Characterizations of HContinuity for Solution Mapping to Parametric Generalized Weak Vector QuasiEquilibrium Problems

SHAO Chongyang; PENG Zaiyun; WANG Jingjing; ZHOU Daqiong

doi:10.21656/1000-0887.390198

Volume 40 Issue 4

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2019 > 40(4): 452-462

SHAO Chongyang, PENG Zaiyun, WANG Jingjing, ZHOU Daqiong. Characterizations of HContinuity for Solution Mapping to Parametric Generalized Weak Vector QuasiEquilibrium Problems[J]. Applied Mathematics and Mechanics, 2019, 40(4): 452-462. doi: 10.21656/1000-0887.390198

Citation:

PDF( 397 KB)

Characterizations of HContinuity for Solution Mapping to Parametric Generalized Weak Vector QuasiEquilibrium Problems

doi: 10.21656/1000-0887.390198

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P.R.China

Funds: The National Natural Science Foundation of China(11431004;11471059)

Received Date: 2018-07-17
Rev Recd Date: 2018-08-31
Publish Date: 2019-04-01

Abstract

Abstract

The stability of a class of parametric generalized weak vector quasi-equilibrium problems (PGWVQEP) in Hausdorff topological vector spaces, were studied. First, a parametric gap function for the problem was given, and the continuity property of the function was studied. Next, a key hypothesis related to the gap function for the considered problem was presented, the characterizations of this hypothesis were discussed, and an equivalence theorem for the key hypothesis was given. Finally, by means of the hypothesis, the sufficient and necessary conditions for the Hausdorff semicontinuity of the solution mapping to PGWVQEP were obtained. Examples were given to verify the obtained results.

FullText(HTML)

References(23)

References

[1]	BLUM E, OETTLI W. From optimization and variational inequalities to equilibrium problems[J]. The Mathmatics Student,1994,63: 123-145.
[2]	BIANCHI M, HADJISAVVAS N, SCHAIBLE S. Vector equilibrium problems with generalized monotone bifunctions[J].Journal of Optimization Theory and Applications,1997,92(3): 527-542.
[3]	ANSARI Q H, OETTLI W, SCHLAGER D. A generalization of vectorial equilibria[J]. Mathematical Methods of Operations Research,1997,46(2): 147-152.
[4]	〖JP2〗LONG X J, HUANG N J, TEO K L. Existence and stability of solutions for generalized strong vector quasi-equilibrium problem[J]. Mathematical and Computer Modelling,2008,47(3/4): 445-451.
[5]	GONG X H. Continuity of the solution set to parametric weak vector equilibrium problems[J]. Journal of Optimization Theory and Applications,2008,139(1): 35-46.
[6]	GONG X H, YAO J C. Lower semicontinuity of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications,2008,138(2): 197-205.
[7]	ANH L Q, KHANH P Q. Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks ii: lower semicontinuities applications[J]. Set-Valued Analysis,2008,16(7/8): 943-960.
[8]	ANH L Q, KHANH P Q. Continuity of solution maps of parametric quasiequilibrium problems[J]. Journal of Global Optimization,2010,46(2): 247-259.
[9]	KIMURA K, YAO J C. Semicontinuity of solution mappings of parametric generalized vector equilibrium problems[J]. Journal of Optimization Theory and Applications,2008,138(3): 429-443.
[10]	KIMURA K, YAO J C. Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems[J]. Journal of Global Optimization,2008,41(2): 187-202.
[11]	曾静, 彭再云, 张石生. 广义强向量拟平衡问题解的存在性和Hadamard适定性[J]. 应用数学和力学, 2015,36(6): 651-658.(ZENG Jing, PENG Zaiyun, ZHANG Shisheng. Existence and Hadamard well-posedness of solutions to generalized strong vector quasi-equilibrium problems[J]. Applied Mathematics and Mechanics,2015,36(6): 651-658.(in Chinese))
[12]	PENG Z Y, PENG J W, LONG X J, et al. On the stability of solutions for semi-infinite vector optimization problems[J]. Journal of Global Optimization,2018,70(1): 55-69.
[13]	PENG Z Y, WANG X F, YANG X M. Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems[J]. Set-Valued and Variational Analysis,2019,27(1): 103-118.
[14]	LI S J, CHEN C R. Stability of weak vector variational inequality[J]. Nonlinear Analysis: Theory, Methods & Applications,2009,70(4): 1528-1535.
[15]	CHEN C R, LI S J. Semicontinuity of the solution set map to a set-valued weak vector variational inequality[J]. Journal of Industrial and Management Optimization,2007,3(3): 519-528.
[16]	CHEN C R, LI S J, FANG Z M. On the solution semicontinuity to a parametric generalized vector quasi- variational inequality[J]. Computers & Mathematics With Applications,2010,60(8): 2417-2425.
[17]	ZHONG R Y, HUANG N J. Lower semicontinuity for parametric weak vector variational inequalities in reflexive Banach spaces[J]. Journal of Optimization Theory and Applications,2011,149(3): 564-579.
[18]	ANH L Q, HUNG N V. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems[J]. Journal of Industrial & Management Optimization,2018,14(1): 65-79.
[19]	ZHONG R Y, HUANG N J. On the stability of solution mapping for parametric generalized vector quasiequilibrium problems[J]. Computers & Mathematics With Applications,2012,63(4): 807-815.
[20]	AUBIN J P, EKELAND I. Applied Nonlinear Analysis [M]. New York: John Wiley and Sons, 1984.
[21]	BERGE C. Topological Spaces [M]. London: Oliver and Boyd, 1963.
[22]	Gerstewitz C. Nichtkonvexe dualitat in der vektaroptimierung[J]. Wissenschafliche Zeitschift der Technischen Hochschule Leuna-Mensehung,1983,25: 357-364.
[23]	LUC D T. Theory of Vector Optimization, Lecture Notes in Economic and Mathematical Systems [M]. Berlin: Springer-Verlag, 1989.