Summary
New oscillation criteria are established for the first order functional differential equation (*) y'(t)+p(t)y(g(t))=0and its nonlinear analogue. The results are presented so that a remarkable duality existing between the case where (*) is retarded (g(t)<t) and the case where(*) is advanced (g(t)>t) is apparent. Possible extension of the results for (*) to equations with several deviating arguments is attempted. Finally, it is shown that there exists a class of autonomous equations for which the oscillation situation can be completely characterized.
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C. H. Anderson,Asymptotic oscillation results for solutions to first-order nonlinear differential-difference equations of advanced type, J. Math. Anal. Appl.,24 (1968), pp. 430–439.
R. D. Driver -D. W. Sasser -M. L. Slater,The equation x′(t)=ax(t)+bx(t−τ) with «small» delay, Amer. Math. Monthly,80 (1973), pp. 990–995.
A. F. Ivanov -V. N. Ševelo,On the oscillation and asymptotic behavior of solutions of differential-functional equations of the first order, Ukrain. Mat. Ž.,33 (1981), pp. 745–751 (Russian).
Y. Kitamura -T. Kusano,Oscillations of first-order nonlinear differential equations with deviating arguments, Proc. Amer. Math. Soc.,78 (1980), pp. 64–68.
R. G. Koplatadze,On oscillatory solutions of first order nonlinear differential equations with retarded argument, Soobšč.Akad. Nauk Gruzin. SSR,70 (1973), pp. 17–20 (Russian).
R. G. Koplatadze -T. A. Čanturija,On Oscillatory Properties of Differential Equations with Deviating Arguments, Tbilisi Univ. Press, Tbilisi (1977) (Russian).
R. G. Koplatadze -T. A. Čanturija,On oscillatory and monotone solutions of first order differential equations with deviating arguments, Differential'nye Uravnenija,18 (1982), pp. 1463–1465 (Russian).
T. Kusano,On even order functional differential equations with advanced and retarded arguments, J. Differential Equations,45 (1982), pp. 75–84.
G. Ladas,Sharp conditions for oscillation caused by delays, Applicable Anal.,9 (1979), pp. 93–98.
G. Ladas -V. Lakshmikantham -J. S. Papadekis,Oscillations of higher-order retarded differential equations generated by the retarded argument, «Delay and Functional Differential Equations and their Applications» (K. Schmitt, Ed.), pp. 219–231, Academic Press, New York (1972).
G. Ladas -I. P. Stavroulakis,On delay differential inequalities of first order, Funkcial. Ekvac.,25 (1982), pp. 105–113.
G. Ladas -I. P. Stavroulakis,Oscillations caused by several retarded and advanced arguments, J. Differential Equations,44 (1982), pp. 134–152.
G. Ladas -Y. G. Sficas -I. P. Stavroulakis,Necessary and sufficient conditions for oscillations, Amer. Math. Monthly,90 (1983), pp. 637–640.
G.Ladas - Y. G.Sficas - I. P.Stavroulakis,Functional differential inequalities and equations with oscillating coefficients, Proceedings of the Fifth International Conference on «Trends in Theory and Practice of Nonlinear Differential Equations» held at Arlington, Texas during June 14–18, 1982 (to appear).
J. C. Lillo,Oscillatory solutions of the equation y′(x)=m(x)(x−n(x)), J. Differential Equations,6 (1969), pp. 1–35.
H. Onose,Oscillation of a functional differential equation arising from an industrial problem, J. Austral. Math. Soc. (Series A),26 (1978), pp. 323–329.
H. Onose,Oscillatory properties of first order differential inequalities with deviating argument, Funkcial. Ekvac.,26 (1983), pp. 189–195.
V. N.Ševelo - A. F.Ivanov,On the asymptotic behavior of solutions of a class of first order differential equations with deviating argument of mixed type, «Asymptotic Behavior of Functional-Differential Equations», Kiev (1978), pp. 145–150 (Russian).
W. E. Shreve,Oscillation in first order nonlinear retarded argument differential equations, Proc. Amer. Math. Soc.,41 (1973), pp. 565–568.
I. P. Stavroulakis,Nonlinear delay differential inequalities, Nonlinear Anal.,6 (1982), pp. 389–396.
A. Tomaras,Oscillations of an equation relevant to an industrial problem, Bull. Austral. Math. Soc.,12 (1975), pp. 425–431.
A. Tomaras,Oscillatory behaviour of an equation arising from an industrial problem, Bull. Austral. Math. Soc.,13 (1975), pp. 255–260.
A. Tomaras,Oscillatory behaviour of first order delay differential equations, Bull. Austral. Math. Soc.,19 (1978), pp. 183–190.
M. I. Tramov,Conditions for oscillatory solutions of first order differential equations with a delayed argument, Izv. Vysš. Učebn. Zaved. Matematika, no. 3 (154) (1975), pp. 92–96 (Russian).
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Fukagai, N., Kusano, T. Oscillation theory of first order functional differential equations with deviating arguments. Annali di Matematica pura ed applicata 136, 95–117 (1984). https://doi.org/10.1007/BF01773379
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DOI: https://doi.org/10.1007/BF01773379