Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems
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Authors
Jing Zeng
- College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China.
Zai-Yun Peng
- College of Mathematics and Statistics, School of Science, Chongqing Jiaotong University, Chongqing 400074, China.
Xiang-Kai Sun
- College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China.
Xian-Jun Long
- College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China.
Abstract
In this paper, we establish an existence result for the (GSVQEP) without assuming that the dual of the
ordering cone has a weak star compact base and give an example to show our existence theorem is different
from the main result of Long et al. [X. J. Long, N. J. Huang, K. L. Teo, Math. Comput. Modelling, 47
(2008), 445-451]. Furthermore, we introduce a concept of Hadamard-type well-posedness for the (GSVQEP)
and establish sufficient conditions of Hadamard-type well-posedness for the (GSVQEP).
Share and Cite
ISRP Style
Jing Zeng, Zai-Yun Peng, Xiang-Kai Sun, Xian-Jun Long, Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 6, 4104--4113
AMA Style
Zeng Jing, Peng Zai-Yun, Sun Xiang-Kai, Long Xian-Jun, Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems. J. Nonlinear Sci. Appl. (2016); 9(6):4104--4113
Chicago/Turabian Style
Zeng, Jing, Peng, Zai-Yun, Sun, Xiang-Kai, Long, Xian-Jun. "Existence of solutions and Hadamard well-posedness for generalized strong vector quasi-equilibrium problems." Journal of Nonlinear Sciences and Applications, 9, no. 6 (2016): 4104--4113
Keywords
- Existence theorem
- Hadamard well-posedness
- fixed-point theorem
- lower semicontinuity
- naturally quasi-convexity.
- Existence theorem
- Hadamard well-posedness
- fixed-point theorem
- lower semicontinuity
- naturally quasi-convexity.
MSC
References
-
[1]
Q. H. Ansari, W. K. Chan, X. Q. Yang, The system of vector equilibrium problems with applications, J. Global Optim., 29 (2004), 45-57.
-
[2]
J. P. Aubin, I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York (1984)
-
[3]
C. Berge, Espaces Topologiques, Dunod, Paris (1959)
-
[4]
B. Chen, N. J. Huang, C. F. Wen, Existence and stability of solutions for generalized symmetric strong vector quasi-equilibrium problems, Taiwanese J. Math., 16 (2012), 941-62.
-
[5]
G. Y. Chen, X. Huang, X. Yang, Vector optimization, Springer-Verlag, Berlin, Heidelberg (2005)
-
[6]
C. R. Chen, M. H. Li, Hölder continuity of solutions to parametric vector equilibrium problems with nonlinear scalarization, Numer. Funct. Anal. Optim., 35 (2014), 685-707.
-
[7]
G. Y. Chen, X. Q. Yang, H. Yu, A nonlinear scalarization function and generalized quasi-vector equilibrium problem, J. Global Optim., 32 (2005), 451-466.
-
[8]
X. P. Ding, New systems of generalized vector quasi-equilibrium problems in product FC-spaces, J. Global Optim., 46 (2010), 133-146.
-
[9]
A. L. Dontchev, T. Zolezzi, Well-posed optimization problems, Springer-Verlag, Berlin (1993)
-
[10]
Y. P. Fang, N. J. Huang, Existence of solutions to generalized vector quasi-equilibrium problems with discontinuous mappings, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1127-1132.
-
[11]
A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 322 (2006), 1099-1110.
-
[12]
Y. Y. Feng, Q. S. Qiu , Optimality conditions for vector equilibrium problems with constraint in Banach spaces, Optim. Lett., 8 (2014), 1931-1944.
-
[13]
J. Y. Fu, S. H. Wang, Generalized strong vector quasi-equilibrium problem with domination structure, J. Global Optim., 55 (2013), 839-847.
-
[14]
F. Giannessi, Vector variational inequilities and vector equilibria: mathematical theories, Kluwer Academic Publishers, Dordrecht (2000)
-
[15]
I. L. Glicksberg, A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.
-
[16]
X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problem, J. Optim. Theory Appl., 108 (2001), 139-154.
-
[17]
X. H. Gong, Strong vector equilibrium problem, J. Global Optim., 36 (2006), 339-349.
-
[18]
S. H. Hou, X. H. Gong, X. M. Yang, Existence and stability of solutions for generalized Ky Fan inequality problems with trifunctions, J. Optim. Theory Appl., 146 (2010), 387-398.
-
[19]
N. J. Huang, J. Li, S. Y. Wu, Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings , J. Global Optim., 41 (2008), 401-415.
-
[20]
N. V. Hung, Existence conditions for symmetric generalized quasi-variational inclusion problems, J. Inequal. Appl., 2013 (2013), 12 pages.
-
[21]
S. J. Li, K. L. Teo, X. Q. Yang, Generalized vector quasi-equilibrium problem, Math. Methods Oper. Res., 61 (2005), 385-397.
-
[22]
S. J. Li, J. Zeng, Existences of solutions for generalized vector quasi-equilibrium problems, Optim. Lett., 2 (2008), 341-349.
-
[23]
S. J. Li, W. Y. Zhang, Hadamard well-posed vector optimization problems, J. Global Optim., 46 (2010), 383-393.
-
[24]
Z. Lin, The study of the system of generalized vector quasi-equilibrium problems, J. Global Optim., 36 (2006), 627-635.
-
[25]
X. J. Long, N. J. Huang, K. L. Teo, Existence and stability of solutions for generalized strong vector quasiequilibrium problem, Math. Comput. Modelling, 47 (2008), 445-451.
-
[26]
R. Luccchetti, J. Revaliski, Recent Developments in Well-posed Variarional Problems, Kluwer Academic Publishers, Dordrecht (1995)
-
[27]
Z. Y. Peng, Y. Zhao, X. M. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500.
-
[28]
X. K. Sun, S. J. Li , Gap functions for generalized vector equilibrium problems via conjugate duality and applications, Appl. Anal., 92 (2013), 2182-2199.
-
[29]
T. Tanaka, Generalized quasiconvexities, cone saddle points and minimax theorems for vector valued functions, J. Optim. Theory Appl., 81 (1994), 355-377.
-
[30]
Q. L. Wang, Second-order optimality conditions for set-valued vector equilibrium problems, Numer. Funct. Anal. Optim., 34 (2013), 94-112.
-
[31]
J. Yu , Essential weak efficient solution in multiobjective optimization problems, J. Math. Anal. Appl., 166 (1992), 230-235.
-
[32]
J. Zeng, S. J. Li, W. Y. Zhang, X. W. Xue, Hadamard well-posedness for a set-valued optimization problem, Optim. Lett., 7 (2013), 559-573.
-
[33]
W. Zhang, J. Chen, S. Xu, W. Dong, Scalar gap functions and error bounds for generalized mixed vector equilibrium problems with applications, Fixed Point Theory Appl., 2015 (2015), 10 pages.
-
[34]
Y. H. Zhou, J. Y. Yu, H. Yang, S. W. Xiang, Hadamard types of well-posedness of non-self set-valued mappings for coincide points, Nonlinear Anal., 63 (2005), 2427-2436.