|
Phelps' lemma, Danes' drop theorem and Ekeland's principle in locally convex spaces
Author(s):
Andreas
H.
Hamel
Journal:
Proc. Amer. Math. Soc.
131
(2003),
3025-3038.
MSC (2000):
Primary 49J40, 46A03
Posted:
April 30, 2003
MathSciNet review:
1993209
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 1.
- ATTOUCH, H. and RIAHI, H.
Stability Results for Ekeland's  -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  -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
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
49J40, 46A03
Retrieve articles in all Journals with
MSC (2000):
49J40, 46A03
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:
10.1090/S0002-9939-03-07066-7
PII:
S 0002-9939(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
Posted:
April 30, 2003
Communicated by:
Jonathan M. Borwein
Copyright of article:
Copyright
2003,
American Mathematical Society
|
