# 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)
A note on the LaSalle-type theorems for stochastic differential delay equations. (English) Zbl 0996.60064

The author presents an improvement of results obtained in an earlier paper [J. Math. Anal. Appl. 236, No. 2, 350-369 (1999; Zbl 0958.60057)]. In the article $n$-dimensional stochastic differential delay equations are considered, which are of the form

$dx\left(t\right)=f\left(x\left(t\right),x\left(t-\tau \right),t\right)dt+g\left(x\left(t\right),x\left(t-\tau \right),t\right)dB\left(t\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $B\left(t\right)$ denotes $m$-dimensional Brownian motion. The main theorem is a stochastic version of the LaSalle theorem, providing criteria for the determination of the almost sure asymptotic behaviour of the solution of (1). The improvement concerns the assumptions on the coefficient functions. The local Lipschitz and local linear growth conditions on $f$ and $g$ are relaxed to local boundedness in the first two arguments and uniform boundedness in the last argument of $f$ and $g$, in addition the existence and uniqueness of a solution of (1) is required. The results can thus be applied to a larger class of equations. The proof of the theorem, some corollaries and an extension to the multiple delay case are given. Several examples are presented, demonstrating the usefulness of the results.

##### MSC:
 60H10 Stochastic ordinary differential equations 34K50 Stochastic functional-differential equations 93D05 Lyapunov and other classical stabilities of control systems