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On monotonic solutions of systems of nonlinear second order differential equations. (English) Zbl 1181.34047

The author discusses the bounded, monotone and asymptotic behavior of solutions for the second order nonlinear differential equation of the form
\[ p(t)f (x'(t))'=q(t)g(x(t)),\quad t\geq a,\tag{*} \]
where the functions \(p,q:[a,\infty)\to\mathbb{R}\) and \(f,g:\mathbb{R}\to\mathbb{R}\) are continuous, \(p(t)> 0\), and \(q(t)\geq 0\) is not eventually vanishing on \([a,\infty),rf(r)>0\) and \(rg(r)>0\) for \(r\neq 0\) and \(f\) is increasing on \(\mathbb{R}\).
The author establishes some necessary and sufficient conditions for boundedness of all solutions of system (*), and proves that the asymptotic properties of the solutions of (*) are characterized by means of the convergence or divergence of some of the following integrals
\[ \int^\infty_0 f^{-1}\left( \pm\frac{1}{p(t)}\int^t_aq(s)\,ds\right)\,dt, \]
\[ \int^\infty_0f^{-1} \left(\pm\frac{1}{p(t)}\int^\infty_tq(s)\,ds\right)\,dt, \]
and
\[ \int^\infty_0f^{-1} \left(\frac{1}{p(t)}\right)dt. \]
The obtained results extend and improve many known results of various authors.

MSC:

34C11 Growth and boundedness of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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