On monotonic solutions of systems of nonlinear second order differential equations
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.
scientific article; zbMATH DE number 5532902