Rychtár, J.
Uniformly Gâteaux differentiable norms in spaces with unconditional basis
(pp. 353358)
A B S T R A C T S
A CLARKELEDYAEV TYPE INEQUALITY
FOR CERTAIN NONCONVEX SETS
M. Ivanov
milen@fmi.unisofia.bg ,
N. Zlateva
zlateva@math.bas.bg
2000 Mathematics Subject Classification: 49J52, 26B05, 58C20, 49J45.
Key words:
Nonsmooth analysis, mean
value theorem, subdifferential.
We consider the question whether the assumption of convexity of the set
involved in
ClarkeLedyaev inequality can be relaxed. In the case when the point is
outside the convex hull of the set we show that ClarkeLedyaev type inequality
holds if and only if there is certain geometrical relation between the point
and the set.
ASPLUND FUNCTIONS AND PROJECTIONAL
RESOLUTIONS OF THE IDENTITY
Martin Zemek
2000 Mathematics Subject Classification: primary 46B20, secondary 46B22.
Key words:
Asplund function, Asplund space,
weakly Lindelöf determined space, projectional resolution of the identity,
locally uniformly rotund norm.
We further develop the theory of the so called Asplund functions,
recently introduced and studied by W. K. Tang. Let f be an Asplund
function on a Banach space X. We prove that (i) the subspace
Y : = [`sp] ¶f(X) has a projectional
resolution of the identity, and that (ii) if X is weakly
Lindelöf determined, then X admits a projectional resolution of the
identity such that the adjoint projections restricted to Y form a
projectional resolution of the identity on Y, and the dual
X^{*} admits an equivalent dual norm such that its restriction to
Y is locally uniformly rotund.
BOUNDARYVALUE PROBLEMS FOR ALMOST NONLINEAR SINGULARLY PERTURBED
SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
L. I. Karandjulov
likar@vmei.acad.bg,
Y. P. Stoyanova
yast@vmei.acad.bg,
2000 Mathematics Subject Classification: 34E15.
Key words:
Boundaryvalue problems, singularly perturbed system,
asymptotic solution, boundary functions.
A boundaryvalue problems for almost nonlinear singularly perturbed
systems of ordinary differential equations are considered. An asymptotic
solution is constructed under some assumption and using boundary functions
and generalized inverse matrix and projectors.
THE SPACE OF DIFFERENCES OF CONVEX FUNCTIONS ON [0, 1]
M. Zippin
zippin@math.huji.ac.il
2000 Mathematics Subject Classification:
46B20.
Key words:
Lpreduals, convex functions.
The space K [0, 1] of
differences of convex functions on the closed interval [0, 1] is
investigated as a dual Banach space. It is proved that a
continuous function f on [0, 1] belongs to K [0 ,1] if, and
only if,

¥ >
 f 
 =  f (0)
 +  f (1)
 + 

2 
sup
n


ì í î 

2^{n1} 1 å
i = 1

i  f (i 2^{n})  
1 2

f ((i  1)2^{n})  
1 2

f ((i + 1) 2^{n} ) 


+ 
2^{n} 1 å
i = 2^{n 1}

(2^{n}  i)  f (i 2^{n})  
1
2

f ((i  1) 2^{n})  
1
2

f ((i + 1) 2^{n})  
ü ý þ  .


 

Under the norm 
,
K [0, 1] has a predual isometric to C (F), the space of continuous
functions on F = {  1} È[0,1] È]
{ 2 }. The isometry between the L_{1} (m)
space C(F)^{*} and K [0, 1] maps the positive
cone of L_{1} (m) onto the set of all non
positive convex functions on [0, 1].
UNIFORMLY GÂTEAUX DIFFERENTIABLE NORMS IN SPACES WITH UNCONDITIONAL BASIS
Jan Rychtár
rychtar@karlin.mff.cuni.cz
2000 Mathematics Subject Classification:
46B03, 46B15, 46B20.
Key words:
Unconditional basis,
uniformly GÂteaux smooth norms, uniform Eberlein
compacts, uniform rotundity in every direction.
It is shown that a Banach space X admits an equivalent
uniformly differentiable norm if it has an unconditional basis and
X^{*} admits an equivalent norm which is uniformly rotund in every
direction.
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