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On asymptotic behaviour of solutions of a first order functional differential equation. (English) Zbl 0615.34071

Necessary and sufficient conditions for the oscillation of solutions to the first order functional differential equation \[ (1)\quad (y'(t)+\gamma f(t,y(t),y(\Delta_ 1(t,y(t)),...,y(\Delta_ n(t,y(t))=Q(t),\quad t\geq t_ 0\in R,\quad \gamma =\pm 1,\quad n\geq 1 \] are obtained in the case Q(t)\(\equiv 0\) on \([t_ 0,\infty)\) and sufficient conditions for oscillation and/or nonoscillation are obtained in the case when Q(t)\(\not\equiv 0\) on \([t_ 0,\infty)\). The functions f, \(\Delta_ i\) and Q are continuous and the conditions (2)-(3) are fulfilled: (2) \(f(t,u_ 0,u_ 1,...,u_ n)>0\) for \(u_ 0u_ i>0\) \((<0)\), (3) \(\Delta_ i(t,\delta)\to \infty\) for \(t\to \infty\), for any fixed \(\delta\in R\), \(\Delta_ i(t,\delta)\leq \Delta_ i(t,{\bar \delta})\) for \(| \delta | \leq | {\bar \delta}|\). The asymptotic behaviour of oscillatory and nonoscillatory solutions of this equation is also studied.
Reviewer: G.Derfel

MSC:

34K25 Asymptotic theory of functional-differential equations
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