JW, Wu SY: The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems. Optim. Lett. 2010, 4: 501–512. 10.1007/s11590-010-0179-9
Chen JW, Wan ZP, Cho YJ: Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems. Math. Methods Oper. Res. 2013. 10.1007/s00186-012-0414-5
Aubin JP, Ekeland I: Applied Nonlinear Analysis. Wiley, New York; 1984.
Lignola MB: Well-posedness and L -well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 2006, 128: 119–138. 10.1007/s10957-005-7561-2
Kuratowski K 1. In Topology. Academic Press, New York; 1966.
Rockafellar RT, Wets RJB: Variational Analysis. Springer, Berlin; 1998.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11171237) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013053358).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Li, Xb., Agarwal, R.P., Cho, Y.J. et al. The well-posedness for a system of generalized quasi-variational inclusion problems. J Inequal Appl 2014, 321 (2014). https://doi.org/10.1186/1029-242X-2014-321
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-321