Oscillation criteria of third-order nonlinear delay differential equations. (English) Zbl 1141.34040

In this paper the self-adjoint nonlinear delay differential equation is studied of the form
\[ \Bigl (c(t)\bigl (a(t)x'(t)\bigl )'\Bigr )'+q(t)f\bigl (x(t-\sigma )\bigr )=0, \]
where \(\sigma \geq 0\) and functions \(c(t),a(t),q(t)\) and \(f(t)\) satisfy addition conditions. Criteria are derived that every solution oscillates or converges to zero. Illustrative examples are presented.


34K11 Oscillation theory of functional-differential equations
Full Text: EuDML


[1] AGARWAL R. P.-GRACE S. R.-O’REGAN D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Acad. PubL, Drdrechet, 2000. · Zbl 0954.34002
[2] BAINOV D. D.-MISHEV D. P.: Oscillation Theory for Neutral Differential Equations with Delay. Adam Hilger, New York, 1991. · Zbl 0747.34037
[3] BARTUŠEK M.: On oscillatory solutions of third order differential equations with quasiderivatives. (Forth Mississippi Conf. Diff. Eqns. and Comp. Simulation), Electron. J. Differential Equations 1999 (1999), 1-11. · Zbl 0971.34016
[4] DZURINA J.: Asymptotic properties of third order delay differential equations. Czechoslovak Math. J. 45(120) (1995), 443-448. · Zbl 0842.34073
[5] DZURINA J.: Property (A) of the third order differential equations with deviating arguments. Math. Slovaca 45 (1995), 395-402.
[6] GYORI I.-LADAS G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford, 1991.
[7] KAMENEV I. V.: Integral criterion for oscillation of linear differential equations of second order. Mat. Zemetki 23 (1978), 249-251. · Zbl 0408.34031 · doi:10.1007/BF01153154
[8] KIGURADZE I. T.-CHATURIA T. A.: Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations. Kluwer Acad. PubL, Drodrcht, 1993.
[9] LADDE G. S.-LAKSHMIKANTHAM V.-ZHANG B. Z.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York, 1987. · Zbl 0832.34071
[10] PARHI N.-DAS P.: Asymptotic property of solutions of a class of third-order differential equations. Proc. Amer. Math. Soc. 110 (1990), 387-393. · Zbl 0721.34025 · doi:10.2307/2048082
[11] PARHI N.-DAS. P.: Oscillation criteria for a class of nonlinear differential equations of third order. Ann. Polon. Math. 57 (1992), 219-229. · Zbl 0771.34024
[12] PARHI N.-DAS P.: Oscillation and nonosdilation of nonhomogeneous third order differential equations. Czechoslovak Math. J. 44 (1994), 443-459. · Zbl 0823.34038
[13] PARHI N.-DAS P.: On the oscillation of a class of linear homogeneous third order differential equations. Arch. Math. (Brno) 34 (1998), 435-443. · Zbl 0973.34023
[14] PARHI N.-DAS P.: On asymptotic behavior of delay-differential equations of third order. Nonlinear Anal. 34 (1998), 391-403. · Zbl 0935.34063 · doi:10.1016/S0362-546X(97)00600-7
[15] PARHI N.-DAS P.: Asymptotic behavior of a class of third order delay differential equations. Math. Slovaca 50 (2000), 315-333. · Zbl 0996.34029
[16] PHILOS, CH. G.: Oscillation theorems for linear differential equation of second order. Arch. Math. (Basel) 53 (1989), 483-492. · Zbl 0661.34030 · doi:10.1007/BF01324723
[17] SAKER S. H.-PANG P. Y. H.-AGARWAL R. P.: Oscillation theorems for second order functional differential equations with damping. Dyn. Syst. Appl. 12 (2003), 307-322. · Zbl 1057.34083
[18] ŠKERLÍK A.: An integral condition of oscillation for equation y”’(t) + p(t)y’(t) + q(t)y(t) = 0\( with nonnegative coefficients. Arch. Math. (Brno) 31 (1995), 155-161.\) · Zbl 0843.34039
[19] ŠKERLÍK A.: Integral criteria of oscillation for a third order linear differential equations. Math. Slovaca 45 (1995), 403-412. · Zbl 0855.34038
[20] YAN J.: A note on an oscillation criterion for an equation with damped term. Proc. Amer. Math. Soc. 90 (1984), 277-280. · Zbl 0542.34028 · doi:10.2307/2045355
[21] YAN J.: Oscillation theorems for second order linear differential equations with damping. Proc. Amer. Math. Soc 98 (1986), 276-282. · Zbl 0622.34027 · doi:10.2307/2045698
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