# 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 exponential stability of a linear delay differential equation with an oscillating coefficient. (English) Zbl 1187.34096

Summary: New explicit exponential stability conditions are obtained for the nonautonomous linear equation

$\stackrel{˙}{x}\left(t\right)=a\left(t\right)x\left(h\left(t\right)\right)=0,$

where $h\left(t\right)\le t$ and $a\left(t\right)$ is an oscillating function.

We apply the comparison method based on the Bohl-Perron type theorem. Coefficients and delays are not assumed to be continuous.

Some real-world applications and several examples are also discussed.

##### MSC:
 34K20 Stability theory of functional-differential equations 34K06 Linear functional-differential equations
##### References:
 [1] Berezansky, L.; Braverman, E.: On stability of some linear and nonlinear delay differential equations, J. math. Anal. appl. 314, No. 2, 391-411 (2006) · Zbl 1101.34057 · doi:10.1016/j.jmaa.2005.03.103 [2] Berezansky, L.; Braverman, E.: On exponential stability of linear differential equations with several delays, J. math. Anal. appl. 324, No. 2, 1336-1355 (2006) · Zbl 1112.34055 · doi:10.1016/j.jmaa.2006.01.022 [3] Berezansky, L.; Braverman, E.: Explicit exponential stability conditions for linear differential equations with several delays, J. math. Anal. appl. 332, No. 1, 246-264 (2007) · Zbl 1118.34069 · doi:10.1016/j.jmaa.2006.10.016 [4] Berezansky, L.; Braverman, E.: Nonoscillation and exponential stability of delay differential equations with oscillating coefficients, J. dyn. Control syst. 15, No. 1, 63-82 (2009) · Zbl 1203.34103 · doi:10.1007/s10883-008-9058-4 [5] L. Berezansky, E. Braverman, New stability conditions for linear differential equations with several delays, arXiv:0806.3234v1[math.DS], (June 20, 2008) [6] Krisztin, T.: On stability properties for one-dimensional functional-differential equations, Funkcial. ekvac. 34, 241-256 (1991) · Zbl 0746.34045 [7] So, J. W. H.; Yu, J. S.; Chen, M. P.: Asymptotic stability for scalar delay differential equations, Funkcial. ekvac. 39, 1-17 (1996) · Zbl 0930.34056 [8] Györi, I.; Hartung, F.: Stability in delay perturbed differential and difference equations, Fields inst. Commun. 29, 181-194 (2001) · Zbl 0990.34066 [9] Gusarenko, S. A.; Domoshnitsky, A. I.: Asymptotic and oscillation properties of first-order linear scalar functional-differential equations, Differential equations 25, No. 12, 1480-1491 (1989) · Zbl 0726.45011 [10] Györi, I.; Pituk, M.: Stability criteria for linear delay differential equations, Differential integral equations 10, 841-852 (1997) · Zbl 0894.34064 [11] Lang, S.: Real and functional analysis, Graduate texts in mathematics 142 (1993) · Zbl 0831.46001 [12] Azbelev, N. V.; Simonov, P. M.: Stability of differential equations with aftereffect, Stability and control: theory, methods and applications 20 (2003) · Zbl 1049.34090 [13] Azbelev, N. V.; Berezansky, L.; Simonov, P. M.; Chistyakov, A. V.: The stability of linear systems with aftereffect, Differential equations 23, No. 5, 493-500 (1987) · Zbl 0652.34079 [14] Hewitt, E.; Stromberg, K. R.: Real and abstract analysis, (1965) · Zbl 0137.03202