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1
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Ekeland's variational principle in Fréchet spaces and the density of extremal points

100%
Qiu J.
Studia Mathematica
|
2005
|
tom 168
|
nr 1
81-94
EN
By modifying the method of Phelps, we obtain a new version of Ekeland's variational principle in the framework of Fréchet spaces, which admits a very general form of perturbations. Moreover we give a density result concerning extremal points of lower semicontinuous functions on Fréchet spaces. Even in the framework of Banach spaces, our result is a proper improvement of the related known result. From this, we derive a new version of Caristi's fixed point theorem and a density result for Caristi fixed points.
2
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On the quasi-weak drop property

100%
Qiu J.
Studia Mathematica
|
2002
|
tom 151
|
nr 2
187-194
EN
A new drop property, the quasi-weak drop property, is introduced. Using streaming sequences introduced by Rolewicz, a characterisation of the quasi-weak drop property is given for closed bounded convex sets in a Fréchet space. From this, it is shown that the quasi-weak drop property is equivalent to weak compactness. Thus a Fréchet space is reflexive if and only if every closed bounded convex set in the space has the quasi-weak drop property.
3
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On weak drop property and quasi-weak drop property

100%
Qiu J.
Studia Mathematica
|
2003
|
tom 156
|
nr 2
189-202
EN
Every weakly sequentially compact convex set in a locally convex space has the weak drop property and every weakly compact convex set has the quasi-weak drop property. An example shows that the quasi-weak drop property is strictly weaker than the weak drop property for closed bounded convex sets in locally convex spaces (even when the spaces are quasi-complete). For closed bounded convex subsets of quasi-complete locally convex spaces, the quasi-weak drop property is equivalent to weak compactness. However, for closed bounded convex sets in sequentially complete locally convex spaces, even the weak drop property does not imply weak compactness. A quasi-complete locally convex space is semi-reflexive if and only if its closed bounded convex subsets have the quasi-weak drop property. For strong duals of quasi-barrelled spaces, semi-reflexivity is equivalent to every closed bounded convex set having the quasi-weak drop property. From this, reflexivity of a quasi-complete, quasi-barrelled space (in particular, a Fréchet space) is characterized by the quasi-weak drop property of the space and of the strong dual.
4
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Weak countable compactness implies quasi-weak drop property

100%
Qiu J.
Studia Mathematica
|
2004
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tom 162
|
nr 2
175-182
EN
Every weakly countably compact closed convex set in a locally convex space has the quasi-weak drop property.
5
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Local completeness of locally pseudoconvex spaces and Borwein-Preiss variational principle

64%
Qiu J., Rolewicz S.
Studia Mathematica
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2007
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tom 183
|
nr 2
99-115
EN
The notion of local completeness is extended to locally pseudoconvex spaces. Then a general version of the Borwein-Preiss variational principle in locally complete locally pseudoconvex spaces is given, where the perturbation is an infinite sum involving differentiable real-valued functions and subadditive functionals. From this, some particular versions of the Borwein-Preiss variational principle are derived. In particular, a version with respect to the Minkowski gauge of a bounded closed convex set in a locally convex space is presented. In locally convex spaces it can be shown that the relevant perturbation only consists of a single summand if and only if the bounded closed convex set has the quasi-weak drop property if and only if it is weakly compact. From this, a new description of reflexive locally convex spaces is obtained.
6
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Ekeland's variational principle in locally p-convex spaces and related results

64%
Qiu J., Rolewicz S.
Studia Mathematica
|
2008
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tom 186
|
nr 3
219-235
EN
In the framework of locally p-convex spaces, two versions of Ekeland's variational principle and two versions of Caristi's fixed point theorem are given. It is shown that the four results are mutually equivalent. Moreover, by using the local completeness theory, a p-drop theorem in locally p-convex spaces is proven.
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