Abstract
This paper aims to introduce a new kind of Variational Inequality, i.e. a ‘Generalized Vector Variational-Like Inequality’ which includes several classic and well-known Variational Inequalities as special cases. As an application of the Knaster-Kuratowski-Mazurkiewicz principle - in the extended form given by Fan in 1961 -, we prove that there exist solutions for our Generalized Vector Variational-Like Inequality under reasonable hypotheses. These results generalize corresponding results given by Chen et al. in (1992), Giannessi (1980), Harker and Pang (1990), Hartman and Stampacchia (1966), Isac (1990), Lee et al. (1993), Noor (1988), Saigal (1976), Siddiqi et al. (1995) and Yang (1993).
Key Words
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aubin J. P., “Applied Functional Analysis”. Wiley-Interscience New York 1975.
Chen G.-Y., “Existence of solutions for a vector variational inequality: An extension of the Hartman-Stampacchia theorem”. Jou. Optimiz. Theory Appls., Vol. 74, No. 3, 1992, pp. 445–456.
Chen G.-Y. and Cheng G. M., “Vector variational inequality and vector optimization”. Lecture Notes in Econ. and Mathem. System Vol.285, 1987, Springer-Verlag, 1987, pp. 408–416.
Chen G.-Y. and Craven B. D., “A vector variational inequality and optimization over an efficient set”. Zeitscrift für Operations Research, Vol. 3, 1990, pp. 1–12.
Chen G.-Y. and Yang X. Q.. “The vector complementarity problem and its equivalence with the weak minimal element in ordered sets”. Jou. Mathem., Analysis, Appls., Vol. 153, 1990, pp. 135–158.
Cottle R.W. and Yao J. C., “Pseudo-monotone complementarity problems in Hilbert spaces”. Jou. Optimiz. Theory. Appas., Vol. 75, No. 2, 1992, pp. 281–295.
Fan K., “A generalization of Tychonoff’s fixed point theorem”. Mathem. Annalen, Vol. 142, 1961, pp. 305–310.
Fang S. C. and Peterson E. L., “Generalized variational inequalities”. Jou. Optimiz. Theory. Appls., Vol. 38, 1982, pp. 363–383.
Giannessi F., “Theorems of alternative, quadratic programs and complementarity problems”. In “Variational Inequalities and Complementarity Problems”, (edited by R. W. Cottle, F. Giannessi and J. -L. Lions) J. Wiley and sons, Chichester, England, 1980, pp. 151–186.
Harker P. T. and Pang J. S., “Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of Theory. Algorithms and Appls.”. Mathem. Programming, Vol. 48, 1990, pp. 161–220.
Hartman G. J. and Stampacchia G., “On some nonlinear elliptic differential functional equations”. Acta Mathematica, Vol. 115 1966, pp. 271–310.
Isac G., “A special variational inequality and the implicit complementarity problem”. Jou. of the Faculty of Sciences, University of Tokyo, Section IA, Mathematics, Vol. 37, 1990, pp. 109–127.
Lee G.M., Kim D. S., Lee B.S. and Cho S. J., “Generalized vector variational inequality and fuzzy extension”. Appl. Mathem. Lett., Vol. 6, No. 6, 1993, pp. 47–51.
Luc D.T., “Theory of Vector Optimization”. Lecture Notes in Econ. and Mathem. Systems“, Vol. 319, Springer-Verlag, Berlin, 1989.
Luc D.T. and Vargas C., “A saddle point theorem for set-valued maps”. Nonlinear Anal. TMA, Vol. 18, 1992, pp. 1–7.
Noor M.A., “General variational inequality”. Applied Mathem. Letters, Vol. 1, 1988, pp. 119–122.
Park S., “On minimax inequalities on spaces having certain contractible subsets”. Bull. Austral. Mathem. Soc., Vol. 47, 1993, pp. 25–40.
Saigal R., “Extension of the generalized complementarity problem”. Mathem.matics of Operations Research, Vol. 1, 1976, pp. 260–266.
Siddiqi A.H., Ansari Q.H. and Khaliq A., “On vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 84, 1995, pp. 171–180.
Tanaka T., “Existence theorems for cone saddle points of vector-valued function in infinite-dimensional spaces”. Jou. Optimiz. Theory Appls., Vol. 62, 1989, pp. 127–138.
Tanaka T., “Generalized quasiconvexities, cone saddle points and a minimax theorem for vector valued functions”. Jou. Optimiz. Theory Appls., Vol. 81, 1994, pp. 355–377.
Yang X. Q., “Vector variational inequality and its duality”. Nonlinear Anal. TMA, Vol. 21, 1993, pp. 869–877.
Yang X.Q., “Generalized convex functions and vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 79, 1993, pp. 563–580.
Lu P. L., “Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives”. Jou. Optimiz. Theory Appls., Vol. 14, 1974, pp. 319–377.
Yuan X. Z., “KKM Theory and Applications in Nonlinear Analysis”.Marcel Dekker, New York, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Kluwer Academic Publishers
About this chapter
Cite this chapter
Chang, SS., Thompson, H.B., Yuan, G.XZ. (2000). Existence of Solutions for Generalized Vector Variational-Like Inequalities. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0299-5_3
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0299-5_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7985-0
Online ISBN: 978-1-4613-0299-5
eBook Packages: Springer Book Archive