On stability of linear neutral differential equations with variable delays

Leonid Berezansky; Elena Braverman

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 3, page 863-891
  • ISSN: 0011-4642

Abstract

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We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays x ˙ ( t ) - a ( t ) x ˙ ( g ( t ) ) + b ( t ) x ( h ( t ) ) = 0 , where | a ( t ) | < 1 , b ( t ) ≥ 0 , h ( t ) ≤ t , g ( t ) ≤ t , and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

How to cite

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Berezansky, Leonid, and Braverman, Elena. "On stability of linear neutral differential equations with variable delays." Czechoslovak Mathematical Journal 69.3 (2019): 863-891. <http://eudml.org/doc/294292>.

@article{Berezansky2019,
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 \[ |a(t)|<1, \quad b(t)\ge 0, \quad h(t)\le t, \quad g(t)\le t, \] and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.},
author = {Berezansky, Leonid, Braverman, Elena},
journal = {Czechoslovak Mathematical Journal},
keywords = {neutral equation; exponential stability; solution estimate; integro-differential equation; distributed delay},
language = {eng},
number = {3},
pages = {863-891},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On stability of linear neutral differential equations with variable delays},
url = {http://eudml.org/doc/294292},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Berezansky, Leonid
AU - Braverman, Elena
TI - On stability of linear neutral differential equations with variable delays
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 3
SP - 863
EP - 891
AB - 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 \[ |a(t)|<1, \quad b(t)\ge 0, \quad h(t)\le t, \quad g(t)\le t, \] and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.
LA - eng
KW - neutral equation; exponential stability; solution estimate; integro-differential equation; distributed delay
UR - http://eudml.org/doc/294292
ER -

