Abstract
In this paper, necessary and sufficient conditions are obtained for every bounded solution of
to oscillate or tend to zero as t → ∞ for different ranges of p(t). It is shown, under some stronger conditions, that every solution of (*) oscillates or tends to zero as t → ∞. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.
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Ming-Po-Chen, Z. C. Wang, J. S. Yu and B. G. Zhang: Oscillation and asymptotic behaviour of higher order neutral differential equations. Bull. Inst. Math. Acad. Sinica 22 (1994), 203–217.
Q. Chuanxi and G. Ladas: Oscillation of higher order neutral differential equations with variable coefficients. Math. Nachr. 150 (1991), 15–24.
D. A. Georgiou and C. Qian: Oscillation criteria in neutral equations of nth order with variable coefficients. Internat. J. Math. Math. Sci. 14 (1991), 689–696.
K. Gopalsamy, B. S. Lalli and B. G. Zhang: Oscillation in odd order neutral differential equations. Czechoslovak Math. J. 42 (1992), 313–323.
K. Gopalsamy, S. R. Grace and B. S. Lalli: Oscillation of even order neutral differential equations. Indian J. Math. 35 (1993), 9–25.
S. R. Grace: On the oscillation of certain forced functional differential equation. J. Math. Anal. Appl. 202 (1996), 555–577.
I. Gyori and G. Ladas: Oscialltion Theory of Delay-Differential Equations with Applications. Clarendon Press, Oxford, 1991.
T. H. Hildebrandt: Introduction to the Theory of Integration. Academic Press, New York, 1963.
I. T. Kiguradze: On the oscillation of solutions of the equation \(\tfrac{{\operatorname{d} ^m u}}{{\operatorname{d} t^m }} + a\left( t \right)u^m\) sign u = 0. Mat. Sb. 65 (1964), 172–187.
G. Ladas and Y. G. Sficas: Oscillations of higher order neutral equations. Austral. Math. Soc. Ser. B 27 (1986), 502–511.
G. Ladas, C. Qian and J. Yan: Oscillations of higher order neutral differential equations. Portugal. Math. 48 (1991), 291–307.
G. S. Ladde, V. Lakshmikantham and B. G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker INC., New York, 1987.
X. Z. Liu, J. S. Yu and B. G. Zhang: Oscillation and nonoscillation for a class of neutral differential equations. Differential Equations Dynam. Systems 1 (1993), 197–204.
N. Parhi and P. K. Mohanty: Oscillation of solutions of forced neutral differential equations of n-th order. Czechoslovak Math. J. 45 (1995), 413–433.
N. Parhi and P. K. Mohanty: Maintenance of oscillation of neutral differential equations under the effect of a forcing term. Indian J. Pure Appl. Math. 26 (1995), 909–919.
N. Parhi and P. K. Mohanty: Oscillatory behaviour of solutions of forced neutral differential equations. Ann. Polon. Math. 65 (1996), 1–10.
N. Parhi and P. K. Mohanty: Oscillations of neutral differential equations of higher order. Bull. Inst. Math. Acad. Sinica 24 (1996), 139–150.
N. Parhi: Oscillation of higher order differential equations of neutral type. Czechoslovak Math. J. 50 (2000), 155–173.
N. Parhi and R. N. Rath: On oscillation criteria for a forced neutral differential equation. Bull. Inst. Math. Acad. Sinica 28 (2000), 59–70.
N. Parhi and R. N. Rath: Oscillation criteria for forced first order neutral differential equations with variable coefficients. J. Math. Anal. Appl. 256 (2001), 525–541.
N. Parhi and R. N. Rath: On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations. Proc. Indian. Acad. Sci. (Math. Sci.), Vol. 111. 2001, pp. 337–350.
H. L. Royden: Real Analysis. 3rd edition, MacMillan Publ. Co., New York, 1989.
J. H. Shen: New oscillation criteria for odd order neutral equations. J. Math. Anal. Appl. 201 (1996), 387–395.
D. Tang: Oscillation of higher order nonlinear neutral functional differential equation. Ann. Differential Equations 12 (1996), 83–88.
J. S. Yu, Z. C. Wang and B. G. Zhang: Oscillation of higher order neutral differential equations. Rocky Mountain J. Math. To appear.
B. G. Zhang and K. Gopalsam: Oscillations and nonoscillations in higher order neutral equations. J. Math. Phys. Sci. 25 (1991), 152–165.
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Parhi, N., Rath, R.N. On Oscillation of Solutions of Forced Nonlinear Neutral Differential Equations of Higher Order. Czech Math J 53, 805–825 (2003). https://doi.org/10.1007/s10587-004-0805-8
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DOI: https://doi.org/10.1007/s10587-004-0805-8