The authors investigate oscillation/nonoscillation properties of the perturbed half-linear Euler differential equation
where the function is piecewise continuous on , , is a fixed real number and . The number is a critical constant, which means that in case the half-linear Euler differential equation is for nonoscillatory while for is oscillatory. Using a Riccati technique and a transformation of the independent variable, there are proved the following main results:
Suppose that there exists the finite limit
such that for .
(a) If and the linear equation
is nonoscillatory then equation (*) is also nonoscillatory.
(b) If and equation (*) is nonoscillatory then the linear equation (**) is also nonoscillatory.
In addition, the authors establish an asymptotic form of the solution to (*) provided that the solutions to (**) satisfy two integral inequalities.