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)
On stability in terms of two measures for impulsive systems of functional differential equations. (English) Zbl 1112.34059
The authors consider the following impulsive functional equations $$ \cases x'(t)=f(t,x_t), &t\in[t_{k-1},t_k),\\ x(t_k)=J_k(x(t^-_k)), &k=1, 2, \dots,\\ x_{t_0}=\phi. \endcases $$ They establish criteria for $(h_0,h)$-uniform stability, $(h_0,h)$-equiasymptotic stability, and $(h_0,h)$-uniform asymptotic stabiliy. The results are illustrated by examples.

MSC:
34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
WorldCat.org
Full Text: DOI
References:
[1] Bainov, D. D.; Simeonov, P. S.: Systems with impulsive effect: stability theory and applications. (1989) · Zbl 0676.34035
[2] Ballinger, G.; Liu, X.: Existence and uniqueness results for impulsive delay differential equations. Dyn. contin. Discrete impuls. Syst. 5, 579-591 (1999) · Zbl 0955.34068
[3] Fu, X.; Zhang, L.: On boundedness and stability in terms of two measures for discrete systems of Volterra type. Commun. appl. Anal. 6, 61-71 (2002) · Zbl 1085.39504
[4] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[5] Ignatyev, A. O.: On the partial equiasymptotic stability in functional differential equations. J. math. Anal. appl. 268, 615-628 (2002) · Zbl 1007.34071
[6] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[7] Lakshmikantham, V.; Liu, X.: Stability criteria for impulsive differential equations in terms of two measures. J. math. Anal. appl. 137, 591-604 (1989) · Zbl 0688.34031
[8] Lakshmikantham, V.; Liu, X.: Stability analysis in terms of two measures. (1993) · Zbl 0797.34056
[9] Liu, X.; Ballinger, G.: Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear anal. 51, 633-647 (2002) · Zbl 1015.34069
[10] Liu, X.; Ballinger, G.: Uniform asymptotic stability of impulsive delay differential equations. Comput. math. Appl. 41, 903-915 (2001) · Zbl 0989.34061
[11] Liu, X.; Xu, D.: Uniform asymptotic stability of abstract functional differential equations. J. math. Anal. appl. 216, 626-643 (1997) · Zbl 0891.34073
[12] Luo, Z.; Shen, J.: New razumikhin type theorems for impulsive functional differential equations. Appl. math. Comput. 125, 375-386 (2002) · Zbl 1030.34078
[13] Shen, J.; Yan, J.: Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear anal. 33, 519-531 (1998) · Zbl 0933.34083
[14] Xu, D.: Uniform asymptotic stability in terms of two measures for functional differential equations. Nonlinear anal. 27, 413-427 (1996) · Zbl 0854.34074
[15] Yuan, C.: Stability in terms of two measures for stochastic differential equations. Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 10, 895-910 (2003) · Zbl 1047.34064