Generalized Quasi Variational-Type Inequalities

Mohammad Kalimuddin Ahmad, Salahuddin .

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DOI: 10.4236/am.2012.31010                

Abstract

In this paper, we define the concepts of (η,h)-quasi pseudo-monotone operators on compact set in locally convex Hausdorff topological vector spaces and prove the existence results of solutions for a class of generalized quasi variational type inequalities in locally convex Hausdorff topological vector spaces.

Keywords

Generalized Quasi Variational Type Inequalities (GQVTI); (η, h)-Quasi Pseudo-Monotone Operator; Locally Convex Hausdorff Topological Vector Spaces; Compact Sets; Bilinear Functional; Lower Semicontinuous; Upper Semicontinuous

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	F. Giannessi, “Theorems of Alternative Quadratic Programs and Complementarity Problems,” In: R. W. Cottle, F. Gianessi and J. L. Lions, Eds., Variational Inequalities and Complementarity Problems, John Wiley and Sons, Chichester, 1980.
[2]	D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications in Pure and Applied Mathematics,” Vol. 88, Academic Press, New York, 1980.
[3]	F. E. Browdev, “Existence and Approximation of Solutions of Nonlinear Variational Inequalities,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 56, No. 4, 1966, pp. 1080-1086.
[4]	G. Y. Chen and G. M. Cheng, “Vector Variational Inequality and Vector Optimizations,” Lecture Notes in Economics and Mathematical Systems, Vol. 285, 1967, pp. 408-456.
[5]	M. S. R. Chowdhury and K. K. Tan, “Generalization of Ky Fan’s Minimax Inequality with Applications to Generalized Variational Inequalities for Pseudomonotone Operators and Fixed Point Theorems,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 3, 1996, pp. 910-929. doi:10.1006/jmaa.1996.0476
[6]	M. S. R. Chowdhury, “The Surjectivity of Upper Hemicontinuous and Pseudomonotone Type II Operators in Reflexive Banach Ppaces,” Journal Bangladesh Mathematical Society, Vol. 20, 2000, pp. 45-53.
[7]	H. Brezis, L. Nirenberg and G. Stampacchia, “A Remark on Ky Fan’s Minimax Principle,” Bollettino Unione Matematica Italiana, Vol. 6, No. 4, 1972, pp.293-300.
[8]	K. Fan, “A Minimax Inequality and Applications,” In: O. Shisha, Ed., Inequalities III, Academic Press, San Diego, 1972, pp. 103-113.
[9]	M. S. R. Chowdhury and K. K. Tan, “Generalized Variational Inequalities for Quasimonotone Operators and Applications,” Bulletin of Polish Academy of Science, Vol. 45, No. 1, 1997, pp. 25-54.
[10]	M. H. Shih and K. K. Tan, “Generalized Bi-Quasi Variational Inequalities,” Journal of Mathematical Analysis and Applications, Vol. 143, No. 1, 1989, pp. 66-85. doi:10.1016/0022-247X(89)90029-2
[11]	M. S. R. Chowdhury and K. K. Tan, “Applications of Upper Hemicontinuous Operators on Generalized Bi-Quasi Variational Inequalities in Locally Convex Topological Vector Spaces,” Positivity, Vol. 3, No. 4, 1999, pp. 333-344. doi:10.1023/A:1009849400516
[12]	M. S. R. Chowdhury and K. K. Tan, “Applications of Pseudomonotone Operators with Some Kind of Upper Semicontinuity in Generalized Quasi Variational Inequalities on Noncompact Sets,” Proceeding of American Mathematica Society, Vol. 126, No. 10, 1998, pp. 2957-2968. doi:10.1090/S0002-9939-98-04436-0
[13]	X. P. Ding and E. Tarafdar, “Generalized Variational Like Inequalities with Pseudomonotone Setvalued Mappings,” Archieve Journal of Mathematics, Vol. 74, No. 4, 2000, pp. 302-313. doi:10.1007/s000130050447
[14]	M. H. Shih and K. K. Tan, “Generalized Quasi Variational Inequalities in Locally Convex Topological Vector Spaces,” Journal of Mathematical Analysis and Applications, Vol. 108, No. 2, 1985, pp. 333-343. doi:10.1016/0022-247X(85)90029-0
[15]	W. Takahashi, “Nonlinear Variational Inequalities and Fixed Point Theorem,” Journal of Mathematical Society of Japan, Vol. 28, No. 1, 1976, pp. 168-181. doi:10.2969/jmsj/02810168
[16]	H. Kneser, “Sur un Theoreme Fundamental de la Theorie des Jeux,” CRAS Paris, Vol. 234, 1952, pp. 2418-2420.
[17]	R. T. Rockafeller, “Convex Analysis,” Princeton University Press, Princeton, 1970.