Existence theorems for generalized vector equilibria with variable ordering relation

Article Sidebar

PDF
Published: Dec 30, 2016
DOI: https://doi.org/10.5556/j.tkjm.47.2016.2163
Keywords:
Generalized vector equilibrium problem Variable ordering relation Cone mapping KKM-Fan theorem Brouwer fixed point theorem Monotonicity Complete continuity

Main Article Content

Lu-Chuan Ceng
Yeong-Cheng Liou
Ching-Feng Wen

Abstract

In this paper we study the solvability of the generalized vector equilibrium problem (for short, GVEP) with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVEPs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVEPs without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.

Article Details

How to Cite
Ceng, L.-C., Liou, Y.-C., & Wen, C.-F. (2016). Existence theorems for generalized vector equilibria with variable ordering relation. Tamkang Journal of Mathematics, 47(4), 455–475. https://doi.org/10.5556/j.tkjm.47.2016.2163
Issue
Vol. 47 No. 4 (2016)
Section
Papers
Author Biographies

Lu-Chuan Ceng

Department ofMathematics, Shanghai Normal University, Shanghai 200234, China.

Yeong-Cheng Liou

Department of Healthcare Administration and Medical Informatics; and Research Center of Nonlinear Analysis and Optimization and Center for Fundamental Science, KaohsiungMedical University, Kaohsiung 807, Taiwan.

Ching-Feng Wen

Center for Fundamental Science, and Research center for Nonlinear and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan.

References

L. E. J. Brouwer, Uber Abbildungen von Mannigfaltigkeiten, Math. Ann., 71 (1912), 97--115.

G. Y. Chen, Existence of solutions for a vector variational inequality: An extension of Hartman-Stampacchia theorem, J. Optim. Theory Appl., 74(1992), 445--456.

G. Y. Chen and X. Q. Yang, The vector complementarity problem and its equivalence with the weak minimal element, J. Math. Anal. Appl., 153(1990), 136--158.

Y.Q. Chen, On the semi-monotone operator theory and applications, J. Math. Anal. Appl., 231(1999), 177--192.

K. Fan, A generalization of Tychonoff's fixed-point theorem, Math. Ann., 142 (1961), 305--310.

F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, In: R.W. Cottle, F. Giannessi, J. L. Lions (eds.) Variational Inequalities and Complementarity Problems, Wiley and Sons, New York, 1980.

F. Giannessi, Vector Variational Inequalities and Vector Equilibria, Kluwer Academic Publisher, Dordrechet, Holland, 2000.

F. Giannessi and A. Maugeri, Variational inequalities and network equilibrium problems, Plenum Press, New York, 1995.

N. J. Huang and Y. P. Fang, On vector variational inequalities in reflexive Banach spaces, J. Glob. Optim., 32(2005), 495--505.

I.V. Konnov and J. C. Yao, On the generalized vector variational inequality problems, J. Math. Anal. Appl., 206(1997), 42--58.

T. C. Lai and J. C. Yao, Existence results for VVIP, Appl. Math. Lett., 9(1996), 17--19.

K. L. Lin, D. P. Yang and J. C. Yao, Generalized vector variational inequalities, J. Optim. Theory Appl., 92(1997), 117-125.

S. B. Jr. Nadler, Multi-valued contraction mappings, Pac. J. Math.,30(1969), 475-488.

A. H. Siddiqi, Q. H. Ansari and A. Khaliq, On vector variational inequalities, J. Optim. Theory Appl., 84(1995), 171--180.

X. Q. Yang, Vector variational inequality and vector pseudolinear optimization, J. Optim. Theory Appl., 95(1997), 729--734.

L. C. Ceng and S. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Glob. Optim., 46(2010), 521--535.

S. J. Yu and J. C. Yao, On vector variational inequalities, J. Optim. Theory Appl., 89(1996), 749--769.

L. C. Ceng and J. C. Yao, Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces, J. Glob. Optim., 36(2006), 483--497.

F. Zheng, Vector variational inequalities with semi-monotone operators, J. Glob. Optim., 32(2005), 633--642.

L. C. Ceng, G. Y. Chen, X. X. Huang and J. C. Yao, Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications, Taiwanese J. Math., 12(2008), 151--172.

L. C. Ceng and S. Kum, On generalized vector implicit variational inequalities and complementarity problems, Taiwanese J. Math., 11(2007), 621--636.

L. C. Ceng, P. Cubiotti and J. C. Yao, Existence of vector mixed variational inequalities in Banach spaces, Nonlinear Anal., 70(2009), 1239--1256.

L. C. Ceng, S. Schaible and J. C. Yao, Existence of solutions for generalized vector variational-like inequalities, J. Optim. Theory Appl., 137(2008), 121--133.

L. C. Zeng and J. C. Yao, Existence theorems for variational inequalities in Banach spaces, J. Optim. Theory Appl.,132(2007), 321--337.

L. C. Ceng, G. Mastroeni and J. C. Yao, Existence of solutions and variational principles for generalized vector systems, J. Optim. Theory Appl., 137(2008), 485--495.

X. Q. Yang and C. J. Goh, On vector variational inequality application to vector traffic equilibria}, J. Optim. Theory Appl., 95(1997), 431--443.

L. C. Zeng and J. C. Yao, Generalized Minty's lemma for generalized vector equilibrium problems, Appl. Math. Lett., 20(2007), 32--37.

L. C. Zeng and J. C. Yao, An existence result for generalized vector equilibrium problems without pseudomonotonicity, Appl. Math. Lett., 19(12)(2006), 1320--1326.