Unbounded solutions of third order delayed differential equations with damping term

• Autorzy: Miroslav Bartušek , Mariella Cecchi , Zuzana Došlá , Mauro Marini
• Języki publikacji: EN

Abstrakty

EN

Globally positive solutions for the third order differential equation with the damping term and delay, $$ x''' + q(t)x'(t) - r(t)f(x(\phi (t))) = 0, $$ are studied in the case where the corresponding second order differential equation $$ y'' + q(t)y = 0 $$ 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.

Słowa kluczowe

EN

Third order differential equation; Delay; Damping term; Globally positive solution; Unbounded solutions; Oscillation

Kategorie tematyczne

• 34C10: Oscillation theory, zeros, disconjugacy and comparison theory
• 34C15: Nonlinear oscillations, coupled oscillators

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 1
• Strony: 184-195

Daty

wydano

2011-02-01

online

2010-12-30

Twórcy

autor

Miroslav Bartušek

• Masaryk University

autor

Mariella Cecchi

• University of Florence

autor

Zuzana Došlá

• Masaryk University

autor

Mauro Marini

• University of Florence

Bibliografia

• [1] Bartušek M., On noncontinuable solutions of differential equations with delay, Electron. J. Qual. Theory Differ. Equ., 2009, Spec. Ed. I, No. 6
• [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
• [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
• [4] Bartušek M., Cecchi M., Došlá Z., Marini M., Positive solutions of third order damped nonlinear differential equations, Math. Bohem. (in press)
• [5] Borůvka O., Linear Differential Transformations of the Second Order, The English Universities Press, London, 1971
• [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
• [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
• [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] 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] Elias U., Oscillation Theory of Two-Term Differential Equations, Math. Appl., 396, Kluwer, Dordrecht, 1997
• [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
• [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
• [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
• [14] Heidel J.W., Qualitative behavior of solutions of a third order nonlinear differential equations, Pacific J. Math., 1968, 27(3), 507–526
• [15] Kiguradze I.T., An oscillation criterion for a class of ordinary differential equations, Differential Equations, 1992, 28(2), 180–190
• [16] Kiguradze I., Chanturia T.A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Math. Appl. (Soviet Ser.), 89, Kluwer, Dordrecht, 1993
• [17] Marini M., Criteri di limitatezza per le soluzioni dell'equazione lineare del secondo ordine, Boll. Un. Mat. Ital., 1975, 11(1), 154–165
• [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
• [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
• [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
• [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
• [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

Identyfikatory

DOI

10.2478/s11533-010-0088-2