# On the asymptotic behavior of a class of third order nonlinear neutral differential equations

Blanka Baculíková; Jozef Džurina

Open Mathematics (2010)

- Volume: 8, Issue: 6, page 1091-1103
- ISSN: 2391-5455

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topBlanka Baculíková, and Jozef Džurina. "On the asymptotic behavior of a class of third order nonlinear neutral differential equations." Open Mathematics 8.6 (2010): 1091-1103. <http://eudml.org/doc/269611>.

@article{BlankaBaculíková2010,

abstract = {The objective of this paper is to study asymptotic properties of the third-order neutral differential equation \[ \left[ \{a\left( t \right)\left( \{\left[ \{x\left( t \right) + p\left( t \right)x\left( \{\sigma \left( t \right)\} \right)\} \right]^\{\prime \prime \} \} \right)^\gamma \} \right]^\prime + q\left( t \right)f\left( \{x\left[ \{\tau \left( t \right)\} \right]\} \right) = 0, t \geqslant t\_0 . \left( E \right) \]
. We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.},

author = {Blanka Baculíková, Jozef Džurina},

journal = {Open Mathematics},

keywords = {Third-order neutral differential equations; Oscillation; Nonoscillation; Comparison theorem; third-order neutral differential equations; oscillation; nonoscillation; comparison theorem},

language = {eng},

number = {6},

pages = {1091-1103},

title = {On the asymptotic behavior of a class of third order nonlinear neutral differential equations},

url = {http://eudml.org/doc/269611},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Blanka Baculíková

AU - Jozef Džurina

TI - On the asymptotic behavior of a class of third order nonlinear neutral differential equations

JO - Open Mathematics

PY - 2010

VL - 8

IS - 6

SP - 1091

EP - 1103

AB - The objective of this paper is to study asymptotic properties of the third-order neutral differential equation \[ \left[ {a\left( t \right)\left( {\left[ {x\left( t \right) + p\left( t \right)x\left( {\sigma \left( t \right)} \right)} \right]^{\prime \prime } } \right)^\gamma } \right]^\prime + q\left( t \right)f\left( {x\left[ {\tau \left( t \right)} \right]} \right) = 0, t \geqslant t_0 . \left( E \right) \]
. We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results.

LA - eng

KW - Third-order neutral differential equations; Oscillation; Nonoscillation; Comparison theorem; third-order neutral differential equations; oscillation; nonoscillation; comparison theorem

