A POROSITY RESULT FOR A SADDLE POINT PROBLEM

Alexander J. Zaslavski

Title	A POROSITY RESULT FOR A SADDLE POINT PROBLEM
Publication Type	Journal Article
Year of Publication	2006
Authors	Zaslavski, AJ
Secondary Title	Communications in Applied Analysis
Volume	10
Issue	1
Start Page	9
Pagination	17
Date Published	01/2006
Type of Work	scientific: mathematics
ISSN	1083–2564
AMS	49J35, 54E52
Abstract	In this paper we consider a complete metric space M of functionsf : X × Y → R¹ which satisfy ${ sup_{y∈Y} inf_{x∈X} f (x, y) = inf_{x∈X} sup_{y∈Y} f (x, y), }$ where ${X}$ and ${Y}$ are complete metric spaces. We establish that the set of all functions from M which have a unique saddle point has a σ-porous complement.
URL	http://www.acadsol.eu/en/articles/10/1/3.pdf
Short Title	Saddle Point Problem
Refereed Designation	Refereed
Full Text	REFERENCES [1] J.P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley-Interscience, New York, 1984. [2] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2000. [3] F.S. De Blasi and J. Myjak, Sur la porosit ́e des contractions sans point fixe, C.R. Acad. Sc. Paris, 308 (1989), 51-54. [4] F.S. De Blasi, J. Myjak, and P.L. Papini, Porous sets in best approximation theory, London Math. Soc., 44 (1991), 135-142. [5] I. Ekeland and R.Temam, Analyse Convexe et Problemes Variationnels, Dunod, Paris, 1974. [6] S. Reich and A.J. Zaslavski, The set of divergent descent methods in a Banach space is σ-porous, SIAM J. Optim., 11 (2001), 1003-1018. [7] S. Reich and A.J. Zaslavski, The set of noncontractive mappings is σ-porous in the space of all nonexpansive mappings, C.R. Acad. Sci. Paris, 333 (2001), 539-544. [8] L. Zajıcek, Porosity and σ-porosity, Real Analysis Exchange, 13 (1987), 314-350. [9] A.J. Zaslavski, Well-posedness and porosity in optimal control without convexity assumptions, Calculus of Variations and Partial Differential Equations, 13 (2001), 265-293. [10] A.J. Zaslavski, Generic existence of a saddle point, Communications in Applied Analysis, 8 (2004), 143-151.