zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Stability and asymptotic behaviour of a two-dimensional differential system with delay. (English) Zbl 1008.34064

The two-dimensional system

x ' (t)=A(t)x(t)+B(t)x(t-r)+h(t,x(t),x(t-r))

is considered, where A(t)=(a jk (t)), B(t)=(b jk (t)), j,k=1,2, are real matrices and h(t,x,y) is a two-dimensional real vector function. It is supposed that the functions a jk are absolutely continuous on [t 0 ,), b jk are locally Lebesgue integrable on [t 0 ,) and the function h satisfies Carathéodory conditions on

[t 0 ,)×{[x 1 ,x 2 ] 2 :x 1 2 +x 2 2 <K}×{[y 1 ,y 2 ] 2 :x 1 2 +x 2 2 <K},

with 0<K. The authors use an original approach for the investigation – with the aid of complex variables the system is rewritten into an equivalent equation with complex-valued coefficients. (This idea was used in a previous paper of the first author, too, see also M. Ráb and J. Kalas [Differ. Integral Equ. 3, No. 1, 127-144 (1990; Zbl 0724.34060)].) Stability and asymptotic stability of the trivial solution, and further asymptotic properties (e.g., the boundedness of all solutions by exponential functions) are studied by means of an appropriate Lyapunov-Krasovskii functional. This approach does not require the uniform stability or uniform asymptotics stability of a corresponding linear system and leads to new, effective and easy applicable results. An illustrative example is considered. The authors discuss possible generalizations, too.

Reviewer: J.Diblík (Brno)
MSC:
34K20Stability theory of functional-differential equations
34K25Asymptotic theory of functional-differential equations
34K12Growth, boundedness, comparison of solutions of functional-differential equations