UR - http://eudml.org/doc/269611

ER -

## References

top- [1] Akın-Bohner E., Došlá Z., Lawrence B., Oscillatory properties for three-dimensional dynamic systems, Nonlinear Anal., 2008, 69(2), 483–494 http://dx.doi.org/10.1016/j.na.2007.05.035 Zbl1161.34018
- [2] Baculíková B., Džurina J., Oscillation of third-order neutral differential equations, Math. Comput. Modelling, 2010, 52(1–2), 215–226 http://dx.doi.org/10.1016/j.mcm.2010.02.011 Zbl1201.34097
- [3] Baculíková B., Elabbasy E.M., Saker S.H., Džurina J., Oscillation criteria for third-order nonlinear differential equations, Math. Slovaca, 2008, 58(2), 201–220 http://dx.doi.org/10.2478/s12175-008-0068-1 Zbl1174.34052
- [4] Baĭnov D.D., Mishev D.P., Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol, 1991 Zbl0747.34037
- [5] Džurina J., Asymptotic properties of third order delay differential equations, Czechoslovak Math. J., 1995, 45(120)(3), 443–448 Zbl0842.34073
- [6] Džurina J., Asymptotic properties of the third order delay differential equations, Nonlinear Anal., 1996, 26(1), 33–39 http://dx.doi.org/10.1016/0362-546X(94)00239-E
- [7] Džurina J., Oscillation theorems for neutral differential equations of higher order, Czechoslovak Math. J., 2004, 54(129)(1), 107–117 http://dx.doi.org/10.1023/B:CMAJ.0000027252.29549.bb Zbl1057.34074
- [8] Džurina J., Baculíková B., Oscillation of third-order functional differential equations, Electron. J. Qual. Theory Differ. Equ., 2010, No. 43 Zbl1211.34077
- [9] Erbe L.H., Kong Q., Zhang B.G., Oscillation Theory for Functional-Differential Equations, Monogr. Textbooks Pure Appl. Math., 190, Marcel Dekker, New York, 1995
- [10] Grace S.R., Agarwal R.P., Pavani R., Thandapani E., On the oscillation of certain third order nonlinear functional differential equations, Appl. Math. Comput., 2008, 202(1), 102–112 http://dx.doi.org/10.1016/j.amc.2008.01.025 Zbl1154.34368
- [11] Greguš M., Third Order Linear Differential Equations, Math. Appl. (East European Ser.), Reidel, Dordrecht, 1987
- [12] Györi I., Ladas G., Oscillation Theory of Delay Differential Equations, Oxford Math. Monogr., Clarendon Press, Oxford, 1991 Zbl0780.34048
- [13] Hassan T.S., Oscillation of third order nonlinear delay dynamic equations on time scales, Math. Comput. Modelling, 2009, 49(7–8), 1573–1586 http://dx.doi.org/10.1016/j.mcm.2008.12.011
- [14] Kiguradze I.T., Chanturia T.A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Math. Appl. (Soviet Ser.), 89, Kluwer Acad. Publ., Dordrecht, 1993
- [15] Kusano T., Naito M., Comparison theorems for functional-differential equations with deviating arguments, J. Math. Soc. Japan, 1981, 33(3), 509–533 http://dx.doi.org/10.2969/jmsj/03330509 Zbl0494.34049
- [16] Lacková D., The asymptotic properties of the solutions of the nth order functional neutral differential equations, Appl. Math. Comput., 2003, 146(2–3), 385–392 http://dx.doi.org/10.1016/S0096-3003(02)00590-8 Zbl1035.34087
- [17] Ladde G.S., Lakshmikantham V., Zhang B.G., Oscillation Theory of Differential Equations with Deviating Arguments, Monogr. Textbooks Pure Appl. Math., 110, Marcel Dekker, New York, 1987
- [18] Lazer A.C., The behavior of solutions of the differential equation y’" + p(x)y’ + q(x)y = 0, Pacific J. Math., 1966, 17(3), 435–466 Zbl0143.31501
- [19] Parhi N., Das P., Asymptotic property of solutions of a class of third-order differential equations, Proc. Amer. Math. Soc., 1990, 110(2), 387–393 Zbl0721.34025
- [20] Parhi N., Padhi S., On asymptotic behavior of delay-differential equations of third order, Nonlinear Anal., 1998, 34(3), 391–403 http://dx.doi.org/10.1016/S0362-546X(97)00600-7 Zbl0935.34063
- [21] Parhi N., Padhi S., Asymptotic behaviour of solutions of third order delay-differential equations, Indian J. Pure Appl. Math., 2002, 33(10), 1609–1620 Zbl1025.34068
- [22] Philos Ch.G., On the existence of nonoscillatory solutions tending to zero at 1 for differential equations with positive delays, Arch. Math. (Basel), 1981, 36(2), 168–178 Zbl0463.34050
- [23] Saker S.H., Oscillation criteria of third-order nonlinear delay differential equations, Math. Slovaca, 2006, 56(4), 433–450 Zbl1141.34040
- [24] Saker S.H., Džurina J., On the oscillation of certain class of third-order nonlinear delay differential equations, Math. Bohemica, 2010, 135(3), 225–237 Zbl1224.34217
- [25] Tanaka K., Asymptotic analysis of odd order ordinary differential equations, Hiroshima Math. J., 1980, 10(2), 391–408 Zbl0453.34033
- [26] Tiryaki A., Aktas M.F., Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl., 2007, 325(1), 54–68 http://dx.doi.org/10.1016/j.jmaa.2006.01.001 Zbl1110.34048
- [27] Zhong J., Ouyang Z., Zou S., Oscillation criteria for a class of third-order nonlinear neutral differential equations, J. Appl. Anal. (in press) Zbl1276.34026