| Title:
|
On stability of linear neutral differential equations with variable delays (English) |
| Author:
|
Berezansky, Leonid |
| Author:
|
Braverman, Elena |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
863-891 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
neutral equation |
| Keyword:
|
exponential stability |
| Keyword:
|
solution estimate |
| Keyword:
|
integro-differential equation |
| Keyword:
|
distributed delay |
| MSC:
|
34K06 |
| MSC:
|
34K20 |
| MSC:
|
34K40 |
| MSC:
|
45J05 |
| idZBL:
|
Zbl 07088821 |
| idMR:
|
MR3989283 |
| DOI:
|
10.21136/CMJ.2019.0534-17 |
| . |
| Date available:
|
2019-07-24T11:21:12Z |
| Last updated:
|
2021-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147794 |
| . |
| Reference:
|
[1] Agarwal, R. P., Grace, S. R.: Asymptotic stability of certain neutral diffferential equations.Math. Comput. Modelling 31 (2000), 9-15. Zbl 1042.34569, MR 1761480, 10.1016/S0895-7177(00)00056-X |
| Reference:
|
[2] Anokhin, A., Berezansky, L., Braverman, E.: Exponential stability of linear delay impulsive differential equations.J. Math. Anal. Appl. 193 (1995), 923-941. Zbl 0837.34076, MR 1341049, 10.1006/jmaa.1995.1275 |
| Reference:
|
[3] Ardjouni, A., Djoudi, A.: Fixed points and stability in neutral nonlinear differential equations with variable delays.Opusc. Math. 32 (2012), 5-19. Zbl 1254.34110, MR 2852465, 10.7494/OpMath.2012.32.1.5 |
| Reference:
|
[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). Zbl 1049.34090, MR 1965019, 10.1201/9781482264807 |
| Reference:
|
[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. Zbl 0612.34069, MR 0846501 |
| Reference:
|
[6] Berezansky, L., Braverman, E.: Oscillation criteria for a linear neutral differential equation.J. Math. Anal. Appl. 286 (2003), 601-617. Zbl 1055.34123, MR 2008851, 10.1016/S0022-247X(03)00502-X |
| Reference:
|
[7] Berezansky, L., Braverman, E.: On stability of some linear and nonlinear delay differential equations.J. Math. Anal. Appl. 314 (2006), 391-411. Zbl 1101.34057, MR 2185238, 10.1016/j.jmaa.2005.03.103 |
| Reference:
|
[8] Berezansky, L., Braverman, E.: Explicit exponential stability conditions for linear differential equations with several delays.J. Math. Anal. Appl. 332 (2007), 246-264. Zbl 1118.34069, MR 2319658, 10.1016/j.jmaa.2006.10.016 |
| Reference:
|
[9] Berezansky, L., Braverman, E.: Linearized oscillation theory for a nonlinear equation with a distributed delay.Math. Comput. Modelling 48 (2008), 287-304. Zbl 1145.45303, MR 2431340, 10.1016/j.mcm.2007.10.003 |
| Reference:
|
[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. Zbl 1208.34115, MR 2532787, 10.1016/j.na.2009.01.147 |
| Reference:
|
[11] Burton, T. A.: Stability by Fixed Point Theory for Functional Differential Equations.Dover Publications, Mineola (2006). Zbl 1160.34001, MR 2281958 |
| Reference:
|
[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. Zbl 1333.34112, MR 3453288 |
| Reference:
|
[13] Gil', M. I.: Stability of Neutral Functional Differential Equations.Atlantis Studies in Differential Equations 3, Atlantis Press, Amsterdam (2014). Zbl 1315.34002, MR 3289984, 10.2991/978-94-6239-091-1 |
| Reference:
|
[14] Gopalsamy, K.: A simple stability criterion for linear neutral differential systems.Funkc. Ekvacioj, Ser. Int. 28 (1985), 33-38. Zbl 0641.34069, MR 0803401 |
| Reference:
|
[15] Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics.Mathematics and Its Applications 74, Kluwer Academic Publishers, Dordrecht (1992). Zbl 0752.34039, MR 1163190, 10.1007/978-94-015-7920-9 |
| Reference:
|
[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. Zbl 0726.45011, MR 1044645 |
| Reference:
|
[17] Györi, I., Ladas, G.: Oscillation Theory of Delay Differential Equations: With Applications.Clarendon Press, Oxford (1991). Zbl 0780.34048, MR 1168471 |
| Reference:
|
[18] Jin, C., Luo, J.: Fixed points and stability in neutral differential equations with variable delays.Proc. Am. Math. Soc. 136 (2008), 909-918. Zbl 1136.34059, MR 2361863, 10.1090/S0002-9939-07-09089-2 |
| Reference:
|
[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). Zbl 0917.34001, MR 1680144, 10.1007/978-94-017-1965-0 |
| Reference:
|
[20] Kolmanovskii, V. B., Nosov, V. R.: Stability of Functional Differential Equations.Mathematics in Science and Engineering 180, Academic Press, London (1986). Zbl 0593.34070, MR 0860947, 10.1016/S0076-5392(08)62051-2 |
| Reference:
|
[21] Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics.Mathematics in Science and Engineering 191, Academic Press, Boston (1993). Zbl 0777.34002, MR 1218880, 10.1016/s0076-5392(08)x6164-8 |
| Reference:
|
[22] Liu, G., Yan, J.: Global asymptotic stability of nonlinear neutral differential equation.Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 1035-1041. MR 3119279, 10.1016/j.cnsns.2013.08.035 |
| Reference:
|
[23] Raffoul, Y. N.: Stability in neutral nonlinear differential equations with functional delays using fixed-point theory.Math. Comput. Modelling 40 (2004), 691-700. Zbl 1083.34536, MR 2106161, 10.1016/j.mcm.2004.10.001 |
| Reference:
|
[24] Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations.Springer, Cham (2013). Zbl 1277.34003, MR 3076210, 10.1007/978-3-319-00101-2 |
| Reference:
|
[25] Tang, X. H., Zou, X.: Asymptotic stability of a neutral differential equations.Proc. Edinb. Math. Soc., II. Ser. 45 (2002), 333-347. Zbl 1024.34070, MR 1912643, 10.1017/S0013091501000396 |
| Reference:
|
[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. Zbl 1054.34128, MR 1970509, 10.1016/S0022-247X(03)00021-0 |
| Reference:
|
[27] Wu, J., Yu, J. S.: Convergence in nonautonomous scalar neutral equations.Dyn. Syst. Appl. 4 (1995), 279-290. Zbl 0830.34066, MR 1338949 |
| Reference:
|
[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. Zbl 0991.34061, MR 1835558, 10.1006/jmaa.2000.7391 |
| Reference:
|
[29] Yu, J. S.: Asymptotic stability for nonautonomous scalar neutral differential equations.J. Math. Anal. Appl. 203 (1996), 850-860. Zbl 0866.34061, MR 1417134, 10.1006/jmaa.1996.0416 |
| Reference:
|
[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. Zbl 1282.34076, MR 2891788, 10.1186/1687-1847-2011-48 |
| . |
