Convex KKM maps, monotone operators and Minty variational inequalities

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• 作者：Marc Lassonde
• 关键词：Primary 47H05 ; 47J20 ; Secondary 49J40 ; KKM principle ; finite intersection property ; monotone operator ; convex set ; Minty variational inequality
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：17
• 期：1
• 页码：137-143
• 全文大小：416 KB
• 参考文献：1.Aussel D., Corvellec J.-N., Lassonde M.: Subdifferential characterization of quasiconvexity and convexity. J. Convex Anal. 1 195-01 (1994)MATH MathSciNet
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  4.Granas A., Lassonde M.: Some elementary general principles of convex analysis. Topol. Methods Nonlinear Anal. 5, 23-7 (1995)MATH MathSciNet
  5.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, 240-46.
  6.Minty G.: On the simultaneous solution of a certain system of linear inequalities. Proc. Amer. Math. Soc. 13, 11-2 (1962)MATH MathSciNet CrossRef
  7.Minty G.: On the generalization of a direct method of the calculus of variations. Bull. Amer. Math. Soc. 73, 315-21 (1967)MATH MathSciNet CrossRef
  8.F. A. Valentine, Convex Sets. Krieger Publishing Co., Huntington, NY, 1976.
  
• 作者单位：Marc Lassonde (1)
  
  1. Université des Antilles, 97159, Pointe à Pitre, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

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. Mathematics Subject Classification Primary 47H05 47J20 Secondary 49J40