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)
Perron-type stability theorems for neutral equations. (English) Zbl 1044.34028

This article deals with asymptotic stability results for a neutral functional-differential equation of the form

x ' (t)=Sx(t)+Px(t-r)+d dtQ(t,x t )+G(t,x t ),(1)

with an asymptotically stable linear part x ' (t)=Sx(t)+Px(t-r). As is well known, the problem of the asymptotical stability of the zero solution to (1) is reduced to the problem of finding a fixed point for some nonlinear integral operator A. The author describes conditions under which the operator A satisfies conditions of the following Krasnosel’skij fixed-point principle: if A=A 1 +A 2 , A 1 is a contraction of a closed convex nonempty subset M in a Banach space S, A 2 is continuous and A 2 M compact, and A 1 M+M 2 MM, then there exists a fixed point of A. The use of this theorem allows the author to consider the case when the functions Q and G are unbounded with respect to t and G is not differentiable or Lipschitzian.

MSC:
34K20Stability theory of functional-differential equations
34K40Neutral functional-differential equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces