×

zbMATH — the first resource for mathematics

On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations. (English) Zbl 0945.34021
The authors investigate oscillatory and asymptotic properties of solutions to the third-order nonlinear differential equation \[ x'''+p(t,x,x',x'')x''+q(t,x,x',x'')x'+r(t,x,x',x'')x=f(t,x,x',x''), \tag{*} \] where \(p,q,r,f\) are continuous functions on \((\alpha,\infty)\times \mathbb{R}^n\), \(\alpha\in \mathbb{R}\). Using the linearization technique combined with a comparison principle conditions on the functions \(p,q,r,f\) are given which guarantee that every solution to (*) with one zero is oscillatory. Asymptotic properties of nonoscillatory solutions are studied, too.

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Greguš, M.; Graef, J.R., On a certain nonautonomous nonlinear third order differential equation, Appl. anal., 58, 175-185, (1995) · Zbl 0880.34035
[2] Ezeilo, J.O.C., A stability result for the solutions of certain third order differential equations, J. London math. soc., 37, 405-409, (1962) · Zbl 0113.07604
[3] Ezeilo, J.O.C., A stability result for a certain third order differential equation, Ann. mat. pura appl., 72, 4, 1-9, (1966) · Zbl 0143.11602
[4] Pliss, V.A., Study of a nonlinear differential equation of the third order, Dokl. akad. nauk. SSSR, 111, 1178-1180, (1956) · Zbl 0072.09303
[5] Reissig, R.; Sansone, G.; Conti, R., Nichtlineare differentialgleichungen höherer ordnung, (1969), Edizione Cremonese Roma · Zbl 0172.10801
[6] Greguš, M., Third order linear differential equations, (1987), D. Reidel Publishing Company Boston · Zbl 0878.34025
[7] Gera, M., Über einige eigenschaften der Lösungen der gleichung x″’ + a(t)x″ + b(t)x′ + c(t)x = 0, c(t) ≥ 0, Mat. C̆asopis, 24, 357-370, (1974) · Zbl 0324.34026
[8] Levin, A.Ju, Nonoscillation of solutions of the equation x(n) + p1(t)x(n−1) + … + pn(t)x = 0, Uspehi mat. nauk., 24, 43-96, (1969)
[9] Gera, M., Bedingungen für die existenz oszillatorischer Lösungen der gleichung x″’ + a(t)x″ + b(t)x′ + c(t)x = 0, c(t) ≥ 0, Mat. C̆asopis, 25, 23-40, (1975) · Zbl 0316.34031
[10] Gera, M., Über die untermengen der Lösungen der gleichung x″’ + a(t)x″ + b(t)x′ + c(t)x = 0, c(t) ≥ 0, Math. slovaca, 30, 313-326, (1980) · Zbl 0443.34034
[11] Hanan, M., Oscillation criteria for third-order linear differential equations, Pacific J. math., 11, 919-944, (1961) · Zbl 0104.30901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.