Abstract
In this paper, we consider two vector versions of Minty’s Lemma and obtain existence theorems for three kinds of vector variational-like inequalities. The results presented in this paper are extension and improvement of the corresponding results of other authors.
Similar content being viewed by others
References
Ansari Q.H., Siddiqi A.H., Yao J.C.A. (2000). Generalized vector variational-like inequalities and their scalarizations. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria, pp 17–37. Kluwer Academic Publishers, Dordrecht, Holland
Baiocchi C., Capelo A. (1984). Variational and Quasi-variational Inequalities, Applications to Free Boundary Problems. Weily, New York
Chen G.Y. (1992). Existence of solutions for a vector variational inequality. An extension of Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74: 445–456
Chen G.Y., Yang X.Q. (1990). The vector complementarity problem and its equivalence with the weak minimal element in ordered spaces. J. Math. Anal. Appl. 153: 136–158
Chiang Y. (2005). Semicontinuous mapping in t. v. s. with applications to mixed vector variational-like inequalities. J. Global Optim. 32: 467–486
Daniilidis A., Hadjisavvas N. (1996). Existence theorems for vector variational inequalities. Bull. Austral. Math. Soc. 54: 473–481
Fakhar M., Zafarani J. (2005). Generalized vector equilibrium problems for pseudomonotone bifunctions. J. Optim. Theory Appl. 126: 109–124
Fang Y.P., Huang N.J. (2003). Variational-like inequalities with generalized monotone mapping in Banach spaces. J. Optim. Theory Appl. 118: 327–338
Giannessi F. (1980). Theorems of the alternative quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds) Variational Inequalities and Complementarity Problems, pp 151–186. John Wiley, Chichester
Giannessi F. (1997). On Minty Variational Principle, New Trends in Mathematical Programming. Kluwer Academic Publishers, Dordrecht, Netherlands
(2000). Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht, Holland
Giannessi F., Maugeri A. (1995). Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York
Giannessi F., Maugeri A. (2005). Variational Analysis and Applications, Non convex Optimization and Its Applications, vol. 79. Springer, New York
Huang N.J., Fang Y.P. (2005). On vector variational-like inequalities in reflexive Banach spaces. J. Global Optim. 32: 495–505
Jabarootian T., Zafarani J. (2006). Generalized invariant monotonicity and invexity of nondifferentiable functions. J. Global Optim. 36: 537–564
Jabarootian, T., Zafarani, J.: Generalized vector variational-like inequalities, J. Optim. Theory Appl. 135(2), November (2007)
Khan M.F., Salahuddin (2004). On generalized vector variational-like inequalities. Nonlinear Anal. 59: 879–889
Konnov I.V., Yao J.C. (1997). On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206: 42–58
Lee B.S., Lee G.M. (1999). A vector version of Minty lemma and applications. Appl. Math. Lett. 12: 43–50
Lin L.J. (1996). Pre-vector variational inequalities. Bull. Austral. Math. Soc. 53: 63–70
Nadler S.B. Jr (1969). Multi-valued contraction mappings. Pacific J. Math. 30: 475–488
Yang X.Q. (1997). On vector variational inequalities: application to vector equilibria. J. Optim. Theory Appl. 95: 729–734
Yang X.Q., Goh C.J. (1997). On vector variational inequality with application to vector traffic equilibria. J. Optim. Theory Appl. 95: 431–443
Zeng L., Yao J. (2006). Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces. J. Global Optim. 36: 483–497
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Zafarani was partially supported by the Center of Excellence for Mathematics (University of Isfahan).
Rights and permissions
About this article
Cite this article
Chinaie, M., Jabarootian, T., Rezaie, M. et al. Minty’s lemma and vector variational-like inequalities. J Glob Optim 40, 463–473 (2008). https://doi.org/10.1007/s10898-007-9177-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-007-9177-6