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

Lalli, B. S.; Grace, S. R.

Arch. Math., Brno 23, 15-22 (1987).

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

Citations:

Zbl 0537.34068