## Oscillatory behaviour of nonlinear differential equations with deviating arguments.(English)Zbl 0639.34068

Some oscillation criteria for $$L_ nx(t)=f(t,x[g_ 1(t)],...,x[g_ m(t)])$$, $$n\geq 2$$ are established. Here $$L_ 0x(t)$$, $$L_ kx(t)=a_ k(t)(L_{k-1}x(t))'$$, $$('=d/dt)$$, $$k=1,2,...,n$$, $$a_ 0=a_ n=1$$. The results generalize those of J. Werbowski [Funkcial Ekvac, Ser. Int. 25, 295-301 (1982; Zbl 0537.34068)]. However, they are not valid for the corresponding ordinary differential equations, which is due to the fact that deviations $$g_ i$$ can destroy oscillations, and also can generate oscillations depending on the “size” of the deviations.

### MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Keywords:

deviating arguments; nonoscillatory solutions

Zbl 0537.34068
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