Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations

• Main Authors: Takaŝi Kusano, Jelena Manojlović
• Format: Article
• Language: English
• Published: University of Szeged 2016-08-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: half-linear differential equations; regularly varying solutions; slowly varying solutions; asymptotic behavior of solutions; positive solutions
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4566

Accurate asymptotic formulas for regularly varying solutions of the second order half-linear differential equation \begin{equation*} (|x'|^{\alpha}\textrm{sgn}\; x')' + q(t)|x|^{\alpha}\textrm{sgn}\; x = 0, \end{equation*} will be established explicitly, depending on the rate of decay toward zero of the function \begin{equation*}Q_c(t) = t^{\alpha}\int_t^{\infty}q(s)ds - c\end{equation*} as $t\to\infty$, where $c<\alpha^\alpha(\alpha+1)^{-\alpha-1}$.