×

Necessary and sufficient conditions for the oscillation a third-order differential equation. (English) Zbl 1208.34039

Summary: We show that, under certain restrictions, the following three conditions are equivalent: the equation
\[ y'''+a(t)y''+b(t)y'+c(t)y=f(t) \]
is oscillatory; the equation
\[ x'''+a(t)x''+b(t)x'+c(t)x=0 \]
is oscillatory; the second-order Riccati equation
\[ z''+3zz'+a(t)z'=z^3+a(t)z^2+b(t)z+c(t) \]
does not admit a non-oscillatory solution that is eventually positive.
Furthermore, we obtain sufficient conditions for the above statements to hold, in terms of the coefficients. These conditions are sharp in the sense that they are both necessary and sufficient when the coefficients \(a(t), b(t), c(t)\) are constant.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: EuDML EMIS