# Unbounded solutions of third order delayed differential equations with damping term

Open Mathematics (2011)

• Volume: 9, Issue: 1, page 184-195
• ISSN: 2391-5455

top

## Abstract

top
Globally positive solutions for the third order differential equation with the damping term and delay, ${x}^{\text{'}\text{'}\text{'}}+q\left(t\right){x}^{\text{'}}\left(t\right)-r\left(t\right)f\left(x\left(\phi \left(t\right)\right)\right)=0,$ are studied in the case where the corresponding second order differential equation ${y}^{\text{'}\text{'}}+q\left(t\right)y=0$ is oscillatory. Necessary and sufficient conditions for all nonoscillatory solutions of (*) to be unbounded are given. Furthermore, oscillation criteria ensuring that any solution is either oscillatory or unbounded together with its first and second derivatives are presented. The comparison of results with those in the case when (**) is nonoscillatory is given, as well.

## How to cite

top

Miroslav Bartušek, et al. "Unbounded solutions of third order delayed differential equations with damping term." Open Mathematics 9.1 (2011): 184-195. <http://eudml.org/doc/269643>.

@article{MiroslavBartušek2011,
abstract = {Globally positive solutions for the third order differential equation with the damping term and delay, $x^\{\prime \prime \prime \} + q(t)x^\{\prime \}(t) - r(t)f(x(\phi (t))) = 0,$ are studied in the case where the corresponding second order differential equation $y^\{\prime \prime \} + q(t)y = 0$ is oscillatory. Necessary and sufficient conditions for all nonoscillatory solutions of (*) to be unbounded are given. Furthermore, oscillation criteria ensuring that any solution is either oscillatory or unbounded together with its first and second derivatives are presented. The comparison of results with those in the case when (**) is nonoscillatory is given, as well.},
author = {Miroslav Bartušek, Mariella Cecchi, Zuzana Došlá, Mauro Marini},
journal = {Open Mathematics},
keywords = {Third order differential equation; Delay; Damping term; Globally positive solution; Unbounded solutions; Oscillation; third order differential equation; delay; damping term; globally positive solution; unbounded solutions; oscillation},
language = {eng},
number = {1},
pages = {184-195},
title = {Unbounded solutions of third order delayed differential equations with damping term},
url = {http://eudml.org/doc/269643},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Miroslav Bartušek
AU - Mariella Cecchi
AU - Zuzana Došlá
AU - Mauro Marini
TI - Unbounded solutions of third order delayed differential equations with damping term
JO - Open Mathematics
PY - 2011
VL - 9
IS - 1
SP - 184
EP - 195
AB - Globally positive solutions for the third order differential equation with the damping term and delay, $x^{\prime \prime \prime } + q(t)x^{\prime }(t) - r(t)f(x(\phi (t))) = 0,$ are studied in the case where the corresponding second order differential equation $y^{\prime \prime } + q(t)y = 0$ is oscillatory. Necessary and sufficient conditions for all nonoscillatory solutions of (*) to be unbounded are given. Furthermore, oscillation criteria ensuring that any solution is either oscillatory or unbounded together with its first and second derivatives are presented. The comparison of results with those in the case when (**) is nonoscillatory is given, as well.
LA - eng
KW - Third order differential equation; Delay; Damping term; Globally positive solution; Unbounded solutions; Oscillation; third order differential equation; delay; damping term; globally positive solution; unbounded solutions; oscillation
UR - http://eudml.org/doc/269643
ER -

## References

top
1. [1] Bartušek M., On noncontinuable solutions of differential equations with delay, Electron. J. Qual. Theory Differ. Equ., 2009, Spec. Ed. I, No. 6 Zbl1195.34093
2. [2] Bartušek M., Cecchi M., Došlá Z., Marini M., On nonoscillatory solutions of third order nonlinear differential equations, Dynam. Systems Appl., 2000, 9(4), 483–499 Zbl1016.34036
3. [3] Bartušek M., Cecchi M., Došlá Z., Marini M., Oscillation for third order nonlinear differential equations with deviating argument, Abstr. Appl. Anal., 2010, Art. ID 278962 Zbl1192.34073
4. [4] Bartušek M., Cecchi M., Došlá Z., Marini M., Positive solutions of third order damped nonlinear differential equations, Math. Bohem. (in press) Zbl1224.34152
5. [5] Borůvka O., Linear Differential Transformations of the Second Order, The English Universities Press, London, 1971 Zbl0222.34002
6. [6] Cecchi M., Došlá Z., Marini M., Asymptotic behavior of solutions of third order delay differential equations, Arch. Math. (Brno), 1997, 33(1–2), 99–108 Zbl0916.34059
7. [7] Cecchi M., Došlá Z., Marini M., On nonlinear oscillations for equations associated to disconjugate operators, Nonlinear Anal., 1997, 30(3), 1583–1594 http://dx.doi.org/10.1016/S0362-546X(97)00028-X Zbl0892.34032
8. [8] Cecchi M., Došlá Z., Marini M., On third order differential equations with property A and B, J. Math. Anal. Appl., 1999, 231(2), 509–525 http://dx.doi.org/10.1006/jmaa.1998.6247
9. [9] 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
10. [10] Elias U., Oscillation Theory of Two-Term Differential Equations, Math. Appl., 396, Kluwer, Dordrecht, 1997 Zbl0878.34022
11. [11] Erbe L., Oscillation, nonoscillation, and asymptotic behavior for third order nonlinear differential equations, Ann. Mat. Pura Appl., 1976, 110(1), 373–391 http://dx.doi.org/10.1007/BF02418014 Zbl0345.34023
12. [12] Erbe L., Peterson A., Saker S.H., Oscillation and asymptotic behavior of a third order nonlinear dynamic equation, Canad. Appl. Math. Q., 2006, 14(2), 124–147 Zbl1145.34329
13. [13] Greguš M., Greguš M., Jr., Asymptotic properties of solutions of a certain nonautonomous nonlinear differential equation of the third order, Boll. Un. Mat. Ital. A, 1993, 7(3), 341–350 Zbl0802.34032
14. [14] Heidel J.W., Qualitative behavior of solutions of a third order nonlinear differential equations, Pacific J. Math., 1968, 27(3), 507–526
15. [15] Kiguradze I.T., An oscillation criterion for a class of ordinary differential equations, Differential Equations, 1992, 28(2), 180–190 Zbl0788.34027
16. [16] Kiguradze I., Chanturia T.A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Math. Appl. (Soviet Ser.), 89, Kluwer, Dordrecht, 1993
17. [17] Marini M., Criteri di limitatezza per le soluzioni dell'equazione lineare del secondo ordine, Boll. Un. Mat. Ital., 1975, 11(1), 154–165 Zbl0332.34027
18. [18] Mojsej I., Ohriska J., Comparison theorems for noncanonical third order nonlinear differential equations, Cent. Eur. J. Math., 2007, 5(1), 154–163 http://dx.doi.org/10.2478/s11533-006-0044-3 Zbl1128.34021
19. [19] Mojsej I., Tartal'ová A., On bounded nonoscillatory solutions of third-order nonlinear differential equations, Cent. Eur. J. Math., 2009, 7(4), 717–724 http://dx.doi.org/10.2478/s11533-009-0054-z Zbl1193.34068
20. [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. [21] Saker S.H., Oscillation criteria of Hille and Nehari types for third order delay differential equations, Commun. Appl. Anal., 2007, 11(3–4), 451–468 Zbl1139.34049
22. [22] Tiryaki A., Aktaş 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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.