Phelps’ lemma, Danes̆’ drop theorem and Ekeland’s principle in locally convex spaces

Hamel, Andreas

doi:10.1090/S0002-9939-03-07066-7

Phelps’ lemma, Danes̆’ drop theorem and Ekeland’s principle in locally convex spaces
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by Andreas H. Hamel

Proc. Amer. Math. Soc. 131 (2003), 3025-3038

DOI: https://doi.org/10.1090/S0002-9939-03-07066-7

Published electronically: April 30, 2003

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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 Danes̆’ 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.

References

Hédy Attouch and Hassan Riahi, Stability results for Ekeland’s $\epsilon$-variational principle and cone extremal solutions, Math. Oper. Res. 18 (1993), no. 1, 173–201. MR 1250113, DOI 10.1287/moor.18.1.173
Errett Bishop and R. R. Phelps, The support functionals of a convex set, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27–35. MR 0154092
Jonathan M. Borwein, On the existence of Pareto efficient points, Math. Oper. Res. 8 (1983), no. 1, 64–73. MR 703826, DOI 10.1287/moor.8.1.64
James Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251. MR 394329, DOI 10.1090/S0002-9947-1976-0394329-4
Li Xin Cheng, Yunchi Zhou, and Fong Zhang, Danes’ drop theorem in locally convex spaces, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3699–3702. MR 1328359, DOI 10.1090/S0002-9939-96-03404-1
Josef Daneš, A geometric theorem useful in nonlinear functional analysis, Boll. Un. Mat. Ital. (4) 6 (1972), 369–375 (English, with Italian summary). MR 0317130
J. Daneš, Equivalence of some geometric and related results of nonlinear functional analysis, Comment. Math. Univ. Carolin. 26 (1985), no. 3, 443–454. MR 817819
Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443–474. MR 526967, DOI 10.1090/S0273-0979-1979-14595-6
Jin-Xuan Fang, The variational principle and fixed point theorems in certain topological spaces, J. Math. Anal. Appl. 202 (1996), no. 2, 398–412. MR 1406237, DOI 10.1006/jmaa.1996.0323
Pando Grigorov Georgiev, The strong Ekeland variational principle, the strong drop theorem and applications, J. Math. Anal. Appl. 131 (1988), no. 1, 1–21. MR 934428, DOI 10.1016/0022-247X(88)90187-4
A. Göpfert and Chr. Tammer, A new maximal point theorem, Z. Anal. Anwendungen 14 (1995), no. 2, 379–390. MR 1337266, DOI 10.4171/ZAA/680
A. Göpfert, Chr. Tammer, and C. Zălinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000), no. 7, Ser. A: Theory Methods, 909–922. MR 1736394, DOI 10.1016/S0362-546X(98)00255-7
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.
G. Isac, Ekeland’s principle and nuclear cones: a geometrical aspect, Math. Comput. Modelling 26 (1997), no. 11, 111–116. MR 1491337, DOI 10.1016/S0895-7177(97)00223-9
Noriko Mizoguchi, A generalization of Brøndsted’s results and its applications, Proc. Amer. Math. Soc. 108 (1990), no. 3, 707–714. MR 991704, DOI 10.1090/S0002-9939-1990-0991704-4
W. Oettli and M. Théra, Equivalents of Ekeland’s principle, Bull. Austral. Math. Soc. 48 (1993), no. 3, 385–392. MR 1248042, DOI 10.1017/S0004972700015847
Sehie Park, On generalizations of the Ekeland-type variational principles, Nonlinear Anal. 39 (2000), no. 7, Ser. A: Theory Methods, 881–889. MR 1736392, DOI 10.1016/S0362-546X(98)00253-3
Jean-Paul Penot, The drop theorem, the petal theorem and Ekeland’s variational principle, Nonlinear Anal. 10 (1986), no. 9, 813–822. MR 856865, DOI 10.1016/0362-546X(86)90069-6
R. R. Phelps, Support cones and their generalizations, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 393–401. MR 0154093
R. R. Phelps, Support cones in Banach spaces and their applications, Advances in Math. 13 (1974), 1–19. MR 338741, DOI 10.1016/0001-8708(74)90062-0
Robert R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR 984602, DOI 10.1007/BFb0089089
François Trèves, Locally convex spaces and linear partial differential equations, Die Grundlehren der mathematischen Wissenschaften, Band 146, Springer-Verlag New York, Inc., New York, 1967. MR 0223939
P. P. Zabreĭko and M. A. Krasnosel′skiĭ, The solvability of nonlinear operator equations, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 42–44 (Russian). MR 0283646