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Oscillation Criteria for Second Order Nonlinear Delay Equations

Published online by Cambridge University Press:  20 November 2018

Lynn Erbe
Lynn Erbe*
Affiliation:
University of Alberta, Edmonton, Alberta
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It is the purpose of this paper to establish oscillation criteria for second order nonlinear differential equations with retarded argument. Specifically, we consider the equation

1.1

where f ∊ C[0, + ∞) x R2, g ∊ C[0, + ∞), and

1.2

We shall restrict attention to solutions of (1.1) which exist on some ray [T, + ∞). A solution of (1.1) is called oscillatory if it has no largest zero.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 1 , March 1973 , pp. 49 - 56
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. El’gol’ts, L.E., Introduction to the theory of differential equations with deviating arguments, Holden-Day, San Francisco, Calif., 1966.Google Scholar
2. Gollwitzer, H.E., On nonlinear oscillations for a second order delay equation, J. Math. Anal. Appl. 26 (1969), 385–389.Google Scholar
3. Bradley, J.S., Oscillation theorems for a second order delay equation, J. Differential Equations, 8 (1970), 397–403.Google Scholar
4. Waltman, P., A note on an oscillation criterion for an equation with a functional argument, Canad. Math. Bull. 11 (1968), 593–595.Google Scholar
5. Staikos, V.A. and Petsoulas, A.G., Some oscillation criteria for second order nonlinear delaydifferential equations, J. Math. Anal. Appl. 30 (1970), 695–701.Google Scholar
6. Ladas, G., Oscillation and asymptotic behavior of solutions of differential equations with retarded argument, Technical Report No. 10, Mathematics Department, Univ. of Rhode Island, July 1970.Google Scholar
7. Atkinson, F.V., On second order nonlinear oscillations. Pacific J. Math. 5 (1955), 643–647.Google Scholar
8. Wong, J.S.W., On second order nonlinear oscillation. Funkcial. Ekvac. 11 (1968), 207–234.Google Scholar
9. Hartman, P., Ordinary differential equations, Wiley, New York, 1964.Google Scholar
10. Hartman, P., On nonoscillatory linear differential equations of second order, Amer. J. Math. 74 (1952), 389–400.Google Scholar
11. Hille, E., Nonoscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234–252.Google Scholar
12. Jackson, L.K., Sub functions and second order differential inequalities, Advances in Math. 2 (1968), 307–363.Google Scholar
13. Belohorec, S., Oscillatory solutions of certain nonlinear differential equations of the second order, (Czech), Mat.-Fyz. Časopis Sloven. Akad. Vied. (4) 11 (1961), 250–255.Google Scholar
14. Jones, J. Jr On the extension of a theorem of Atkinson, Quart. J. Math. Oxford Ser. 7 (1956), 306–309.Google Scholar