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Article

Berezansky, Leonid ; Braverman, Elena
On stability of linear neutral differential equations with variable delays. (English). Czechoslovak Mathematical Journal, vol. 69 (2019), issue 3, pp. 863-891
MSC: 34K06, 34K20, 34K40, 45J05 | MR 3989283 | Zbl 07088821 | DOI: 10.21136/CMJ.2019.0534-17
Keywords:
neutral equation; exponential stability; solution estimate; integro-differential equation; distributed delay
Summary:
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 $$ |a(t)|<1, \quad b(t)\geq 0, \quad h(t)\leq t, \quad g(t)\leq t, $$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.
References:
[1] Agarwal, R. P., Grace, S. R.: 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] Anokhin, A., Berezansky, L., Braverman, E.: 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] Ardjouni, A., Djoudi, A.: 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] Azbelev, N. V., Simonov, P. M.: 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] Berezansky, L. M.: 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 Russian. MR 0846501 | Zbl 0612.34069
[6] Berezansky, L., Braverman, E.: 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] Berezansky, L., Braverman, E.: 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] Berezansky, L., Braverman, E.: 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] Berezansky, L., Braverman, E.: 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] Berezansky, L., Braverman, E.: 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] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, Mineola (2006). MR 2281958 | Zbl 1160.34001
[12] Cahlon, B., Schmidt, D.: 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] Gil', M. I.: 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] Gopalsamy, K.: A simple stability criterion for linear neutral differential systems. Funkc. Ekvacioj, Ser. Int. 28 (1985), 33-38. MR 0803401 | Zbl 0641.34069
[15] Gopalsamy, K.: 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] Gusarenko, S. A., Domoshnitsky, A. I.: 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 Russian. MR 1044645 | Zbl 0726.45011
[17] Györi, I., Ladas, G.: Oscillation Theory of Delay Differential Equations: With Applications. Clarendon Press, Oxford (1991). MR 1168471 | Zbl 0780.34048
[18] Jin, C., Luo, J.: 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] Kolmanovskii, V. B., Myshkis, A.: 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] Kolmanovskii, V. B., Nosov, V. R.: 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] Kuang, Y.: 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] Liu, G., Yan, J.: 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] Raffoul, Y. N.: 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] Shaikhet, L.: 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] Tang, X. H., Zou, X.: 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] Wang, X., Liao, L.: 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] Wu, J., Yu, J. S.: Convergence in nonautonomous scalar neutral equations. Dyn. Syst. Appl. 4 (1995), 279-290. MR 1338949 | Zbl 0830.34066
[28] Ye, H., Gao, G.: 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] Yu, J. S.: 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] Zhao, D.: 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
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