On oscillation of nonlinear delay differential equations
Authors:
M. R. S. Kulenović, G. Ladas and A. Meimaridou
Journal:
Quart. Appl. Math. 45 (1987), 155-164
MSC:
Primary 34K15
DOI:
https://doi.org/10.1090/qam/885177
MathSciNet review:
885177
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Abstract: Necessary, sufficient, and necessary and sufficient conditions are obtained for all solutions of the nonlinear differential equation \[ \frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}f\left ( {y\left ( {t - {\tau _j}} \right )} \right ) = 0,\qquad t \ge 0,} \qquad \left ( * \right )\] to be oscillatory. These conditions are expressed in terms of the characteristic equation of the corresponding linear “variational” equation \[ \frac {{dy}}{{dt}} + \sum \limits _{j = 1}^n {{q_j}y\left ( {t - {\tau _j}} \right ) = 0, \qquad t \ge 0. \qquad \left ( { * * } \right )} \] Our results show that for a certain class of nonlinear functions $f,\left ( * \right )$ oscillates if and only if $\left ( { * * } \right )$ oscillates. As an application of our results, we obtain simple sufficient and necessary and sufficient conditions for the oscillation of several nonlinear delay differential equations which appear in applications.
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O. Arino, G. Ladas, and Y. G. Sficas, On oscillations of some retarded differential equations, SIAM J. Math. Anal. (to appear)
S. A. Chapin and R. D. Nussbaum, Asymptotic estimates for the periods of periodic solutions of differential-delay equations, Mich. Math. J. 31, 215–229 (1984)
K. Gopalsamy, Oscillations in a delay-logistic equation, Quart. Appl. Math. 44, 447–461 (1986)
K. P. Hadeler, Delay equations in biology, in Functional differential equations and approximation of fixed points, pp. 136–156, Lecture Notes in Math., 730, Springer, Berlin, 1979
J. K. Hale, Theory of functional-differential equations, Springer-Verlag, New York, 1977
A. F. Ivanov and V. N. Shevelo, On the oscillation and asymptotic behavior of solutions of first order functional differential equations (Russian), Uk. Mat. Zh. 33, 745–751 (1981)
B. R. Hunt and J. A. Yorke, When all solutions of $\dot x\left ( t \right ) = - \sum \nolimits _{i = 1}^n {{q_i}\left ( t \right )x\left ( {t - {T_i}\left ( t \right )} \right )}$ oscillate, J. Differential Equations 53, 139–145 (1984)
S. Kakutani and L. Markus, On the non-linear difference-differential equation $\dot y\left ( t \right ) = \left [ {A - By\left ( {t - \tau } \right )} \right ]y\left ( t \right )$, in Contribution to the theory of nonlinear oscillations, Vol. 4, pp. 1–18, Princeton Univ. Press, 1958
J. L. Kaplan and J. A. Yorke, On the nonlinear differential delay equation $\dot x\left ( t \right ) = - f\left ( {x\left ( t \right ),x\left ( {t - 1} \right )} \right )$, J. Differential Equations 23, 293–314 (1977)
G. Ladas and I. P. Stavroulakis, Oscillations caused by several retarded and advanced arguments, J. Differential Equations 44, 134–152 (1982)
G. Ladas, Y. G. Sficas, and I. P. Stavroulakis, Necessary and sufficient conditions for oscillations, Amer. Math. Monthly 90, 637–640 (1983)
E. M. Wright, A nonlinear difference-differential equation, J. Reine Angew. Math. 194, 66–87 (1955)
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© Copyright 1987
American Mathematical Society