References

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  1. Agarwal, R. P., Grace, S. R., 10.1016/S0895-7177(00)00056-X, Math. Comput. Modelling 31 (2000), 9-15. (2000) Zbl1042.34569MR1761480DOI10.1016/S0895-7177(00)00056-X
  2. Anokhin, A., Berezansky, L., Braverman, E., 10.1006/jmaa.1995.1275, J. Math. Anal. Appl. 193 (1995), 923-941. (1995) Zbl0837.34076MR1341049DOI10.1006/jmaa.1995.1275
  3. Ardjouni, A., Djoudi, A., 10.7494/OpMath.2012.32.1.5, Opusc. Math. 32 (2012), 5-19. (2012) Zbl1254.34110MR2852465DOI10.7494/OpMath.2012.32.1.5
  4. Azbelev, N. V., Simonov, P. M., 10.1201/9781482264807, Stability and Control: Theory, Methods and Applications 20, Taylor and Francis, London (2003). (2003) Zbl1049.34090MR1965019DOI10.1201/9781482264807
  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. (1986) Zbl0612.34069MR0846501
  6. Berezansky, L., Braverman, E., 10.1016/S0022-247X(03)00502-X, J. Math. Anal. Appl. 286 (2003), 601-617. (2003) Zbl1055.34123MR2008851DOI10.1016/S0022-247X(03)00502-X
  7. Berezansky, L., Braverman, E., 10.1016/j.jmaa.2005.03.103, J. Math. Anal. Appl. 314 (2006), 391-411. (2006) Zbl1101.34057MR2185238DOI10.1016/j.jmaa.2005.03.103
  8. Berezansky, L., Braverman, E., 10.1016/j.jmaa.2006.10.016, J. Math. Anal. Appl. 332 (2007), 246-264. (2007) Zbl1118.34069MR2319658DOI10.1016/j.jmaa.2006.10.016
  9. Berezansky, L., Braverman, E., 10.1016/j.mcm.2007.10.003, Math. Comput. Modelling 48 (2008), 287-304. (2008) Zbl1145.45303MR2431340DOI10.1016/j.mcm.2007.10.003
  10. Berezansky, L., Braverman, E., 10.1016/j.na.2009.01.147, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 2614-2624. (2009) Zbl1208.34115MR2532787DOI10.1016/j.na.2009.01.147
  11. Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Mineola (2006). (2006) Zbl1160.34001MR2281958
  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. (2016) Zbl1333.34112MR3453288
  13. Gil', M. I., 10.2991/978-94-6239-091-1, Atlantis Studies in Differential Equations 3, Atlantis Press, Amsterdam (2014). (2014) Zbl1315.34002MR3289984DOI10.2991/978-94-6239-091-1
  14. Gopalsamy, K., A simple stability criterion for linear neutral differential systems, Funkc. Ekvacioj, Ser. Int. 28 (1985), 33-38. (1985) Zbl0641.34069MR0803401
  15. Gopalsamy, K., 10.1007/978-94-015-7920-9, Mathematics and Its Applications 74, Kluwer Academic Publishers, Dordrecht (1992). (1992) Zbl0752.34039MR1163190DOI10.1007/978-94-015-7920-9
  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. (1989) Zbl0726.45011MR1044645
  17. Györi, I., Ladas, G., Oscillation Theory of Delay Differential Equations: With Applications, Clarendon Press, Oxford (1991). (1991) Zbl0780.34048MR1168471
  18. Jin, C., Luo, J., 10.1090/S0002-9939-07-09089-2, Proc. Am. Math. Soc. 136 (2008), 909-918. (2008) Zbl1136.34059MR2361863DOI10.1090/S0002-9939-07-09089-2
  19. Kolmanovskii, V. B., Myshkis, A., 10.1007/978-94-017-1965-0, Mathematics and Its Applications 463, Kluwer Academic Publishers, Dordrecht (1999). (1999) Zbl0917.34001MR1680144DOI10.1007/978-94-017-1965-0
  20. Kolmanovskii, V. B., Nosov, V. R., 10.1016/S0076-5392(08)62051-2, Mathematics in Science and Engineering 180, Academic Press, London (1986). (1986) Zbl0593.34070MR0860947DOI10.1016/S0076-5392(08)62051-2
  21. Kuang, Y., 10.1016/s0076-5392(08)x6164-8, Mathematics in Science and Engineering 191, Academic Press, Boston (1993). (1993) Zbl0777.34002MR1218880DOI10.1016/s0076-5392(08)x6164-8
  22. Liu, G., Yan, J., 10.1016/j.cnsns.2013.08.035, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 1035-1041. (2014) MR3119279DOI10.1016/j.cnsns.2013.08.035
  23. Raffoul, Y. N., 10.1016/j.mcm.2004.10.001, Math. Comput. Modelling 40 (2004), 691-700. (2004) Zbl1083.34536MR2106161DOI10.1016/j.mcm.2004.10.001
  24. Shaikhet, L., 10.1007/978-3-319-00101-2, Springer, Cham (2013). (2013) Zbl1277.34003MR3076210DOI10.1007/978-3-319-00101-2
  25. Tang, X. H., Zou, X., 10.1017/S0013091501000396, Proc. Edinb. Math. Soc., II. Ser. 45 (2002), 333-347. (2002) Zbl1024.34070MR1912643DOI10.1017/S0013091501000396
  26. Wang, X., Liao, L., 10.1016/S0022-247X(03)00021-0, J. Math. Anal. Appl. 279 (2003), 326-338. (2003) Zbl1054.34128MR1970509DOI10.1016/S0022-247X(03)00021-0
  27. Wu, J., Yu, J. S., Convergence in nonautonomous scalar neutral equations, Dyn. Syst. Appl. 4 (1995), 279-290. (1995) Zbl0830.34066MR1338949
  28. Ye, H., Gao, G., 10.1006/jmaa.2000.7391, J. Math. Anal. Appl. 258 (2001), 556-564. (2001) Zbl0991.34061MR1835558DOI10.1006/jmaa.2000.7391
  29. Yu, J. S., 10.1006/jmaa.1996.0416, J. Math. Anal. Appl. 203 (1996), 850-860. (1996) Zbl0866.34061MR1417134DOI10.1006/jmaa.1996.0416
  30. Zhao, D., 10.1186/1687-1847-2011-48, Adv. Difference Equ. 2011 (2011), Paper No. 48, 11 pages. (2011) Zbl1282.34076MR2891788DOI10.1186/1687-1847-2011-48

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