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We investigate the criticality of the one term $2n$-order difference operators $l{\left(y\right)}_k={\Delta }^n\left(r_k{\Delta }^n{y}_k\right)$. We explicitly determine the recessive and the dominant system of solutions of the equation $l{\left(y\right)}_k=0$. Using their structure we prove a criticality criterion.

We show that the half-linear differential equation $${\left[r\left(t\right)\Phi \left(x^{\text{'}}\right)\right]}^{\text{'}}+\frac{s\left(t\right)}{t^p}\Phi \left(x\right)=0*$$ with $\alpha$-periodic positive functions $r,s$ is conditionally oscillatory, i.e., there exists a constant $K>0$ such that () with $\frac{\gamma s\left(t\right)}{t^p}$ instead of $\frac{s\left(t\right)}{t^p}$ is oscillatory for $\gamma >K$ and nonoscillatory for $\gamma <K$.