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GENERALIZED IMPLICIT INCLUSION PROBLEMS ON NONCOMPACT SETS WITH APPLICATIONS

Part of: Existence theories

Published online by Cambridge University Press:  15 March 2011

SAN-HUA WANG  and
NAN-JING HUANG
SAN-HUA WANG
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, PR China
NAN-JING HUANG*
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, a class of generalized implicit inclusion problems is introduced, which can be regarded as a generalization of variational inequality problems, equilibrium problems, optimization problems and inclusion problems. Some existence results of solutions for such problems are obtained on noncompact subsets of Hausdorff topological vector spaces using the famous FKKM theorem. As applications, some existence results for vector equilibrium problems and vector variational inequalities on noncompact sets of Hausdorff topological vector spaces are given.

Keywords

generalized implicit inclusion problemnoncompact setFKKM theoremvector equilibrium problemvector variational inequality

MSC classification

Secondary: 49J40: Variational methods including variational inequalities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 2 , October 2011 , pp. 261 - 279
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This work was supported by the National Natural Science Foundation of China (70831005, 10671135) and the Natural Science Foundation of Jiangxi Province (2007GZS2120).

References

[1]Ansari, Q. A., Oettli, W. and Schlager, D., ‘A generalization of vectorial equilibria’, Math. Methods Oper. Res. 46 (1997), 147–152.CrossRefGoogle Scholar
[2]Aubin, J. P. and Ekeland, I., Applied Nonlinear Analysis (Wiley, New York, 1984).Google Scholar
[3]Blum, E. and Oettli, W., ‘From optimization and variational inequalities to equilibrium problems’, Math. Student 63 (1994), 123–145.Google Scholar
[4]Chen, X. H., ‘Existences of solution for the implicit multi-valued vector equilibrium problem’, J. Appl. Math. Comput. 30 (2009), 469–478.CrossRefGoogle Scholar
[5]Chen, G. Y., Yang, X. Q. and Yu, H., ‘A nonlinear scalarization function and generalized quasivector equilibrium problems’, J. Global Optim. 32 (2005), 451–466.CrossRefGoogle Scholar
[6]Di Bella, B., ‘An existence theorem for a class of inclusions’, Appl. Math. Lett. 13 (2000), 15–19.CrossRefGoogle Scholar
[7]Fan, K., ‘Some properties of convex sets related to fixed point theorems’, Math. Ann. 266 (1984), 519–537.CrossRefGoogle Scholar
[8]Fang, Y. P. and Huang, N. J., ‘An existence result for a class of extended inclusion problems with applications to equilibrium problems’, J. Anal. Appl. 25 (2006), 257–264.Google Scholar
[9]Fang, Y. P. and Huang, N. J., ‘Strong vector variational inequalities in Banach spaces’, Appl. Math. Lett. 19 (2006), 362–368.CrossRefGoogle Scholar
[10]Fu, J. Y., ‘Stampacchia generalized vector quasiequilibrium problems and vector saddle points’, J. Optim. Theory Appl. 128 (2006), 605–619.CrossRefGoogle Scholar
[11]Fu, J. Y. and Wang, S. H., ‘Stampacchia vector equilibrium problems and vector complementarity problems’, Adv. Math. 36(3) (2007), 339–348.Google Scholar
[12]Fu, J. Y., Wang, S. H. and Huang, Z. D., ‘New type of generalized vector quasiequilibrium problem’, J. Optim. Theory Appl. 135 (2007), 643–652.CrossRefGoogle Scholar
[13]Giannessi, F., ‘Theorems of alternative, quadratic programs and complementarity problems’, in: Variational Inequalities and Complementarity Problems (eds. Cottle, R. W., Giannessi, F. and Lions, J. L.) (Wiley, New York, 1980).Google Scholar
[14]Gong, X. H., ‘Symmetric strong vector quasi-equilibrium problems’, Math. Methods Oper. Res. 65 (2007), 305–314.CrossRefGoogle Scholar
[15]Hai, N. X. and Khanh, P. Q., ‘Systems of set-valued quasivariational inclusion problems’, J. Optim. Theory Appl. 135 (2007), 55–67.CrossRefGoogle Scholar
[16]Hou, S. H., Yu, H. and Chen, G. Y., ‘On system of generalized vector variational inequalities’, J. Global Optim. 40 (2008), 739–749.CrossRefGoogle Scholar
[17]Huang, N. J. and Fang, Y. P., ‘On vector variational inequalities in reflexive Banach spaces’, J. Global Optim. 32 (2005), 495–505.CrossRefGoogle Scholar
[18]Huang, N. J. and Li, J., ‘On vector implicit variational inequalities and complementarity problems’, J. Global Optim. 34 (2006), 399–408.CrossRefGoogle Scholar
[19]Huang, N. J., Li, J. and Thompson, H. B., ‘Generalized vector F-variational inequalities and vector F-complementarity problems for point-to-set mappings’, Math. Comput. Modelling 48 (2008), 908–917.CrossRefGoogle Scholar
[20]Huang, N. J., Li, J. and Yao, J. C., ‘Gap functions and existence of solutions for a system of vector equilibrium problems’, J. Optim. Theory Appl. 133 (2007), 201–212.CrossRefGoogle Scholar
[21]Kazmi, K. R. and Khan, S. A., ‘Existence of solutions to a generalized system’, J. Optim. Theory Appl. 142 (2009), 355–361.CrossRefGoogle Scholar
[22]Kim, J. K., Fang, Y. P. and Huang, N. J., ‘An existence result for a system of inclusion problems with applications’, Appl. Math. Lett. 21 (2008), 1209–1214.CrossRefGoogle Scholar
[23]Konnov, I. V. and Yao, J. C., ‘Existence of solutions for generalized vector equilibrium problems’, J. Math. Anal. Appl. 233 (2009), 328–335.CrossRefGoogle Scholar
[24]Lin, Y. C., ‘On F-implicit generalized vector variational inequalities’, J. Optim. Theory Appl. 142 (2009), 557–568.CrossRefGoogle Scholar
[25]Lin, L. J., Chuang, C. S. and Wang, S. Y., ‘From quasivariational inclusion problems to Stampacchia vector quasiequilibrium problems, Stampacchia set-valued vector Ekeland’s variational principle and Caristi’s fixed point theorem’, Nonlinear Anal. TMA 71 (2009), 179–185.CrossRefGoogle Scholar
[26]Lin, L. J. and Tu, C. I., ‘The studies of systems of variational inclusions problems and variational disclusions problems with applications’, Nonlinear Anal. TMA 69 (2008), 1981–1998.CrossRefGoogle Scholar
[27]Park, S., ‘Fixed points and quasi-equilibrium problems’, Math. Comput. Modelling 34 (2001), 947–954.CrossRefGoogle Scholar
[28]Ricceri, B., ‘Un théorème d’existence pour les inéquations variationnelles’, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 885–888.Google Scholar
[29]Sach, P. H. and Tuan, L. A., ‘Generalizations of vector quasivariational inclusion problems with set-valued maps’, J. Global Optim. 43 (2009), 23–45.CrossRefGoogle Scholar
[30]Wang, S. H. and Fu, J. Y., ‘Stampacchia generalized vector quasi-equilibrium problem with set-valued mapping’, J. Global Optim. 44 (2009), 99–110.CrossRefGoogle Scholar
[31]Yang, X. Q. and Zheng, X. Y., ‘Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces’, J. Global Optim. 40 (2008), 455–462.CrossRefGoogle Scholar