Oscillation and nonoscillation in a nonautonomous delay-logistic equation

Zhang, B. G.; Gopalsamy, K.

doi:10.1090/qam/950601

Authors: B. G. Zhang and K. Gopalsamy
Journal: Quart. Appl. Math. 46 (1988), 267-273
MSC: Primary 34K15; Secondary 34C10, 92A15
DOI: https://doi.org/10.1090/qam/950601
MathSciNet review: 950601
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Abstract: Sufficient conditions are obtained for the delay-logistic equation $\dot x\left ( t \right ) = \\ r\left ( t \right )x\left ( t \right )\left [ {1 - x\left ( {t - \tau \left ( t \right )} \right )/K} \right ]$ to be respectively oscillatory and nonoscillatory.

L. E. Èl′sgol′ts and S. B. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Translated from the Russian by John L. Casti; Mathematics in Science and Engineering, Vol. 105. MR 0352647
K. Gopalsamy, Oscillations in a delay-logistic equation, Quart. Appl. Math. 44 (1986), no. 3, 447–461. MR 860898, DOI https://doi.org/10.1090/S0033-569X-1986-0860898-5
G. Stephen Jones, The existence of periodic solutions of $f^{\prime } (x)=-\alpha f(x-1)\{1+f(x)\}$, J. Math. Anal. Appl. 5 (1962), 435–450. MR 141837, DOI https://doi.org/10.1016/0022-247X%2862%2990017-3
S. Kakutani and L. Markus, On the non-linear difference-differential equation $y^{\prime } (t)=[A-By(t-\tau )]y(t)$, Contributions to the theory of nonlinear oscillations, Vol. IV, Annals of Mathematics Studies, no. 41, Princeton University Press, Princeton, N.J., 1958, pp. 1–18. MR 0101953
R. G. Koplatadze and T. A. Chanturiya, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differentsial′nye Uravneniya 18 (1982), no. 8, 1463–1465, 1472 (Russian). MR 671174
M. R. S. Kulenović, G. Ladas, and A. Meimaridou, On oscillation of nonlinear delay differential equations, Quart. Appl. Math. 45 (1987), no. 1, 155–164. MR 885177, DOI https://doi.org/10.1090/S0033-569X-1987-0885177-5
G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation theory of differential equations with deviating arguments, Monographs and Textbooks in Pure and Applied Mathematics, vol. 110, Marcel Dekker, Inc., New York, 1987. MR 1017244
V. N. Shevelo, Ostsillyatsiya resheniĭ differentsial′nykh uravneniĭ s otklonyayushchimsya argumentom, Izdat. “Naukova Dumka”, Kiev, 1978 (Russian). MR 0492732
E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math. 194 (1955), 66–87. MR 72363, DOI https://doi.org/10.1515/crll.1955.194.66