Nonzero Solutions of Generalized Variational Inequalities


The existence of nonzero solutions for a class of generalized variational inequalities is studied by ?xed point index approach for multivalued mappings in ?nite dimensional spaces and re?exive Banach spaces. Some new existence theorems of nonzero solutions for this class of generalized variational inequalities are established.

Share and Cite:

J. Li and Y. Lai, "Nonzero Solutions of Generalized Variational Inequalities," Applied Mathematics, Vol. 1 No. 1, 2010, pp. 81-86. doi: 10.4236/am.2010.11010.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] G. X. Z. Yuan, “KKM Theory and Applications in Non- linear Analysis,” Marcel Dekker, New York, 1999.
[2] D. Kinderlehrer and G. Stampacchia, “An Introduction to Variational Inequalities and Their Applications,” Acade- mic Press, New York, 1980.
[3] S. S. Chang, “Variational Inequality and Complemen- tarity Problem Theory with Applications,” Shanghai Scienti?c Technology and Literature Press, Shanghai, 1991.
[4] F. Facchinei and J. S. Pang, “Finite-dimensional Varia- tional Inequality and Complementarity Problems,” Springer- Verlag, New York, 2003.
[5] A. Szulkin, “Positive Solutions of Variational Inequalities: A Degree Theoretic Approch,” Journal of Differential Equations, Vol. 57, No. 1, 1985, pp. 90-111.
[6] Y.G. Zhu, “Positive Solutions to A System of Variational Inequalities,” Applied Mathematics Letters, Vol. 11, No. 4, 1998, pp. 63-70.
[7] J. Mawhin, “Equivalence Theorems for Nonlinear Operator Equations and Coincidence Degree Theory for Some Mappings in Locally Convex Topological Vector Spaces,” Journal of Differential Equations, Vol. 12, 1972, pp. 610-636.
[8] Y. Lai, “Existence of Nonzero Solutions for A Class of Generalized Variational Inequalities,” Positivity, Vol. 12, No. 4, 2008, pp. 667-676.
[9] K. Q. Wu and N. J. Huang, “Non-Zero Solutions for A Class of Generalized Variational Inequalities in Re?exive Banach Spaces,” Applied Mathematics Letters, Vol. 20, No. 2, 2007, pp. 148-153.
[10] Y. Lai and Y. G. Zhu, “Existence Theorems for Solutions of Variational Inequalities,” Acta Mathematica Hunga- rica, Vol. 108, No. 1-2, 2005, pp. 95-103.
[11] J. H Fan and W. H. Wei, “Nonzero Solutions for A Class of Set-Valued Variational Inequalities in Re?exive Banach Spaces,” Computers Mathematics with Applica- tions, Vol. 56, No. 1, 2008, pp. 233-241.
[12] D. D. Ang, K. Schmitt and L. K. Vy, “Noncoercive Variational Inequalities: Some Applications,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 15, No. 6, 1990, pp. 497-512.
[13] P. M. Fitzpatrick and W. V. Petryshyn, “Fixed Point Theorems and The Fixed Point Index for Multivalued Mappings in Cones,” Journal of the London Mathema- tical Society, Vol. 12, No. 2, 1975, pp. 75-85.
[14] M. S. R. Chowdhury and K. K. Tan, “Generalization of Ky Fan’s Minimax Inequality with Applications to Generalized Variational Inequalities for Pseudo-Mono- tone Operators and Fixed point theorems,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 3, 1996, pp. 910-929.
[15] K. Deimling, “Nonlinear Functional Applications,” Springer- Verlag, New York, 1985.
[16] D. Pascali and S. Sburlan, “Nonlinear Mappings of Mo- notone Type,” Sijtho? & Noordhoff International Publishers, Bucuresti, 1976.
[17] W. V. Petryshyn, “Multiple Positive Fxed Points of Mul-tivalued Condensing Mappings with Some Applications,” Journal of Mathematical Analysis and Applications, Vol. 124, 1987, pp. 237-253.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.