Semicontinuity of solution mappings for a class of parametric generalized vector equilibrium problems

Volume 9, Issue 9, pp 5449--5462 http://dx.doi.org/10.22436/jnsa.009.09.13

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1691 Downloads
• 2302 Views

Authors

Jue Lu - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China. - School of Mathematics, Physics and Information Science, Shaoxing University, Shaoxing, Zhejiang 312000, China. Yu Han - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China. Nan-Jing Huang - Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China.

Abstract

In this paper, we discuss the upper and lower semicontinuity of the strong efficient solution mapping, the weakly efficient solution mapping and the efficient solution mapping to a class of parametric generalized vector equilibrium problems by using scalarization methods and a new density result.

Share and Cite

Keywords

• Parametric generalized vector equilibrium problem
• solution mapping
• lower semicontinuity
• upper semicontinuity.

MSC

•  49J45
•  49J40
•  90C29

References

• [1] L. Q. Anh, P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699--711


• [2] L. Q. Anh, P. Q. Khanh, On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 135 (2007), 271--284


• [3] L. Q. Anh, P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Global Optim., 46 (2010), 247--259


• [4] Q. H. Ansari, Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems, J. Math. Anal. Appl., 341 (2008), 1271--1283


• [5] Q. H. Ansari, I. V. Konnov, J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481--492


• [6] J. P. Aubin, I. Ekeland, Applied nonlinear analysis, Reprint of the (1984) original, Dover Publications, Inc, Mineola (1984)


• [7] E. Blum, W. Oettli , From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123--145


• [8] G. Y. Chen, Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem, J. Optim. Theory Appl., 74 (1992), 445--456


• [9] B. Chen, N.-J. Huang , Continuity of the solution mapping to parametric generalized vector equilibrium problems , J. Global Optim., 56 (2013), 1515--1528


• [10] C. R. Chen, S. J. Li, K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Global Optim., 45 (2009), 309--318


• [11] Y. H. Cheng, D. L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim., 32 (2005), 543--550


• [12] Y. Chiang, O. Chadli, J. C. Yao, Generalized vector equilibrium problems with trifunctions, J. Global Optim., 30 (2004), 135--154


• [13] X. D. Fan, C. Z. Cheng, H. J. Wang, Stability analysis for vector quasiequilibrium problems, Positivity, 17 (2013), 365--379


• [14] Y.-P. Fang, N.-J. Huang, Strong vector variational inequalities in Banach spaces, Appl. Math. Lett., 19 (2006), 362--368


• [15] J.-Y. Fu , Generalized vector quasi-equilibrium problems, Math. Methods Oper. Res., 52 (2000), 57--64


• [16] F. Giannessi , Theorems of alternative, quadratic programs and complementarity problems, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, (1978)), Wiley, Chichester, 1980 (1980), 151--186


• [17] F. Giannessi, Vector variational inequalities and vector equilibria, Mathematical theories, Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht (2000)


• [18] X. H. Gong , Strong vector equilibrium problems , J. Global Optim., 36 (2006), 339--349


• [19] X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35--46


• [20] X. H. Gong, J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197--205


• [21] A. Göpfert, H. Riahi, C. Tammer, C. Zălinescu, Variational methods in partially ordered spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer-Verlag, New York (2003)


• [22] Y. Han, X.-H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38--41


• [23] Y. Han, N.-J. Huang, Existence and stability of solutions for a class of generalized vector equilibrium problems, Positivity, 2015 (2015), 18 pages


• [24] Y. Han, N.-J. Huang, Some characterizations of the approximate solutions to generalized vector equilibrium problems , J. Ind. Manag. Optim., 12 (2016), 1135--1151


• [25] Y. Han, N.-J. Huang, Stability of efficient solutions to parametric generalized vector equilibrium problems, (Chinese), Sci. Sin. Math., 46 (2016), 1--12


• [26] S. H. Hou, H. Yu, G. Y. Chen, On vector quasi-equilibrium problems with set-valued maps, J. Optim. Theory Appl., 119 (2003), 485--498


• [27] N.-J. Huang, J. Li, H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling, 43 (2006), 11--12


• [28] I. V. Konnov, J. C. Yao , Existence of solutions for generalized vector equilibrium problems, J. Math. Anal. Appl., 233 (1999), 328--335


• [29] S. J. Li, Z. M. Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507--515


• [30] X. B. Li, S. J. Li, Existence of solutions for generalized vector quasi-equilibrium problems, Optim. Lett., 4 (2010), 17--28


• [31] S. J. Li, H. M. Liu, Y. Zhang, Z. M. Fang, Continuity of the solution mappings to parametric generalized strong vector equilibrium problems, J. Global Optim., 55 (2013), 597--610


• [32] S. J. Li, K. L. Teo, X. Q. Yang, Generalized vector quasi-equilibrium problems , Math. Methods Oper. Res., 61 (2005), 385--397


• [33] Y. D. Xu, S. J. Li, On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem, Positivity, 17 (2013), 341--353


• [34] W. Y. Zhang, Z. M. Fang, Y. Zhang, A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems, Appl. Math. Lett., 26 (2013), 469--472


• [35] W.-B. Zhang, S.-Q. Shan, N.-J. Huang, Existence of solutions for generalized vector quasi-equilibrium problems in abstract convex spaces with applications, Fixed Point Theory Appl., 2015 (2015), 23 pages