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

• Gilles Lancien. On Uniformly Convex and Uniformly Kadec-Klee Renomings. (pp. 1-18)
• Alex Khazanov. Fermat's Equation in Matrices. (pp. 19-40)
• D. Goeleven and M. Théra. Nonlinear Variational Inequalities depending on a parameter. (pp. 41-58)
• Robert Deville A mean value theorem for non differentiable mappings in Banach spaces. (pp. 59-66)
• Julian Dontchev and Maximilian Ganster. Some Comments on q-irresolute and Quasi-irresolute Functions. (pp. 67-74)
• Shlomo Reisner. Operators Factoring through Banach Lattices and Ideal Norms. (pp. 75-89)

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 C¹, 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.