On a certain nonautonomous nonlinear third-order differential equation. (English) Zbl 0880.34035

The authors consider third-order nonlinear differential equations of the form \[ x'''+ p_1(t)x''+ p_2(t)g(x)x'+ p_3(t)f(x)= 0,\tag{1} \] where
(i) \(p_i\in C((a,\infty))\), \(i=1,2,3\), for some \(a\in\mathbb{R}\),
(ii) \(g,f\in C(\mathbb{R})\), \(xf(x)>0\) for \(x\neq 0\), and \(\lim_{x\to 0}{f(x)\over x}=\theta\), where \(0<\theta<\infty\),
(iii) there exists a constant \(k\) with \(0<k\leq\theta\) such that \(|f(u)|\geq k|u|\) for all \(u\in\mathbb{R}\).
By an oscillatory solution, the authors mean a nontrivial solution \(x\) of (1) that has arbitrarily large zeros. Otherwise, the solution is said to be nonoscillatory. Sufficient conditions are presented under which any solution \(x_1\) of \[ x'''+ p_1(t)x''+ p_2(t)x'+ p_3(t)f(x)= 0\tag{2} \] defined on \([t_0,\infty)\), \(t_0>a\), and having one (or at least one) zero in \([t_0,\infty)\) is oscillatory.
Moreover, some asymptotic properties of solutions \(x\) of (1) in a neighbourhood of infinity are given. In the proofs given, some results and techniques from the theory of linear differential equations are applied.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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[1] DOI: 10.1112/jlms/s1-37.1.405 · Zbl 0113.07604
[2] DOI: 10.1007/BF02414323 · Zbl 0143.11602
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