Half-linear oscillation criteria: Perturbation in term involving derivative

Half-linear Euler differential equations in the critical case

“…We suppose that the perturbation termsr,c are continuous functions such that r(t) +r(t) > 0 for large t. Let x be a solution of (2) such that x(t) = 0, then w = rΦ…”

“…It is proved in [2] that (6) is oscillatory if µ − λγ p > µ p := nonoscillatory if µ − λγ p < µ p . It was conjectured in [2] that (6) is nonoscillatory also in the limiting case µ − λγ p = µ p .…”

“…We also discuss some general aspects of the problem. Next we derive the modified Riccati equation in a more general setting than in [2]. Let h(t) = 0 be a positive differentiable function, denote…”

“…We will also need the following statement which is a slight modification of [2, Theorem 3] (here we do not require that the function h is a solution of (2)). So we omit its proof since it is the same as that of [2,Theorem 3] and it based on the fact that solvability of (10) implies nonoscillation of (4).…”

Solutions of Riemann–Weber type half-linear differential equation

“…More precisely, we consider equation (3) which is supposed to be nonoscillatory with a solution h satisfying h (t) = 0 for large t, say t ≥ T . A prototype is the Euler equation (4) and its solution h(t) = t p−1 p .…”