We consider the half-linear second order differential equation which is viewed as a perturbation of the so-called Riemann-Weber half-linear differential equation. We present a comparison theorem with respect to the power of the half-linearity in the equation under consideration. Our research is motivated by the recent results published by J. Sugie, N. Yamaoka, Acta Math. Hungar. 111 (2006), 165–179.
In this paper we investigate oscillatory properties of the second order half-linear equation \[ (r(t)\Phi (y^{\prime }))^{\prime }+c(t)\Phi (y)=0, \quad \Phi (s):= |s|^{p-2}s. \qquad \mathrm{{(*)}}\] Using the Riccati technique, the variational method and the reciprocity principle we establish new oscillation and nonoscillation criteria for (*). We also offer alternative methods of proofs of some recent oscillation results.
Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation \[ (-1)^n(t^{\alpha }y^{(n)})^{(n)}+q(t)y=0 \qquad \mathrm{(*)}\] are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation \[ (-1)^n(t^{\alpha }y^{(n)})^{(n)}- \frac{\mu _{n,\alpha }}{t^{2n-\alpha }}y=0, \] where $\mu _{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.
Some recent results concerning properties of solutions of the half-linear second order differential equation (∗) (r(t)Φ(x' ))' + c(t)Φ(x)=0, Φ(x) := |x| p−2x, p > 1, are presented. A particular attention is paid to the oscillation theory of (∗). Related problems are also discussed.
The Picone-type identity for the half-linear second order partial differential equation n∑ i=1 ∂ ⁄ ∂xi Φ (∂u ⁄ ∂xi) + c(x)Φ(u) = 0, Φ(u) := |u| p−2 u, p > 1, is established and some applications of this identity are suggested.