Abstract
It is known that for convex sets, the Knaster–Kuratowski–Mazurkiewicz (KKM) condition is equivalent to the finite intersection property. We use this equivalence to obtain a characterization of monotone operators in terms of convex KKM maps and in terms of the existence of solutions to Minty variational inequalities. The latter result provides a converse to the seminal theorem of Minty.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aussel D., Corvellec J.-N., Lassonde M.: Subdifferential characterization of quasiconvexity and convexity. J. Convex Anal. 1 195–201 (1994)
A. Granas and J. Dugundji, Fixed Point Theory. Springer Monographs in Mathematics, Springer, New York, 2003.
Granas A., Lassonde M.: Sur un principe géométrique en analyse convexe. Studia Math. 101, 1–18 (1991)
Granas A., Lassonde M.: Some elementary general principles of convex analysis. Topol. Methods Nonlinear Anal. 5, 23–37 (1995)
R. John, A note on Minty variational inequalities and generalized monotonicity. In: Generalized Convexity and Generalized Monotonicity (Karlovassi, 1999), Lecture Notes in Econom. and Math. Systems 502, Springer, Berlin, 2001, 240–246.
Minty G.: On the simultaneous solution of a certain system of linear inequalities. Proc. Amer. Math. Soc. 13, 11–12 (1962)
Minty G.: On the generalization of a direct method of the calculus of variations. Bull. Amer. Math. Soc. 73, 315–321 (1967)
F. A. Valentine, Convex Sets. Krieger Publishing Co., Huntington, NY, 1976.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Andrzej Granas, with admiration and friendship
Rights and permissions
About this article
Cite this article
Lassonde, M. Convex KKM maps, monotone operators and Minty variational inequalities. J. Fixed Point Theory Appl. 17, 137–143 (2015). https://doi.org/10.1007/s11784-015-0231-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-015-0231-6