Serdica Mathematical Journal

Volume 21, Number 2, 1995

C O N T E N T S

• Christov, O., D. Dragnev. Perturbations of Systems Describing the Motion of a Particle in Central Fields. (pp. 91-108)
• Stamov, G. Integral Manifolds and Perturbations of the Nonlinear Part of Systems of Autonomous Differential Equations with Impulses at Fixed Moments. (pp. 109-122)
• Dion, J.-P., G. Gauthier, A. Latour. Branching Processes with Immigration and Integer-valued Time Series. (pp. 123-136)
• Gonska, H., Ding-Xuan Zhou. On an Extremal Problem concerning Bernstein Operators. (pp. 137-150)
• Aboussoror, A., P. Loridan. Strong-weak Stackelberg Problems in Finite Dimensional Spaces. (pp. 151-170)

A B S T R A C T S

PERTURBATIONS OF SYSTEMS DESCRIBING THE MOTION OF A PARTICLE IN CENTRAL FIELDS
Ognyan Christov, Dragomir Dragnev

1991 Mathematics Subject Classification: 70D10, 70H05, 70K89, 58F07, 34D10.
Key words: particle in central field, KAM-theory, Abelian integrals, Picard-Fuchs equations.

The present paper deals with the KAM-theory conditions for systems describing the motion of a particle in central field.

INTEGRAL MANIFOLDS AND PERTURBATIONS OF THE NONLINEAR PART OF SYSTEMS OF AUTONOMOUS DIFFERENTIAL EQUATIONS WITH IMPULSES AT FIXED MOMENTS
G. T. Stamov

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

Sufficient conditions are obtained for the existence of local integral manifolds of autonomous systems of differential equations with impulses at fixed moments. In case of perturbations of the nonlinear part an estimate of the difference between the manifolds is obtained.

BRANCHING PROCESSES WITH IMMIGRATION AND INTEGER-VALUED TIME SERIES
J.-P. Dion, G. Gauthier and A. Latour

1991 Mathematics Subject Classification:62M10, 60J80.
Key words: integer-valued time series, branching processes with immigration, estimation, consistency, asymptotic normality.

In this paper, we indicate how integer-valued autoregressive time series Ginar(d) of ordre d, d ³ 1, are simple functionals of multitype branching processes with immigration. This allows the derivation of a simple criteria for the existence of a stationary distribution of the time series, thus proving and extending some results by Al-Osh and Alzaid [1], Du and Li [9] and Gauthier and Latour [11]. One can then transfer results on estimation in subcritical multitype branching processes to stationary Ginar(d) and get consistency and asymptotic normality for the corresponding estimators. The technique covers autoregressive moving average time series as well.

ON AN EXTREMAL PROBLEM CONCERNING BERNSTEIN OPERATORS
Heinz H. Gonska and Ding-Xuan Zhou

1991 Mathematics Subject Classification: 41A15, 41A17.
Key words: Bernstein operators, best constant, second modulus of smoothness, K-functional.

The best constant problem for Bernstein operators with respect to the second modulus of smoothness is considered. We show that for any ¹/₂ £ a < 1, there is an N(a) Î N such that for n ³ N(a),

sup 1-a £ k/n £ a  ê ê ê Bn æ ç è f, k n ö ÷ ø -f æ ç è k n ö ÷ ø ê ê ê £ c w2 æ ç è f, 1 Ön ö ÷ ø ,

where c is a constant, 0 < c < 1.

STRONG-WEAK STACKELBERG PROBLEMS IN FINITE DIMENSIONAL SPACES
Abdelmalek Aboussoror and Pierre Loridan

1991 Mathematics Subject Classification: 90D40, 90C31, 90C25.
Key words: Marginal functions, two-level optimization, limits of sets, stability, convex analysis.

We are concerned with two-level optimization problems called strong-weak Stackelberg problems, generalizing the class of Stackelberg problems in the strong and weak sense. In order to handle the fact that the considered two-level optimization problems may fail to have a solution under mild assumptions, we consider a regularization involving e-approximate optimal solutions in the lower level problems. We prove the existence of optimal solutions for such regularized problems and present some approximation results when the parameter e goes to zero. Finally, as an example, we consider an optimization problem associated to a best bound given in [2] for a system of nondifferentiable convex inequalities.