Half-linear Euler differential equations in the critical case. (English) Zbl 1265.34126

Došlý, Ondřej; Haladová, Hana

Tatra Mt. Math. Publ. 48, 41-49 (2011).

The authors study oscillatory properties of the perturbed half-linear differential equation \[ \Bigl (r(t)\Phi (x')\Bigr)^{\hskip -2pt{\prime }}+c(t)\Phi (x)=0\,, \] where \(\Phi (x)=| x| ^{p-2}x\), \(p>1\), \(r\), \(c\) are continuous functions and \(r(t)>0\). The perturbation is allowed in both the coefficients \(r\) and \(c\). The authors extend the results of O.Došlý and S. Fišnarová [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 12, 3756–3766 (2010; Zbl 1207.34041)] and show that the perturbed half-linear Euler equation \[ \left [\left (1+\frac {\lambda }{\log ^2 t}\right)\Phi (x')\right ]' +\left [\frac {\gamma _p}{t^p}+\frac {\mu }{t^p\log ^2 t}\right ]\Phi (x) =0\,, \qquad \gamma _p:=\left (\frac {p-1}{p}\right)^p, \] considered in the limiting case \(\mu -\lambda \gamma _p=\frac {1}{2}\bigl (\frac {p-1}{p}\bigr)^{p-1}\), is nonoscillatory.

Reviewer: Simona Fišnarová (Brno)

MSC:

34C10

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

Keywords:

half-linear differential equation; asymptotic formula; Euler equation; oscillation

Citations:

Zbl 1207.34041