Asymptotics of some classes of nonoscillatory solutions of second-order half-linear differential equations. (English) Zbl 1062.34035

Takaŝi, K.; Marić, V.; Tanigawa, T.

Bull., Cl. Sci. Math. Nat., Sci. Math. 127, No. 28, 61-74 (2003).

Let \(u^{\gamma*}\) stand for \(| u| ^{\gamma-1}\operatorname{sgn}u\), for \(u\in\mathbb{R}\) and \(\gamma>0\). By using Banach’s fixed-point theorem, the authors prove that under some conditions the following half-linear differential equation \[ (| y'| ^{\alpha-1}y')'+q(t)| y| ^{\alpha-1}y=0,\quad t\geq0, \] has a nonoscillatory solution of the form \[ y(t)=\exp\left(\int_{t_0}^t(v(s)+Q(s))^{1/\alpha*}ds\right),\quad t\geq t_0, \] where \(v(t)\) is a solution of the integral equation \[ v(t)=\alpha\int_t^\infty(v(s)+Q(s))^{1+1/\alpha}ds,\quad t\geq t_0, \] and \(Q(t)=\int_t^\infty q(s)ds\).
They also find an asymptotic formula for the solution above. Some applications of the results are provided.

Reviewer: Stevo Stević (Beograd)

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations