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Oscillation criteria for second order nonlinear delay inequalities

Published online by Cambridge University Press:  17 April 2009

S. Nababan  and
E.S. Noussair
S. Nababan
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
E.S. Noussair
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
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Abstract

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Oscillation criteria are obtained, for the nonlinear delay differential inequality u(u″+f(t, u(t), u(g(t)))) ≤ 0. The main theorems give sufficient conditions (and in some cases sufficient and. necessary conditions) for all solutions u(t) to have arbitrary large zeros. Generalizations to more general cases are discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 3 , June 1976 , pp. 331 - 341
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Belohorec, Štefan, “Oscillatory solutions of certain nonlinear differential equations of second order” (Czech), Mat.-Fyz. Časopis Sloven. Akad. Vied. 11 (1961), 250–255.Google Scholar
[2]Bradley, John S., “Oscillation theorems for a second-order delay equation”, J. Differential Equations 8 (1970), 397–403.CrossRefGoogle Scholar
[3]El'sgol'ts, L.E., Introduction to the theory of differential equations with deviating arguments (translated by Mclaughlin, Robert J.. Holden-Day, San Francisco, California; London; Amsterdam; 1966).Google Scholar
[4]Erbe, Lynn, “Oscillation criteria for second order nonlinear delay equations”, Canad. Math. Bull. 16 (1973), 49–56.CrossRefGoogle Scholar
[5]Gollwitzer, H.E., “On nonlinear oscillations for a second order delay equation”, J. Math. Anal. Appl. 26 (1969), 385–389.CrossRefGoogle Scholar
[6]Ladas, G., “Oscillation and asymptotic behavior of solutions of differential equations with retarded argument”, J. Differential Equations 10 (1971), 281–290.CrossRefGoogle Scholar
[7]Lillo, James C., “Oscillatory solutions of the equation y′(x) = m(x)y(x-n(x))”, J. Differential Equations 6 (1969), 1–35.CrossRefGoogle Scholar
[8]Норкин, С.Б., Длфференциальные уравнения второго порядка с запаздывающим аргументом (Izdat. “Nauka”, Moscow, 1965). S.B. Norkin, Differential equations of the second order with retarded argument. Some problems of the theory of vibrations of systems with retardation (Translations of Mathematical Monographs, 31. Amer. Math. Soc., Providence, Rhode Island, 1972).Google Scholar
[9]Staïkos, V.A., “Oscillatory property of certain delay differential equations”, Bull. Soc. Math. Grèce 11 (1970), 1–5.Google Scholar
[10]Waltman, Paul, “A note on an oscillation criterion for an equation with a functional argument”, Canad. Math. Bull. 11 (1968), 593–595.CrossRefGoogle Scholar
[11]Wong, James S.W., “Second order oscillation with retarded arguments”, Ordinary differential equations, 581–596 (Proc. Conf. Ordinary Differential Equations, Washington, 1971. Academic Press, New York and London, 1972).Google Scholar