Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification for spaces, as well as the Freudenthal-Morita Theorem for spaces, can be obtained from our frame constructions.
In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation \[ (r(t)\Phi (u^{\prime }(t)))^{\prime }+c(t)\Phi (u(t))=0, \] where (i) $r,c\in C([t_{0}, \infty )$, $\mathbb{R}:=(-\infty , \infty ))$ and $r(t)>0$ on $[t_{0},\infty )$ for some $t_{0}\ge 0$; (ii) $\Phi (u)=|u|^{p-2}u$ for some fixed number $p> 1$. We also generalize some results of Hille-Wintner, Leighton and Willet.