Oscillation criteria for a difference equation (Q1386816)

Consider the difference inequality \[ y(t)- y(t-\tau)+ p\cdot y(t- \sigma)\leq 0,\tag{i} \] where \(\tau\), \(\sigma\), \(p\in\mathbb{R}^+\) and \(\sigma>\tau\). The inequality (i) has an eventually positive solution if and only if the difference equation \[ y(t)- y(t-\tau)+ p\cdot y(t-\sigma)= 0\tag{ii} \] has an eventually positive solution. If \[ \liminf_{t\to\infty} \Biggl[\limsup_{t\to\infty}\Biggr]p(t)> [<]\;{(\sigma/\tau- 1)^{\sigma/\tau- 1}\over (\sigma/\tau)^{\sigma/\tau}} \] then all solutions of (ii) are oscillatory [(ii) has a nonoscillatory solution] (\(p:=p(t)\in C(\mathbb{R}^+,\mathbb{R}^+)\)).