Jules, F., M. Lassonde.
Dense subdifferentiability and trustworthiness
for arbitrary subdifferentials
(pp. 387-402)
A B S T R A C T S
SELECTIONS, PARACOMPACTNESS AND COMPACTNESS
Mitrofan M. Choban
mmchoban@mail.md
Ekaterina P. Mihaylova
katiamih@fmi.uni-sofia.bg
Stoyan I. Nedev
nedev@math.bas.bg
2000 Mathematics Subject Classification:
54C60, 54C65, 54D20, 54D30.
Key words:
Set-valued mapping, Selection, Cozero-dimensional
kernel, Compactness degree, Lindelöf number, Paracompact space,
Shrinking.
In the present paper the Lindelöf number and the
degree of compactness of spaces and of the cozero-dimensional kernel
of paracompact spaces are characterized in terms of selections of
lower semi-continuous closed-valued mappings into complete
metrizable (or discrete) spaces.
LOCAL ENERGY DECAY IN EVEN DIMENSIONS FOR THE WAVE EQUATION WITH A TIME-PERIODIC NON-TRAPPING METRIC AND APPLICATIONS TO STRICHARTZ
ESTIMATES
Yavar Kian
Yavar.Kian@math.u-bordeaux1.fr
2000 Mathematics Subject Classification:
35B40, 35L15.
Key words:
time-dependent perturbation, non-trapping metric, local energy decay, Strichartz estimates.
We obtain local energy decay as well as global Strichartz estimates for the solutions u of the wave equation ∂_{t}^{2} u-div_{x}(a(t,x)∇_{x}u) = 0, t ∈ R, x ∈ R^{n}, with time-periodic non-trapping metric a(t,x) equal to 1 outside a compact set with respect to x. We suppose that the cut-off resolvent R_{χ}(θ) = χ(U(T, 0)− e^{−iθ})^{−1}χ, where U(T, 0) is the monodromy operator and T the period of a(t,x), admits an holomorphic continuation to {θ ∈ C : Im(θ) ≥ 0}, for n ≥ 3, odd, and to
{θ ∈ C : Im(θ) ≥ 0, θ ≠ 2kπ − iμ, k ∈ Z, μ ≥ 0} for n ≥ 4, even, and for n ≥ 4 even R_{χ}(θ) is bounded in a neighborhood of θ = 0.
PLUS-MINUS PROPERTY AS A GENERALIZATION OF THE DAUGAVET PROPERTY
Varvara Shepelska
shepelskaya@yahoo.com
2000 Mathematics Subject Classification:
Primary 46B20. Secondary 47A99, 46B42.
Key words:
Daugavet equation, operator norm, unital Banach algebra.
It was shown in [2] that the most natural equalities valid for every rank-one operator T in real Banach spaces lead either to the Daugavet equation ||I+T|| = 1 + ||T|| or to the equation ||I − T|| = ||I+T||. We study if the spaces where the latter condition is satisfied for every finite-rank operator inherit the properties of Daugavet spaces.
DENSE SUBDIFFERENTIABILITY AND TRUSTWORTHINESS FOR ARBITRARY SUBDIFFERENTIALS
Florence Jules
florence.jules@univ-ag.fr
Marc Lassonde
marc.lassonde@univ-ag.fr
2000 Mathematics Subject Classification:
49J52, 49J50, 58C20, 26B09.
Key words:
lower semicontinuous function, inf-convolution, subdifferential, approximate sum rule, Asplund space, subdifferentiability space, trustworthy space, variational analysis.
We show that the properties of dense subdifferentiability and of trustworthiness are equivalent for any subdifferential satisfying a small set of natural axioms. The proof relies on a remarkable property of the subdifferential of the inf-convolution of two (non necessarily convex) functions. We also show the equivalence of the dense subdifferentiability property with other basic properties of subdifferentials such as a weak^{*} Lipschitz Separation property, a strong Compact Separation property and a Minimal property for the analytic approximate subdifferential of Ioffe.
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