Serdica Mathematical Journal

Volume 24, Number 2, 1998

C O N T E N T S

• Stamov, G. T. Integral manifolds and bounded solutions of singularly perturbed systems of impulsive differential equations (pp. 135-144)
• Kadets, V. M., O. I. Vladimirskaya. Three spaces problem for Lyapunov theorem on vector measure (pp. 145-152)
• Plotnikov, V. A., P. M. Kitanov. Continuous dependence of solutions of quasidifferential equations with non-fixed time of impulses (pp. 153-162)
• Chowdhury, M. S. R. Generalized variational inequalities for upper hemicontinuous and demi operators with applications to fixed point theorems in Hilbert spaces (pp. 163-178)
• Landjev, I., S. Topalova. New symmetric (61,16,4) designs invariant under the dihedral group of order 10 (pp. 179-186)
• Kortezov, I. Fragmentability of the dual of a Banach space with smooth bump (pp. 187-198)
• Bossard, B., G. López. The point of continuity property: descriptive complexity and ordinal index (pp. 199-214)

A B S T R A C T S

INTEGRAL MANIFOLDS AND BOUNDED SOLUTIONS OF SINGULARLY PERTURBED SYSTEMS OF IMPULSIVE DIFFERENTIAL EQUATIONS
Gani T. Stamov

1991 Mathematics Subject Classification: 34A37. Key words: integral manifolds, singularly perturbed impulsive differential equations.

Sufficient conditions for the existence of bounded solutions of singularly perturbed impulsive differential equations are obtained. For this purpose integral manifolds are used.

THREE SPACES PROBLEM FOR LYAPUNOV THEROEM ON VECTOR MEASURE
V. M. Kadets, O. I. Vladimirskaya

1991 Mathematics Subject Classification: 46B20, 46G10. Key words: Banach space, vector measure, three spaces problem, Lyapunov convexity theorem.

It is proved that a Banach space X has the Lyapunov property if its subspace Y and the quotient space X/Y have it.

CONTINUOUS DEPENDENCE OF SOLUTIONS OF QUASIDIFFERENTIAL EQUATIONS WITH NON-FIXED TIME OF IMPULSES
V. A. Plotnikov, P. M. Kitanov

1991 Mathematics Subject Classification: Primary 34A37, Secondary 34A60. Key words: quasidifferential equation, impulses, continuous dependence, differential inclusion, R-solution, Hukuhara derivative.

In this article on quasidifferential equation with non-fixed time of impulses we consider the continuous dependence of the solutions on the initial conditions as well as the mappings defined by these equations. We prove general theorems for quasidifferential equations from which follows corresponding results for differential equations, differential inclusion and equations with Hukuhara derivative.

GENERALIZED VARIATIONAL INEQUALITIES FOR UPPER HEMICONTINUOUS AND DEMI OPERATORS WITH APPLICATIONS TO FIXED POINT THEOREMS IN HILBERT SPACES
Mohammad S. R. Chowdhury msrc@maths.uq.edu.au

1991 Mathematics Subject Classification: 47H04, 47H05, 47H09, 47H10, 49J35, 49J40, 54C60. Key words: Kneser's minimax theorem, generalized variational inequalities, generalized complementarity problems, h-demi operator, lower semicontinuous, upper semicontinuous, lower hemicontinuous, upper hemicontinuous, demi operator, monotone and semi-monotone maps.

Existence theorems of generalized variational inequalities and generalized complementarity problems are obtained in topological vector spaces for demi operators which are upper hemicontinuous along line segments in a convex set X. Fixed point theorems are also given in Hilbert spaces for set-valued operators T which are upper hemicontinuous along line segments in X such that I-T are demi operators.

NEW SYMMETRIC (61,16,4) DESIGNS INVARIANT UNDER THE DIHEDRAL GROUP OF ORDER 10
Ivan Landjev, Svetlana Topalova

1991 Mathematics Subject Classification: 05B05, 94B05. Key words: symmetric design, residual design, automorphism group of a design, self-dual code.

In this note we construct five new symmetric 2-(61,16,4) designs invariant under the dihedral group of order 10. As a by-product we obtain 25 new residual 2-(45,12,4) designs. The automorphism groups of all new designs are computed.

FRAGMENTABILITY OF THE DUAL OF A BANACH SPACE WITH SMOOTH BUMP
I. Kortezov kortezov@math.bas.bg

1991 Mathematics Subject Classification: 46B20. Key words: smooth bump, fragmentability, sigma-fragmentability.

We prove that if a Banach space X admits a Lipschitz b-smooth bump function, then (X^*,weak^*) is fragmented by a metric, generating a topology, which is stronger than the t_b-topology. We also use this to prove that if X^* admits a Lipschitz Gâteaux-smooth bump function, then X is sigma-fragmentable.

THE POINT OF CONTINUITY PROPERTY: DESCRIPTIVE COMPLEXITY AND ORDINAL INDEX
Benoit Bossard,
Ginés López glopezp@goliat.ugr.es

1991 Mathematics Subject Classification: 46B22, 04A15. Key words: point of continuity property, Borel set, ordinal index.

Let X be a separable Banach space without the Point of Continuity Property. When the set of closed subsets of its closed unit ball is equipped with the standard Effros-Borel structure, the set of those which have the Point of Continuity Property is non-Borel. We also prove that, for any separable Banach space X, the oscillation rank of the identity on X (an ordinal index which quantifies the Point of Continuity Property) is determined by the subspaces of X with a finite-dimensional decomposition. If X does not contain l₁, subspaces with basis suffice. If X^* is separable, one can even restrict to subspaces with shrinking basis.