Abstract
As shown by F. Sullivan (Proc. Am. Math. Soc. 83:345–346, 1981), validity of the weak Ekeland variational principle implies completeness of the underlying metric space. In this note, we show that what really forces completeness in Sullivan’s argument is an even simpler geometric property of lower bounded Lipschitz functions. We derive the weak Ekeland principle from this new property, and use the new property to directly obtain an omnibus non-empty intersection result for decreasing sequences of closed sets that yields as special cases the theorems of Cantor and Kuratowski valid in complete metric spaces
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References
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1, 443–474 (1979)
Phelps, R.: Convex Functions, Monotone Operators and Differentiability. Springer, Berlin (1989)
de Figueiredo, D.: Lectures on the Ekeland Variational Principle with Applications and Detours. Springer, Berlin (1989)
Meghea, I.: Ekeland Variational Principle with Generalizations and Variants. Éditions des archives contemporaines, Paris (2009)
Sullivan, F.: A characterization of complete metric spaces. Proc. Am. Math. Soc. 83, 345–346 (1981)
Lucchetti, R.: Convexity and Well-Posed Problems. Springer, New York (2006)
Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic, Dordrecht (1993)
Klein, E., Thompson, A.: Theory of Correspondences. Wiley, New York (1984)
Hiriart-Urruty, J.-B.: Extension of Lipschitz functions. J. Math. Anal. Appl. 77, 539–544 (1980)
Willard, S.: General Topology. Addison-Wesley, Reading (1970)
Cornet, B.: Topologies sur les fermés d’un espace métrique. Cahiers de mathématiques de la decision Report No. 7309, Université de Paris Dauphine (1973)
Beer, G., Lechicki, A., Levi, S., Naimpally, S.: Distance functionals and the suprema of hyperspace topologies. Ann. Mat. Pura Appl. 162, 367–381 (1992)
Lechicki, A., Levi, S.: Wijsman convergence in the hyperspace of a metric space. Boll. Unione Mat. Ital. (7) 1-B, 439–452 (1987)
Attouch, H., Lucchetti, R., Wets, R.: The topology of the ρ-Hausdorff distance. Ann. Mat. Pura Appl. 160, 303–320 (1991)
Kuratowski, K.: Topology, vol. 1. Academic Press, New York (1966)
Banaś, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Dekker, New York (1980)
Lowen, R., Sioen, M.: A unified functional look at completion in MET, UNIF, and AP. Appl. Categ. Struct. 8, 447–461 (2000)
Penot, J.-P., Zalinescu, C.: Bounded (Hausdorff) convergence: basic facts and applications. In: Giannessi, F., Maugeri, A. (eds.) Variational Analysis and Applications. Kluwer Academic, Dordrecht (2005)
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Communicated by Michel Théra.
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Beer, G., Ceniceros, J. Lipschitz Functions and Ekeland’s Theorem. J Optim Theory Appl 152, 652–660 (2012). https://doi.org/10.1007/s10957-011-9942-z
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DOI: https://doi.org/10.1007/s10957-011-9942-z