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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.

##### MSC:
 34K11 Oscillation theory of functional-differential equations
##### Keywords:
oscillation; Kamenev-type criteria
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##### References:
  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  BAINOV D. D.-MISHEV D. P.: Oscillation Theory for Neutral Differential Equations with Delay. Adam Hilger, New York, 1991. · Zbl 0747.34037  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  DZURINA J.: Asymptotic properties of third order delay differential equations. Czechoslovak Math. J. 45(120) (1995), 443-448. · Zbl 0842.34073 · eudml:31493  DZURINA J.: Property (A) of the third order differential equations with deviating arguments. Math. Slovaca 45 (1995), 395-402.  GYORI I.-LADAS G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon Press, Oxford, 1991.  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  KIGURADZE I. T.-CHATURIA T. A.: Asymptotic Properties of Solutions of Nonatunomous Ordinary Differential Equations. Kluwer Acad. PubL, Drodrcht, 1993.  LADDE G. S.-LAKSHMIKANTHAM V.-ZHANG B. Z.: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York, 1987. · Zbl 0832.34071  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  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  PARHI N.-DAS P.: Oscillation and nonosdilation of nonhomogeneous third order differential equations. Czechoslovak Math. J. 44 (1994), 443-459. · Zbl 0823.34038 · eudml:31430  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 · eudml:233601  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  PARHI N.-DAS P.: Asymptotic behavior of a class of third order delay differential equations. Math. Slovaca 50 (2000), 315-333. · Zbl 0996.34029 · eudml:34517  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  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  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 · eudml:18431  ŠKERLÍK A.: Integral criteria of oscillation for a third order linear differential equations. Math. Slovaca 45 (1995), 403-412. · Zbl 0855.34038  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  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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