Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 863-891, 2019


On stability of linear neutral differential equations with variable delays

Leonid Berezansky, Elena Braverman

Received November 24, 2017.   Published online March 22, 2019.

Abstract:  We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $\dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0$, where $|a(t)|<1$, $b(t)\geq0$, $h(t)\leq t$, $g(t)\leq t$, and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.
Keywords:  neutral equation; exponential stability; solution estimate; integro-differential equation; distributed delay
Classification MSC:  34K40, 34K20, 34K06, 45J05


References:
[1] R. P. Agarwal, S. R. Grace: Asymptotic stability of certain neutral diffferential equations. Math. Comput. Modelling 31 (2000), 9-15. DOI 10.1016/S0895-7177(00)00056-X | MR 1761480 | Zbl 1042.34569
[2] A. Anokhin, L. Berezansky, E. Braverman: Exponential stability of linear delay impulsive differential equations. J. Math. Anal. Appl. 193 (1995), 923-941. DOI 10.1006/jmaa.1995.1275 | MR 1341049 | Zbl 0837.34076
[3] A. Ardjouni, A. Djoudi: Fixed points and stability in neutral nonlinear differential equations with variable delays. Opusc. Math. 32 (2012), 5-19. DOI 10.7494/OpMath.2012.32.1.5 | MR 2852465 | Zbl 1254.34110
[4] N. V. Azbelev, P. M. Simonov: Stability of Differential Equations with Aftereffect. Stability and Control: Theory, Methods and Applications 20, Taylor and Francis, London (2003). DOI 10.1201/9781482264807 | MR 1965019 | Zbl 1049.34090
[5] L. M. Berezansky: Development of N. V. Azbelev's $W$-method in problems of the stability of solutions of linear functional-differential equations. Differ. Equations 22 (1986), 521-529; translation from Differ. Uravn. 22 (1986), 739-750. (In Russian.) MR 0846501 | Zbl 0612.34069
[6] L. Berezansky, E. Braverman: Oscillation criteria for a linear neutral differential equation. J. Math. Anal. Appl. 286 (2003), 601-617. DOI 10.1016/S0022-247X(03)00502-X | MR 2008851 | Zbl 1055.34123
[7] L. Berezansky, E. Braverman: On stability of some linear and nonlinear delay differential equations. J. Math. Anal. Appl. 314 (2006), 391-411. DOI 10.1016/j.jmaa.2005.03.103 | MR 2185238 | Zbl 1101.34057
[8] L. Berezansky, E. Braverman: Explicit exponential stability conditions for linear differential equations with several delays. J. Math. Anal. Appl. 332 (2007), 246-264. DOI 10.1016/j.jmaa.2006.10.016 | MR 2319658 | Zbl 1118.34069
[9] L. Berezansky, E. Braverman: Linearized oscillation theory for a nonlinear equation with a distributed delay. Math. Comput. Modelling 48 (2008), 287-304. DOI 10.1016/j.mcm.2007.10.003 | MR 2431340 | Zbl 1145.45303
[10] L. Berezansky, E. Braverman: Global linearized stability theory for delay differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 2614-2624. DOI 10.1016/j.na.2009.01.147 | MR 2532787 | Zbl 1208.34115
[11] T. A. Burton: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, Mineola (2006). MR 2281958 | Zbl 1160.34001
[12] B. Cahlon, D. Schmidt: An algorithmic stability test for neutral first order delay differential equations with $M$ commensurate delays. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 23 (2016), 1-26. MR 3453288 | Zbl 1333.34112
[13] M. I. Gil': Stability of Neutral Functional Differential Equations. Atlantis Studies in Differential Equations 3, Atlantis Press, Amsterdam (2014). DOI 10.2991/978-94-6239-091-1 | MR 3289984 | Zbl 1315.34002
[14] K. Gopalsamy: A simple stability criterion for linear neutral differential systems. Funkc. Ekvacioj, Ser. Int. 28 (1985), 33-38. MR 0803401 | Zbl 0641.34069
