Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations
Some results about the asymptotic behavior of the non-oscillatory solutions of the half-linear equations of the form \[ (p(t)|x'|^{\alpha-1}x')'+q(t)|x|^{\alpha-1}x=0,\qquad t\geq t_0,\tag{1.1} \] are presented, where \(\alpha>0\) and \(p,q\in C([t_0,\infty),{\mathbb R})\) with \(p(t)>0\) for \(t\geq t_0\). Since the results given are related to the non-oscillatory solutions, it is assumed that the solution \(x(t)\) of (1.1) satisfies \(x(t)>0\) for \(t\geq T\) (\(\geq t_0\)) throughout the paper. In the classical sense, \textit{J. Jaroš} and \textit{T. Kusano} [Bull., Cl. Sci. Math. Nat., Sci. Math. 129, No. 29, 25--60 (2004; Zbl 1091.34521)] generalized that the asymptotic behavior of a non-oscillatory solution of (1.1) is strongly determined by the condition on \(p(t)\), i.e., \[ \lim_{t\to\infty}P(t)=\infty\quad\mbox{or}\quad\lim_{t\to\infty}P(t)<\infty, \] where \[ P(t)=\int_{t_0}^tp^{-1/\alpha}(s)\,\mathrm{d}s. \] In this paper, it is shown that, for any solution \(x(t)\) of Eq. (1.1), there are only two types of asymptotic behavior as \(t\to\infty\) which are \[ \lim_{t\to\infty}\frac{p^{1/\alpha}(t)x'(t)}{x(t)}P(t)=|\rho_1|^{1/\alpha-1}\rho_1 \] or \[ \lim_{t\to\infty}\frac{p^{1/\alpha}(t)x'(t)}{x(t)}P(t)=|\rho_2|^{1/\alpha-1}\rho_2, \] where \(\rho_1,\rho_2\) (\(\rho_1<\rho_2\)) are the real roots of the equation \[ |\rho|^{1+1/\alpha}-\rho+c=0;\qquad c<\frac{\alpha^\alpha}{(\alpha+1)^{\alpha+1}}. \]
scientific article; zbMATH DE number 7343935