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Comparison theorems for noncanonical third order nonlinear differential equations. (English) Zbl 1128.34021
The 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.

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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