Serdica Mathematical Journal
Volume 21, Number 3, 1995
C O N T E N T S

Trencevski, K.
Solutions of Analytical Systems of Partial Differential
Equations.
(pp. 171184)

Ndoutoume, J. L.
Sufficient Conditions of Optimality for Control Pproblem
Governed by Variational Inequalities.
(pp. 185200)

Tsvetkov, D.
MeanPeriodic Solutions of Retarded Functional Differential
Equations.
(pp. 201218)

Kolev, E., I. Landgev.
On a Twodimensional Search Problem.
(pp. 219230)

Boyvalenkov, P., P. Kazakov.
New Upper Bounds for Some Spherical Codes.
(pp. 231238)

Ribarska, N., Ts. Tsachev, M. Krastanov.
Deformation Lemma, LjusternikSchnirellmann Theory
and Mountain Pass Theorem on C^{1}Finsler Manifolds.
(pp. 239266)
A B S T R A C T S
SOLUTIONS OF ANALYTICAL SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
K. Trencevski
1991 Mathematics Subject Classification: 35C10, 35F05, 35F20.
Key words:
linear analytical systems, nonlinear
analytical systems, partial differential equations, compatibility conditions.
In this paper are examined
some classes of linear and nonlinear 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: setvalued mapping, protoderivative, subgradient
operator, pseudoconvexity, closed convex process, optimality condition,
variational inequality.
The author recently introduced a regularity assumption for derivatives
of setvalued 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 wellknown 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 pseudoconvexity assumption is made for
the subgradient operator governing the control equation. Some examples
of application to concrete situations are presented involving obstacle
problems.
MEANPERIODIC SOLUTIONS OF RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
Dimitar Tsvetkov
1991 Mathematics Subject Classification:34K05, 34K15.
Key words: meanperiodic, retarded functional differential
equations, spectralmapping theorems.
In this paper we present a spectral criterion for existence of meanperiodic
solutions of retarded functional differential equations with a
timeindependent
main part.
ON A TWODIMENSIONAL SEARCH PROBLEM
Emil Kolev
emil@moi.math.bas.bg
Ivan Landgev
ivan@moi.math.bas.bg
1991 Mathematics Subject Classification: 05A05.
Key words: twodimensional search problem.
In this article we explore the socalled
twodimensional tree^{} search problem. We prove that for integers m
of the form
m = (2^{st}1)/(2^{s}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
A(m,n) is
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
Peter Boyvalenkov
peter@moi.math.bas.bg
Peter Kazakov
1991 Mathematics Subject Classification: 94B65, 52C17, 05B40.
Key words: spherical codes, linear programming bounds,
distance distribution.
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, LJUSTERNIKSCHNIRELLMANN THEORY
AND MOUNTAIN PASS THEOREM ON C^{1}FINSLER MANIFOLDS
Nadezhda Ribarska
ribarska@fmi.unisofia.bg
Tsvetomir Tsachev
tsachev@math.bas.bg
Michail Krastanov
krast@math.bas.bg
1991 Mathematics Subject Classification:58E05.
Key words: deformation lemma, LjusternikSchnirelmann theory,
mountain pass theorem, C^{1}Finsler manifold, locally Lipschitz
functions.
Let M be a complete C^{1}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 LjusternikSchnirelmann 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
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