Comparison theorems for forced functional differential equations. (English) Zbl 0683.34040
Summary: We establish comparison results by which oscillating and asymptotic properties of solutions of forced differential equations with deviating arguments of the form
\[
(*)\quad \frac{d}{dt}\frac{1}{a_{n- 1}(t)}\frac{d}{dt}\cdot \cdot \cdot \frac{d}{dt}\frac{1}{a_ 1(t)}\frac{d}{dt}x(t)+H(t,x[g(t)])=Q(t)
\]
are inherited from the same properties of solutions of the unforced ordinary equation
\[
(**)\quad \frac{d}{dt}\frac{1}{a_{n-1}(t)}\frac{d}{dt}\cdot \cdot \cdot \frac{d}{dt}\frac{1}{a_ 1(t)}\frac{d}{dt}x(t)+H(t,x(t))=0.
\]
The obtained results extend the oscillation criteria of Trench and Kusano and Naito to the more general equation (*).
MSC:
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
References:
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