Serdica Mathematical Journal
Volume 21, Number 1, 1995
Starting from the present volume Serdica Mathematical Journal
will have a new international Editorial Board which reflects our
endeavour to go on improving the quality of the journal. Peer reviewing
will be used in order to reach high international standards of
its publications. We do hope that the mathematical community
will back us in achieving these goals.
It is a pleasure for us to thank the colleagues who kindly accepted
our invitation to become members of the new Editorial Board.
Last but not least we would like to express our gratitude to
Professor L.Iliev for his contribution to the journal as its former
Editor-in-Chief since 1975.
V. Kanev, S. L. Troyanski
Sofia, January 12, 1995
C O N T E N T S
A B S T R A C T S
ON UNIFORMLY CONVEX AND UNIFORMLY KADEC-KLEE RENORMINGS
Gilles Lancien
1991 Mathematics Subject Classification: 46B20.
Key words:
renorming, Szlenk index, dentability, uniformly convex, Kadec-Klee,
super-reflexive, scattered compact, L^{p} spaces.
We give a new construction of uniformly convex norms with a power type
modulus on super-reflexive spaces based on the notion of dentability index.
Furthermore, we prove that if the Szlenk index of a Banach space is less than
or equal to w (first infinite ordinal) then
there is an equivalent weak^{*} lower semi-continuous positively
homogeneous functional on X^{*} satisfying the uniform Kadec-Klee
Property for the weak^{*}-topology (UKK^{*}). Then we
solve the UKK or UKK^{*} renorming problems for L^{p} spaces and
C(K) spaces for K scattered compact space.
FERMAT'S EQUATION IN MATRICES
Alex Khazanov
1991 Mathematics Subject Classification: 11D41,15A24,15A36,14J50.
Key words: matrix equations, Fermat equation,
simultaneous similarity, automorphisms on affine varieties.
The Fermat equation is solved in integral two by two matrices
of determinant one as well as in finite order integral three
by three matrices.
NONLINEAR VARIATIONAL INEQUALITIES DEPENDING ON A PARAMETER
D. Goeleven and M. Théra
1991 Mathematics Subject Classification: primary - 47H19, 49J40,
secondary - 47H11, 47H15, 58C40, 58E07.
Key words: variational inequality, elastic plate, post-buckling.
This paper develops the results announced in the Note
[15]. Using an eigenvalue problem governed by a variational
inequality, we try to unify the theory concerning the post-critical
equilibrium state of
a thin elastic plate subjected to unilateral conditions.
A MEAN VALUE THEOREM FOR NON DIFFERENTIABLE
MAPPINGS IN BANACH SPACES
Robert Deville
1991 Mathematics Subject Classification: 49J50, 49J52, 49L25.
Key words: mean value theorem, smooth variational principle,
non smooth analysis, viscosity solutions.
We prove that if f is a real valued lower semicontinuous
function on a Banach space X and if there exists a
C^{1},
real valued
Lipschitz continuous function on X with bounded support and which is not
identically equal to zero,
then f is Lipschitz continuous of constant K
provided all lower subgradients of f are bounded by K.
As an application, we give a regularity result of viscosity supersolutions
(or subsolutions) of Hamilton-Jacobi equations in
infinite dimensions which satisfy a coercive condition.
This last result slightly improves some earlier work by G. Barles and H. Ishii.
SOME COMMENTS ON q-IRRESOLUTE AND
QUASI-IRRESOLUTE FUNCTIONS
Julian Dontchev and Maximilian Ganster
1991 Mathematics Subject Classification: primary - 54C10, secondary - 54H05.
Key words: q-irresolute, quasi-irresolute,
semi-open, q-semi-open, regular closed, semi-regular,
R-map.
The aim of this paper is to continue the study of
q-irresolute and quasi-irresolute functions as well as to
give an example of a function which is q-irresolute but
neither quasi-irresolute nor an R-map and thus give an answer to
a question posed by Ganster, Noiri and Reilly. We prove that
RS-compactness is preserved under open, quasi-irresolute
surjections.
OPERATORS FACTORING THROUGH BANACH LATTICES AND
IDEAL NORMS
Shlomo Reisner
1991 Mathematics Subject Classification: 46B20, 46B30, 47D30.
Key words: Banach lattices, operator ideals, tensor norms.
A new, unified
presentation of the ideal norms of factorization of operators through
Banach lattices and related ideal norms is given.
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