Oscillations of nonlinear differential equations caused by deviating arguments. (English) Zbl 0537.34068

In four theorems the author describes the oscillatory behavior of the nonlinear differential equation with deviating arguments of the form \[ x^{(n)}(t)=f(t,x(g_ 1(t)),...,x(g_ m(t))), \] where f and \(g_ k\) are continuous functions on \([0,\infty)\), and \(\lim_{t\to \infty}g_ k(t)=\infty\). Only solutions defined for all large t are considered. Nonlinear oscillations studied in the paper are generated by deviating arguments, and the corresponding assertions are generally not valid for ordinary differential equations. For the first order differential equations with delays related results were obtained by Y. Kitamura and T. Kusano [Proc. Am. Math. Soc. 78, 64-68 (1980; Zbl 0433.34051)] and V. N. Shevelo and A. F. Ivanov [Asymptotic behavior of solutions of functional-differential equations, Kiev 1978, 143-150 (1978; Zbl 0412.34070)].
Reviewer: F.Neuman


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems