Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions
In this article, the authors consider the following second-order half-linear differential equation: \[(p(t)|x'|^\alpha \mathrm{sgn} x')'+q(t)|x|^\alpha \mathrm{sgn} x=0,\tag{1}\] where \(\alpha>0, a>0\), \(q\colon[a;+\infty)\rightarrow \mathbb R\) is a continuous function which may take both positive and negative values in any neighborhood of infinity, and \(p\colon[a;+\infty)\rightarrow(0;+\infty)\) is a continuous function satisfying one of the conditions \[ \int\limits_a^{\infty}\frac{ds}{p(s)^{\frac{1}{\alpha}}}=\infty, \qquad \int\limits_a^{\infty}\frac{ds}{p(s)^{\frac{1}{\alpha}}}<\infty. \tag{2}\] The authors, referring to previous works, argue that equation (1) is nonoscillatory if and only if the generalized Riccati differential equation \[u'=\frac{\alpha}{p(t)^{\frac{1}{\alpha}}}|u|^{1+\frac{1}{\alpha}}+q(t)=0 \tag{3}\] has a solution defined in some neighborhood of infinity. For both cases (2), there were obtained conditions for the existence of so called generalized regularly varying solutions of the equation (1) and its asymptotic representations. Examples of specific equations are given for which the conditions of the theorems are verified and the corresponding asymptotics is written also. In my opinion, the results presented in the article are very important and actual.
scientific article; zbMATH DE number 7379142