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Phelps' lemma, Danes' drop theorem and Ekeland's principle in locally convex spaces


Author: Andreas H. Hamel
Journal: Proc. Amer. Math. Soc. 131 (2003), 3025-3038
MSC (2000): Primary 49J40, 46A03
DOI: https://doi.org/10.1090/S0002-9939-03-07066-7
Published electronically: April 30, 2003
MathSciNet review: 1993209
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Abstract: A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997.

We show that a different formulation of Ekeland's principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.


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  • 1. ATTOUCH, H. and RIAHI, H.
    Stability Results for Ekeland's $\varepsilon$-Variational Principle and Cone Extremal Solutions.
    Mathematics of Operations Research, 18(4):173-201, 1993. MR 94k:49011
  • 2. BISHOP, E. and PHELPS, R. R.
    The Support Functionals of Convex Sets.
    In KLEE, V., editor, Convexity, volume VII of Proceedings of Symposia in Pure Mathematics, pages 27-35. American Mathematical Society, 1963. MR 27:4051
  • 3. BORWEIN, J. M.
    On the Existence of Pareto Efficient Points.
    Mathematics of Operations Research, 8(1):64-73, 1983. MR 85f:90083
  • 4. CARISTI, J.
    Fixed Point Theorems for Mappings Satisfying Inwardness Conditions.
    Transactions of the American Mathematical Society, 215:241-251, 1976. MR 52:15132
  • 5. CHENG, L., ZHOU, Y. and ZHANG, F.
    Danes' Drop Theorem in Locally Convex Spaces.
    Proceedings of the American Mathematical Society, 124(12):3699-3702, 1996. MR 97b:46013
  • 6. DANES, J.
    A Geometric Theorem Useful in Nonlinear Functional Analysis.
    Bulletino U.M.I., 6(4):369-375, 1972. MR 47:5678
  • 7. DANES, J.
    Equivalence of some Geometric and Related Results of Nonlinear Functional Analysis.
    Commentationes Mathematicae Universitatis Carolinae, 26(3):443-454, 1985. MR 88a:47053
  • 8. EKELAND, I.
    Nonconvex Minimization Problems.
    Bulletin of the American Mathematical Society, 1(3):443-474, 1979. MR 80h:49007
  • 9. FANG, J.-X.
    The Variational Principle and Fixed Point Theorem in Certain Topological Spaces.
    Journal of Mathematical Analysis and Applications, 202:398-412, 1996. MR 97d:54070
  • 10. GEORGIEV, P. G.
    The Strong Ekeland Variational Principle, the Strong Drop Theorem and Applications.
    Journal of Mathematical Analysis and Application, 131:1-21, 1988. MR 89c:46019
  • 11. GÖPFERT, A. and TAMMER, CHR.
    A New Maximal Point Theorem.
    Journal for Mathematical Analysis and its Applications, 15(2):379-390, 1995. MR 96e:90035
  • 12. GÖPFERT, A., TAMMER, CHR. and ZALINESCU, C.
    On the vectorial Ekeland's variational principle and minimal points in product spaces.
    Nonlinear Analysis. Theory, Methods & Applications, 39:909-922, 2000. MR 2001a:49019
  • 13. HAMEL, A.
    Equivalents to Ekeland's Variational Principle in $\mathcal{F}$-type Topological Spaces.
    Report of the Institut of Optimization and Stochastics 9, Martin-Luther-University Halle-Wittenberg, Department of Mathematics and Computer Science, 2001.
  • 14. ISAC, G.
    Ekeland's Principle and Nuclear Cones: A Geometrical Aspect.
    Mathematical and Computer Modelling, 26(11):111-116, 1997. MR 99j:49026
  • 15. MIZOGUCHI, N.
    A Generalization of Brøstedt's Result and its Application.
    Proceedings of the American Mathematical Society, 108(3):707-714, 1990. MR 90e:47068
  • 16. OETTLI, W. and THÉRA, M.
    Equivalents of Ekeland's Principle.
    Bulletin of the Australian Mathematical Society, 48(3):385-392, 1993. MR 94k:49005
  • 17. PARK, S.
    On generalizations of the Ekeland-type variational principles.
    Nonlinear Analysis. Theory, Methods & Application, 39:881-889, 2000. MR 2000m:49010
  • 18. PENOT, J.-P.
    The Drop Theorem, the Petal Theorem and Ekeland's Variational Principle.
    Nonlinear Analysis. Theory, Methods & Applications, 10(9):813-822, 1986. MR 87j:49031
  • 19. PHELPS, R. R.
    Support Cones and their Generalization.
    In KLEE, V., editor, Convexity, volume VII of Proceedings of Symposia in Pure Mathematics, pages 393-401. American Mathematical Society, 1963. MR 27:4052
  • 20. PHELPS, R. R.
    Support Cones in Banach Spaces and Their Applications.
    Advances in Mathematics, 13:1-19, 1974. MR 49:3505
  • 21. PHELPS, R. R.
    Convex Functions, Monotone Operators and Differentiability, volume 1364 of Lecture Notes in Mathematics.
    Springer-Verlag, 1989. MR 90g:46063
  • 22. TREVES, F.
    Locally Convex Spaces and Linear Partial Differential Equations, volume 146 of Die Grundlehren der mathematischen Wissenschaften.
    Springer-Verlag, 1967. MR 36:6986
  • 23. ZABREIKO, P. P. and KRASNOSEL'SKII, M. A.
    Solvability of Nonlinear Operator Equations.
    Functional Analysis and Applications, 5:206-208, 1971. MR 44:876

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Additional Information

Andreas H. Hamel
Affiliation: Department of Mathematics and Computer Sciences, Martin-Luther-University Halle-Wittenberg, Theodor-Lieser-Str. 5, D-06099 Halle, Germany
Email: hamel@mathematik.uni-halle.de

DOI: https://doi.org/10.1090/S0002-9939-03-07066-7
Keywords: Phelps' lemma, Ekeland's variational principle, Dane\u{s}' drop theorem, efficiency, locally convex space
Received by editor(s): May 17, 2001
Published electronically: April 30, 2003
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society

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