Two-parametric conditionally oscillatory half-linear differential equations. (English) Zbl 1217.34054

Došlý, Ondřej; Fišnarová, Simona

Abstr. Appl. Anal. 2011, Article ID 182827, 16 p. (2011).

Summary: We study perturbations of the nonoscillatory half-linear differential equation
\[ (r(t)\Phi(x'))'+c(t)\Phi(x)=0, \]
\(\Phi(x):=|x|^{p-2}x,\) \(p>1\). We find explicit formulas for the functions \(\widehat r\), \(\widehat c\) such that the equation
\[ [(r(t)+\lambda\widehat r(t)) \Phi(x')]'+[c(t)+\mu\widehat c(t)] \Phi (x)=0 \]
is conditionally oscillatory, that is, there exists a constant \(\gamma\) such that the previous equation is oscillatory if \(\mu-\lambda>\gamma\) and nonoscillatory if \(\mu-\lambda <\gamma\). The obtained results extend previous results concerning two-parametric perturbations of the half-linear Euler differential equation.

MSC:

34C10

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

Keywords:

nonoscillatory half-linear differential equation; two-parametric perturbations; Euler differential equation

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