Oscillation theorems for third order nonlinear differential equations.(English)Zbl 0760.34031

The paper deals with oscillation and asymptotic behaviour of the nonlinear third order equation $$(*)\;(r_ 2(t)(r_ 1(t)y')')'+p(t)y'+q(t)f(y)=0$$, where $$p,q$$ are nonnegative functions and $$r_ 1(t)$$, $$r_ 2(t)>0$$ for $$t\in[a,\infty)$$. Conditions on the functions $$r_ 1,r_ 2,p,q$$ are given which guarantee that no nonoscillatory solution of $$(*)$$ has property $$V_ 2$$ provided certain (nonlinear) second order equation associated with $$(*)$$ is oscillatory (a solution $$y$$ of $$(*)$$ is said to possess property $$V_ 2$$ if $$y(t)L_ ky(t)>0$$, $$k=1,2$$, $$y(t)L_ 3y(t)\leq 0$$ for large $$t$$, where $$L_ 0y=y$$, $$L_ iy=r_ i(L_{i-1}y)'$$, $$i=1,2$$, $$L_ 3y=(L_ 2y)')$$. As a corollary of these results the following oscillation criterion is proved.
Theorem. Let any condition which implies that no nonoscillatory solution of $$(*)$$ has property $$V_ 2$$ be satisfied. A solution $$y$$ of $$(*)$$ which exists on an interval $$[T,\infty)$$ is oscillatory if and only if there exists $$t_ 0\in[T,\infty)$$ such that $$2yL_ 2y-{r_ 2\over r_ 1}(L_ 1y)^ 2+py^ 2\leq 0$$ for $$t=t_ 0$$.
Reviewer: O.Došlý (Brno)

MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
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References:

 [1] COPPEL W. A.: Disconjugacy. Vol. 220, Springer-Verlag, Berlin-Heidelberg-New York, 1971. · Zbl 0224.34003 [2] ERBE L.: Oscillation, nonoscillation and asymptotic behavior for third order nonlinear differential equations. Ann. Mat. Pura Appl. 110 (1976), 373-391. · Zbl 0345.34023 · doi:10.1007/BF02418014 [3] HEIDEL J. W.: Qualitative behavior of solutions of a third order nonlinear differential equation. Pacific J. Math. 27 (1968), 507-526. · Zbl 0172.11703 · doi:10.2140/pjm.1968.27.507 [4] LAZER A. C.: The behavior of solutions of the differential equation y”’+p(x)y’+q(x)y=0. Pacific J. Math. 17 (1966), 435-466. · Zbl 0143.31501 · doi:10.2140/pjm.1966.17.435 [5] OHRISKA J.: On the oscillation a linear differential equation of second order. Czechoslovak Math. J. 39(114) (1989), 16-23. · Zbl 0673.34043 [6] PHILOS, CH. G.: Oscillation and asymptotic behavior of third order linear differential equations. Bull. Inst. Math. Acad. Sinica 11 (1983), 141-160. · Zbl 0523.34028 [7] PHILOS, CH. G., SFICAS Y. G.: Oscillatory and asymptotic behavior of second and third order retarded differential equations. Czechoslovak Math. J. 32(107) (1982), 169-182. · Zbl 0507.34062 [8] RAO V. S. H., DAHIYA R. S.: Properties of solutions of a class of third-order linear differential equations. Period. Math. Hungar. 20(3) (1989), 177-184. · Zbl 0705.34032 · doi:10.1007/BF01848121 [9] SEMAN J.: Oscillation and Asymptotic Properties of Solutions of Differential Equations with Deviating Argument. (Slovak) Dissertation, Department of Mathematics and Physics VŠT, Prešov, 1988. [10] SEMAN J.: Oscillation theorems for second order delay inequalities. Math. Slovaca 39 (1989), 313-322. · Zbl 0698.34058 [11] SINGH Y. P.: Some oscillation theorems for third order nonlinear differential equations. Yokohama Math. J. 18 (1970), 77-86. · Zbl 0216.36901 [12] SWANSON C. A.: Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New.York-London, 1968. · Zbl 0191.09904 [13] ŠEDA V.: Nonoscilatory solutions of differential equations with deviating argument. Czechoslovak Math. J. 36(111) (1986), 93-107. · Zbl 0603.34064 [14] ŠKERLÍK A.: Oscillatory properties of solutions of a third-order nonlinear differential equation. (Slovak) In: Zborník ved. prác VŠT v Košiciach, 1987, pp. 365-375. [15] ŠOLTÉS V.: Oscillatory properties of solutions of a third order nonlinear differential equation. (Russian), Math. Slovaca 26 (1976), 217-227. [16] ŠVIDROŇOVA E., ŠOLTÉS P., SEILER J.: Oscillatory properties of solutions of the third-order nonlinear differential equation. (Slovak) In: Zborník ved. prác VŠT v Košiciach, 1982, pp. 33-44. [17] WALTMAN P.: Oscillation criteria for third order nonlinear differential equations. Pacific J. Math. 18 (1966), 386-389. · Zbl 0144.11403 · doi:10.2140/pjm.1966.18.385 [18] WONG J. S. W.: On the generalized Emden-Fowler equation. SIAM Rev. 17 (1975), 339-360. · Zbl 0295.34026 · doi:10.1137/1017036
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