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GENERALIZED MIXED QUASI-COMPLEMENTARITY PROBLEMS IN TOPOLOGICAL VECTOR SPACES

Part of: Existence theories

Published online by Cambridge University Press:  01 April 2008

ALI P. FRAJZADEH  and
MUHAMMAD ASLAM NOOR
ALI P. FRAJZADEH*
Affiliation:
Mathematics Department, Razi University, Kermanshah, 67149, Iran (email: ali-ff@sci.razi.ac.ir)
MUHAMMAD ASLAM NOOR
Affiliation:
Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan (email: noormaslam@hotmail.com)
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Abstract

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In this paper, we introduce and consider a new class of complementarity problems, which are called the generalized mixed quasi-complementarity problems in a topological vector space. We show that the generalized mixed quasi-complementarity problems are equivalent to the generalized mixed quasi variational inequalities. Using a new type of KKM mapping theorem, we study the existence of a solution of the generalized mixed quasi-variational inequalities and generalized mixed quasi-complementarity problems. Several special cases are also discussed. The results obtained in this paper can be viewed as extension and generalization of the previously known results.

Keywords

complementarity problemsvariational inequalitiesexistence results

MSC classification

Secondary: 49J40: Variational methods including variational inequalities 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions)
Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 4 , April 2008 , pp. 525 - 531
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Aslam Noor, M., “Mixed quasi variational inequalities”, Appl. Math. Comput. 146 (2003) 553–578.Google Scholar
[2]Aslam Noor, M., “Fundamentals of mixed quasi variational inequalities”, Int. J. Pure Appl. Math. 15 (2004) 138–257.Google Scholar
[3]Aslam Noor, M., “Some developments in general variational inequalities”, Appl. Math. Comput. 152 (2004) 199–277.Google Scholar
[4]Aslam Noor, M., Inayat Noor, K. and Rassias, Th. M., “Some aspects of variational inequalities”, J. Comput. Appl. Math. 47 (1993) 285–312.CrossRefGoogle Scholar
[5]Baiochhi, C. and Capelo, A., Variational and quasivariational inequalities (John Wiley and Sons, New York, 1984).Google Scholar
[6]Blum, E. and Oettli, W., “From optimization and variational inequalities to equilibrium problems”, Math. Stud. 63 (1994) 123–145.Google Scholar
[7]Bnouhachem, A. and Aslam Noor, M., “A new predictor-corrector method for pseudomonotone nonlinear complementarity problem”, Int. J. Comput. Math. (2007) in press.Google Scholar
[8]Cho, Y. J., Li, J. and Huang, N. J., “Solvability of implicit complementarity problems”, Math. Comput. Modelling 45 (2007) 1001–1009.CrossRefGoogle Scholar
[9]Cottle, R. W., “Complementarity and variational problems”, Sympos. Math. 19 (1976) 177–208.Google Scholar
[10]Cottle, R. W. and Dantzig, G. B., “Complementarity pivot theory of mathematical programming”, Linear Algebra. Appl. 1 (1968) 163–185.CrossRefGoogle Scholar
[11]Fakhar, M. and Zafarani, J., “Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions”, J. Optim. Theory Appl. 126 (2005) 109–124.CrossRefGoogle Scholar
[12]Farajzadeh, A. P., Amini-Harandi, A. and Aslam Noor, M., “On the generalized vectorF-implicit complementarity problems and vector F-implicit variational inequality problems”, Math. Commun. (2007) in press.Google Scholar
[13]Glowinski, R., Lions, J. L. and Tremolieres, R., Numerical analysis of variational inequalities (North-Holland, Amsterdam, 1981).Google Scholar
[14]Huang, N. J., Li, J. and O’Regan, D., “Generalized f-complementarity problems in Banch Spaces”, Nonlinear Anal. (2007) doi.10.1016/j.na.2007.04.022.Google Scholar
[15]Inoan, D. and Kolumban, J., “On pseudomonotone set-valued mappings”, Nonlinear Anal. (2007) in press.Google Scholar
[16]Itoh, S., Takahashi, W. and Yanagi, K., “Variational inequalities and complementarity problems”, J. Math. Soc. Japan 30 (1978) 23–28.Google Scholar
[17]Karamardian, S., “Generalized complementarity problems”, J. Optim. Theory Appl. 8 (1971) 223–239.CrossRefGoogle Scholar
[18]Lemke, C. E., “Bimatrix equilibrium point and mathematical programming”, Manag. Sci. 11 (1965) 681–689.CrossRefGoogle Scholar
[19]Mosco, U., “Implicit variational problems and quasi variational inequalities”, in Nonlinear operators and the calculus of variations, Volume 543 of Lect. Notes Math. (Springer, Berlin, 1976) 83–156.CrossRefGoogle Scholar