Asymptotic behavior of solutions to third-order delay differential equations. (English) Zbl 0916.34059

The authors study asymptotic and oscillatory properties of solutions to third-order differential equations \[ \left (\frac {1}{p(t)}\left (\frac {1}{r(t)} x'(t)\right)'\right)' +q(t)f(x(h(t)))=0 \tag{\(N,h\)} \] and \[ \left (\frac {1}{r(t)}\left (\frac {1}{p(t)} z'(t)\right)'\right)' -q(t)f(z(h(t)))=0,\tag{\(N^A,h\)} \] with \(r,p,q,h\in C^{0}([a,\infty),\mathbb{R}),\;r(t)>0,\;p(t)>0,\^^Mq(t)> 0\), \(f\in C^0(\mathbb{R}, \mathbb{R}),\) \(f(u)u> 0\) for \(u\neq 0\), \hbox {\(h(t)\leq t\)}, \(\lim _{t\to \infty }h(t)=\infty \) and \[ \int _a^\infty r(t) dt=\int _a^\infty p(t) dt=\infty . \] The equation \((N,h)\) is said to have property A if any proper solution \(x\) to \((N,h)\) is either oscillatory or satisfies \[ | x^{[i]}(t)| \downarrow 0\;\;\text{as }t\to \infty ,\;i=0,1,2, \] and \((N^{A},h)\) is said to have property B if any proper solution \(z\) to \((N^{A},h)\) is either oscillatory or satisfies \[ | z^{[i]}(t)| \uparrow \infty \quad\text{as }t\to \infty ,\;i=0,1,2. \] Here, \(x^{[0]}=x\), \(x^{[1]}={x'\over r}\), \(x^{[2]}={1\over p}\left ( {x'\over r}\right)\), the quasiderivatives \(z^{[i]}\) are defined in a similar way.
In the first part of the paper, two comparison theorems on property A and property B in case \(f(x)\equiv x\) (i.e., for linear equations) are given, and in the second part, the authors study relations (concerning properties A, B) between equations (\(N,h\)), (\(N^A,h\)) and corresponding linear equations.
Reviewer: O.Došlý (Brno)


34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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