ELBERT, Á A., half-linear second order differential equation, Qualitative theory of differential equations (Szeged, 1979), Colloq. Math. Soc. János Bolyai , 30 , North-Holland, Amsterdam - New York, 1981, 153
is conditionally oscillatory. Then we study oscillations of the perturbed equation (**). The obtained (non)oscillation criteria
extend existing results for perturbed half-linear Euler and Euler-Weber equations.
We establish new comparison theorems on the oscillation of solutions of a class of perturbed half-linear differential equations.
These improve the work of Elbert and Schneider  in which connections are found between half-linear differential equations
and linear differential equations. Our comparison theorems are not of Sturm type or Hille--Wintner type which are very famous.
We can apply the main results in combination with Sturm's or Hille--Wintner's comparison theorem to a half-linear differential
equation of the general form (|x'|α-1x')' + a(t) |x|α-1x = 0.
Several comparison theorems with respect to powers in nonlinearities for half-linear differential equations are presented.
The Riccati transformation and the reciprocity principle are utilized. Some examples and an integral extension of the classical
comparison result are presented as well.
Here ϕp(z):= |z|p−2z is the so-called one-dimensional p-Laplacian operator. Our main purpose is to establish new criteria for all nontrivial solutions to be oscillatory and for
those to be nonoscillatory. In our theorems, the parametric curve given by (a(t), b(t)) plays a critical role in judging whether all solutions are oscillatory or nonoscillatory. This paper takes a di-erent approach
from most of the previous research. Our results are new even in the linear case (p = 2). The method used here is mainly phase plane analysis for a system equivalent to the half-linear differential equation.
Some suitable examples are included to illustrate the main results. Global phase portraits are also attached.