Lipschitz Functions and Ekeland’s Theorem

Gerald Beer () and Jose Ceniceros ()
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Gerald Beer: California State University Los Angeles
Jose Ceniceros: California State University Los Angeles

Journal of Optimization Theory and Applications, 2012, vol. 152, issue 3, No 5, 652-660

Abstract: 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

Keywords: Ekeland variational principle; Lipschitz function; Complete metric space; Lipschitz regularization; Hausdorff distance; Attouch-Wets convergence (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10957-011-9942-z

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