# 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)
Exponential stability for impulsive delay differential equations by Razumikhin method. (English) Zbl 1084.34066
This paper deals with the exponential stablity of the solutions of the following impulsive delay differential equation $$\dot x(t)=f(t,x_t), \ \ t\not=t_{k},$$ $$\Delta x(t)=I_{k}(t,x_{t^{-}}), \ t=t_{k}, \ k\in \bbfN,$$ $$x_{t_{0}}=\phi.$$ The proofs are based on the Razumikhin method. Some examples illustrating the results are presented.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses
Full Text:
##### References:
 [1] Anokhin, A.; Berezansky, L.; Braverman, E.: Exponential stability of linear delay impulsive differential equations. J. math. Anal. appl. 193, 923-941 (1995) · Zbl 0837.34076 [2] Ballinger, G.; Liu, X.: Existence and uniqueness results for impulsive delay differential equations. Dynam. contin. Discrete impuls. Systems 5, 579-591 (1999) · Zbl 0955.34068 [3] Ballinger, G.; Liu, X.: Practical stability of impulsive delay differential equations and applications to control problems. Optimization methods and applications (2001) · Zbl 0986.93062 [4] Berezansky, L.; Idels, L.: Exponential stability of some scalar impulsive delay differential equation. Comm. appl. Math. anal. 2, 301-309 (1998) · Zbl 0901.34068 [5] Gopalsamy, K.; Zhang, B. G.: On delay differential equations with impulses. J. math. Anal. appl. 139, 110-122 (1989) · Zbl 0687.34065 [6] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations. (1993) · Zbl 0787.34002 [7] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002 [8] Kolmanovskii, V. B.; Nosov, V. R.: Stability of functional differential equations. (1986) · Zbl 0593.34070 [9] Lakshmikantham, V.; Leela, S.; Martynyuk, A. A.: Stability analysis of nonlinear systems. (1989) · Zbl 0676.34003 [10] 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 [11] Liu, X.: Impulsive stabilization of nonlinear systems. IMA J. Math. control inform. 10, 11-19 (1993) · Zbl 0789.93101 [12] Liu, X.: Stability results for impulsive differential systems with applications to population growth models. Dynam. stability systems 9, 163-174 (1994) · Zbl 0808.34056 [13] 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 [14] Liu, X.; Ballinger, G.: Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear anal. 53, 1041-1062 (2003) · Zbl 1037.34061 [15] Liu, X.; Ballinger, G.: Uniform asymptotic stability of impulsive delay differential equations. Comput. math. Appl. 41, 903-915 (2001) · Zbl 0989.34061 [16] Liu, X.; Shen, X.; Zhang, Y.: Exponential stability of singularly perturbed systems with time delay. Appl. anal. 82, 117-130 (2003) · Zbl 1044.34031 [17] Liu, Y.; Ge, W.: Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients. Nonlinear anal. 57, 363-399 (2004) · Zbl 1064.34051 [18] Shen, J.; Yan, J.: Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear anal. 33, 519-531 (1998) · Zbl 0933.34083 [19] Stamova, I. M.; Stamov, G. T.: Lyapunov -- razumikhin method for impulsive functional equations and applications to the population dynamics. J. comput. Appl. math. 130, 163-171 (2001) · Zbl 1022.34070 [20] Yan, J.; Shen, J.: Impulsive stabilization of functional differential equations by Lyapunov -- razumikhin functions. Nonlinear anal. 37, 245-255 (1999) · Zbl 0951.34049