Kneser-type oscillation criteria for second-order half-linear delay differential equations

In the paper, we establish a variant of Kneser oscillation theorem for the second-order half-linear delay differential equation(r(t)|y′(t)|α−1y′(t))′+q(t)|y(τ(t))|α−1y(τ(t))=0,t≥t0>0,under the assumption∫t0∞r−1/α(s)ds=∞,which improves a plenty of results reported in the literature. In case of α ≥ 1, the obtained criterion is sharp in the sense that the oscillation constant is optimal for the corresponding half-linear Euler type equation with delay, while the result for α < 1 is slightly worse. Our methodology differs from the previous ones, since it is only based on sequentially improved monotonicities of the nonoscillatory solution.