Serdica Mathematical Journal
Volume 21, Number 3, 1995
C O N T E N T S
Solutions of Analytical Systems of Partial Differential
Ndoutoume, J. L.
Sufficient Conditions of Optimality for Control Pproblem
Governed by Variational Inequalities.
Mean-Periodic Solutions of Retarded Functional Differential
Kolev, E., I. Landgev.
On a Two-dimensional Search Problem.
Boyvalenkov, P., P. Kazakov.
New Upper Bounds for Some Spherical Codes.
Ribarska, N., Ts. Tsachev, M. Krastanov.
Deformation Lemma, Ljusternik-Schnirellmann Theory
and Mountain Pass Theorem on C1-Finsler Manifolds.
A B S T R A C T S
SOLUTIONS OF ANALYTICAL SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
1991 Mathematics Subject Classification: 35C10, 35F05, 35F20.
linear analytical systems, non-linear
analytical systems, partial differential equations, compatibility conditions.
In this paper are examined
some classes of linear and non-linear analytical systems
of partial differential equations. Compatibility
conditions are found and if they are satisfied, the
solutions are given as functional series in a neighborhood of
a given point (x=0).
SUFFICIENT CONDITIONS OF OPTIMALITY FOR CONTROL PROBLEMS
GOVERNED BY VARIATIONAL INEQUALITIES
James Louis Ndoutoume
1991 Mathematics Subject Classification: 49B99, 49A29.
Key words: set-valued mapping, proto-derivative, subgradient
operator, pseudo-convexity, closed convex process, optimality condition,
The author recently introduced a regularity assumption for derivatives
of set-valued mappings, in order to obtain first order necessary
conditions of optimality, in some generalized sense, for
nondifferentiable control problems governed by variational inequalities.
It was noticed that this regularity assumption can be viewed as a symmetry
condition playing a role parallel to that of the well-known symmetry property
of the Hessian of a function at a given point. In this paper, we elaborate
this point in a more detailed way and discuss some related questions. The
main issue of the paper is to show (using this symmetry condition)
that necessary conditions of optimality alluded above can be shown to
be also sufficient if a weak pseudo-convexity assumption is made for
the subgradient operator governing the control equation. Some examples
of application to concrete situations are presented involving obstacle
MEAN-PERIODIC SOLUTIONS OF RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
1991 Mathematics Subject Classification:34K05, 34K15.
Key words: mean-periodic, retarded functional differential
equations, spectral-mapping theorems.
In this paper we present a spectral criterion for existence of mean-periodic
solutions of retarded functional differential equations with a
ON A TWO-DIMENSIONAL SEARCH PROBLEM
1991 Mathematics Subject Classification: 05A05.
Key words: two-dimensional search problem.
In this article we explore the so-called
two-dimensional tree- search problem. We prove that for integers m
of the form
m = (2st-1)/(2s-1) the rectangles
A(m,n) are all tight,
no matter what n is. On the other hand,
we prove that there exist infinitely many integers
m for which there is an infinite number of n's such that
loose. Furthermore, we determine the smallest loose rectangle as
well as the smallest loose square (A(181,181)).
It is still undecided whether there exist infinitely many loose squares.
NEW UPPER BOUNDS FOR SOME SPHERICAL CODES
1991 Mathematics Subject Classification: 94B65, 52C17, 05B40.
Key words: spherical codes, linear programming bounds,
The maximal cardinality of a code W on the unit sphere in n dimensions
with (x,y) £ s whenever x,y
Î W, x ¹ y, is denoted by
A(n,s). We use two methods for obtaining new upper bounds on A(n,s)
for some values of n and s. We find new linear programming bounds by
suitable polynomials of degrees which are higher than the degrees of the
previously known good polynomials due to Levenshtein [11, 12].
Also we investigate the possibilities for attaining the Levenshtein bounds
[11, 12]. In such cases we find the distance distributions of the
corresponding feasible maximal spherical codes. Usually this leads to a
contradiction showing that such codes do not exist.
DEFORMATION LEMMA, LJUSTERNIK-SCHNIRELLMANN THEORY
AND MOUNTAIN PASS THEOREM ON C1-FINSLER MANIFOLDS
1991 Mathematics Subject Classification:58E05.
Key words: deformation lemma, Ljusternik-Schnirelmann theory,
mountain pass theorem, C1-Finsler manifold, locally Lipschitz
Let M be a complete C1-Finsler manifold without boundary
and f:M ® R be a locally Lipschitz function.
The classical proof of
the well known deformation lemma can not be extended in this case
because integral lines may not exist. In this paper we establish
existence of deformations generalizing the classical result. This
allows us to prove some known results in a more general setting
(minimax theorem, a theorem of Ljusternik-Schnirelmann type,
mountain pass theorem). This approach enables us to drop the
compactness assumptions characteristic for recent papers in the field
using the Ekeland's variational principle as the main tool