Geodesic
Oscillation of third order differential equation with damping term
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 301-316 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$ x'''(t)+q(t)x'(t)+r(t)|x|^{\lambda }(t)\mathop {\rm sgn} x(t)=0 ,\quad t\geq 0. $$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \leq 1$ and if the corresponding second order differential equation $h''+q(t)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.
We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$ x'''(t)+q(t)x'(t)+r(t)|x|^{\lambda }(t)\mathop {\rm sgn} x(t)=0 ,\quad t\geq 0. $$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \leq 1$ and if the corresponding second order differential equation $h''+q(t)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.
DOI : 10.1007/s10587-015-0176-3
Classification : 34C10, 34C15
Keywords: third order nonlinear differential equation; vanishing at infinity solution; Kneser solution; oscillatory solution 
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Bartušek, Miroslav; Došlá, Zuzana. Oscillation of third order differential equation with damping term. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 301-316. doi: 10.1007/s10587-015-0176-3

[1] Agarwal, R. P., Baculíková, B., Džurina, J., Li, T.: Oscillation of third-order nonlinear differential equations with mixed arguments. Acta Math. Hung. 134 (2012), 54-67. | DOI | MR

[2] Agarwal, R. P., O'Regan, D.: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers Dordrecht (2001). | MR | Zbl

[3] Astashova, I.: On the asymptotic behavior at infinity of solutions to quasi-linear differential equations. Math. Bohem. 135 (2010), 373-382. | MR | Zbl

[4] Astashova, I. V.: Uniform estimates for positive solutions of quasi-linear ordinary differential equations. Russian Izv. Ross. Akad. Nauk, Ser. Mat. 72 (2008), 85-104 translation in Izv. Math. 72 (2008), 1141-1160. | MR | Zbl

[5] Baculíková, B., Džurina, J., Rogovchenko, Y. V.: Oscillation of third order trinomial delay differential equations. Appl. Math. Comput. 218 (2012), 7023-7033. | DOI | MR | Zbl

[6] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: Asymptotics for higher order differential equations with a middle term. J. Math. Anal. Appl. 388 (2012), 1130-1140. | DOI | MR | Zbl

[7] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: Oscillation for third-order nonlinear differential equations with deviating argument. Abstr. Appl. Anal. 2010 (2010), Article ID 278962, 19 pages. | MR | Zbl

[8] Bartušek, M., Cecchi, M., Došlá, Z., Marini, M.: On nonoscillatory solutions of third order nonlinear differential equations. Dyn. Syst. Appl. 9 (2000), 483-499. | MR

[9] Bartušek, M., Cecchi, M., Marini, M.: On Kneser solutions of nonlinear third order differential equations. J. Math. Anal. Appl. 261 (2001), 72-84. | DOI | MR

[10] Cecchi, M., Došlá, Z., Marini, M.: On third order differential equations with property A and B. J. Math. Anal. Appl. 231 (1999), 509-525. | DOI | MR

[11] Cecchi, M., Došlá, Z., Marini, M.: Some properties of third order differential operators. Czech. Math. J. 47 (1997), 729-748. | DOI | MR | Zbl

[12] Elias, U.: Oscillation Theory of Two-Term Differential Equations. Mathematics and Its Applications 396 Kluwer Academic Publishers, Dordrecht (1997). | MR | Zbl

[13] Greguš, M.: Third Order Linear Differential Equations. Mathematics and Its Applications (East European Series) 22 Reidel Publishing Company, Dordrecht (1987). | MR

[14] Kiguradze, I. T.: An oscillation criterion for a class of ordinary differential equations. Differ. Uravn. 28 (1992), 207-219 Russian translation in Differ. Equ. 28 (1992), 180-190. | MR | Zbl

[15] Kiguradze, I. T., Chanturiya, T. A.: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Mathematics and Its Applications. Soviet Series 89 Kluwer Academic Publishers, Dordrecht (1993). | MR | Zbl

[16] Kusano, T., Švec, M.: On unbounded positive solutions of nonlinear differential equations with oscillating coefficients. Czech. Math. J. 39 (1989), 133-141. | MR | Zbl

[17] Ohriska, J.: Oscillatory and asymptotic properties of third and fourth order linear differential equations. Czech. Math. J. 39 (1989), 215-224. | MR | Zbl

[18] Padhi, S., Pati, S.: Theory of Third-Order Differential Equations. Springer New Delhi (2014). | MR | Zbl

[19] Švec, M.: Behavior of nonoscillatory solutions of some nonlinear differential equations. Acta Math. Univ. Comenianae 39 (1980), 115-130. | MR | Zbl

[20] Švec, M.: Sur une propriété intégrales de l'équation $y^{(n)}+Q(x)y=0$, $n=3,4$. Czech. Math. J. 7 (1957), 450-462 French. | MR

[21] Tiryaki, A., Aktaş, M. F.: Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping. J. Math. Anal. Appl. 325 (2007), 54-68. | DOI | MR | Zbl

[22] Villari, G.: Contributi allo studio asintotico dell'equazione $x'''(t)+p(t)x(t)=0$. Italian Ann. Mat. Pura Appl., IV. Ser. 51 (1960), 301-328. | MR | Zbl

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