Comparison theorems for noncanonical third order nonlinear differential equations.

*(English)*Zbl 1128.34021The third-order nonlinear differential equations that have quasiderivatives of the form
\[
\left(\frac{1}{p(t)}\left(\frac{1}{r(t)} x'(t)\right)'\right)'+q(t) f(x(t))=0,\;t\geq 0\tag{N}
\]
are considered. Investigations are based on a study of the asymptotic behavior of nonoscillatory solutions of equation (N) as well as a linearization device. Some established results for linear equation (N) (the case \(f(x(t))=x(t))\) are summarized. Comparison theorems on property A (that means: any proper solution \(x\) of (N) is either oscillatory or satisfies the condition \(|x^{[i]}(t)|\to 0\) as \(t\to\infty,\;\tau=0,1,2)\), between the linear and nonlinear equations are proved. As a result, sufficient conditions that ensure property A for equation (N) are obtained. Several examples illustrating the main theorems are also provided. The assumptions on the nonlinearity \(f\) are restricted to its behavior only in a neighborhood of zero and a neighborhood of infinity.

Reviewer: Vasile Marinca (Timişoara)

##### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

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\textit{I. Mojsej} and \textit{J. Ohriska}, Cent. Eur. J. Math. 5, No. 1, 154--163 (2007; Zbl 1128.34021)

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