[15] K. Gopalsamy: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Mathematics and Its Applications 74, Kluwer Academic Publishers, Dordrecht (1992). DOI 10.1007/978-94-015-7920-9 | MR 1163190 | Zbl 0752.34039
[16] S. A. Gusarenko, A. I. Domoshnitsky: Asymptotic and oscillation properties of first-order linear scalar functional-differential equations. Differ. Equations 25 (1989), 1480-1491; translation from Differ. Uravn. 25 (1989), 2090-2103. (In Russian.) MR 1044645 | Zbl 0726.45011
[17] I. Györi, G. Ladas: Oscillation Theory of Delay Differential Equations: With Applications. Clarendon Press, Oxford (1991). MR 1168471 | Zbl 0780.34048
[18] C. Jin, J. Luo: Fixed points and stability in neutral differential equations with variable delays. Proc. Am. Math. Soc. 136 (2008), 909-918. DOI 10.1090/S0002-9939-07-09089-2 | MR 2361863 | Zbl 1136.34059
[19] V. B. Kolmanovskii, A. Myshkis: Introduction to the Theory and Applications of Functional Differential Equations. Mathematics and Its Applications 463, Kluwer Academic Publishers, Dordrecht (1999). DOI 10.1007/978-94-017-1965-0 | MR 1680144 | Zbl 0917.34001
[20] V. B. Kolmanovskii, V. R. Nosov: Stability of Functional Differential Equations. Mathematics in Science and Engineering 180, Academic Press, London (1986). DOI 10.1016/S0076-5392(08)62051-2 | MR 0860947 | Zbl 0593.34070
[21] Y. Kuang: Delay Differential Equations: With Applications in Population Dynamics. Mathematics in Science and Engineering 191, Academic Press, Boston (1993). DOI 10.1016/s0076-5392(08)x6164-8 | MR 1218880 | Zbl 0777.34002
[22] G. Liu, J. Yan: Global asymptotic stability of nonlinear neutral differential equation. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 1035-1041. DOI 10.1016/j.cnsns.2013.08.035 | MR 3119279
[23] Y. N. Raffoul: Stability in neutral nonlinear differential equations with functional delays using fixed-point theory. Math. Comput. Modelling 40 (2004), 691-700. DOI 10.1016/j.mcm.2004.10.001 | MR 2106161 | Zbl 1083.34536
[24] L. Shaikhet: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Cham (2013). DOI 10.1007/978-3-319-00101-2 | MR 3076210 | Zbl 1277.34003
[25] X. H. Tang, X. Zou: Asymptotic stability of a neutral differential equations. Proc. Edinb. Math. Soc., II. Ser. 45 (2002), 333-347. DOI 10.1017/S0013091501000396 | MR 1912643 | Zbl 1024.34070
[26] X. Wang, L. Liao: Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients. J. Math. Anal. Appl. 279 (2003), 326-338. DOI 10.1016/S0022-247X(03)00021-0 | MR 1970509 | Zbl 1054.34128
[27] J. Wu, J. S. Yu: Convergence in nonautonomous scalar neutral equations. Dyn. Syst. Appl. 4 (1995), 279-290. MR 1338949 | Zbl 0830.34066
[28] H. Ye, G. Gao: Stability theorem for a class of nonautonomous neutral differential equations with unbounded delay. J. Math. Anal. Appl. 258 (2001), 556-564. DOI 10.1006/jmaa.2000.7391 | MR 1835558 | Zbl 0991.34061
[29] J. S. Yu: Asymptotic stability for nonautonomous scalar neutral differential equations. J. Math. Anal. Appl. 203 (1996), 850-860. DOI 10.1006/jmaa.1996.0416 | MR 1417134 | Zbl 0866.34061
[30] D. Zhao: New criteria for stability of neutral differential equations with variable delays by fixed points method. Adv. Difference Equ. 2011 (2011), Paper No. 48, 11 pages. DOI 10.1186/1687-1847-2011-48 | MR 2891788 | Zbl 1282.34076

Affiliations:   Leonid Berezansky, Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel, e-mail: brznsky@math.bgu.ac.il; Elena Braverman, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB T2N 1N4, Canada, e-mail: maelena@ucalgary.ca


 
PDF available at: