Summary: Consider the following pair of even order linear dynamic equations on a time scale \[ x^{\Delta^n}(t)+ p(t) x(t)= 0,\tag{1} \]
\[ x^{\Delta^n}(t)+ q(t)x(t)= 0,\tag{2} \] where \(p,q\in C_{rd}(\mathbb{T},\mathbb{R}^+)\), \(n\) is even, \(\mathbb{T}\) is a time scale. In this paper, we obtain some pointwise and integral comparison theorems for the above equations. These will be shown to be “sharp” by means of specific examples.
Theorem 1. Suppose that \(p(t)\geq q(t)\) for all large \(t\). Then if the equation (2) is oscillatory, it follows that the equation (1) is oscillatory.
Theorem 2. Suppose that \(\int^\infty_t p(s)\Delta s\geq \int^\infty_t q(s)\Delta s\) for all large \(t\). Then if the equation (2) is oscillatory, it follows that the equation (1) is oscillatory.
As applications, we get that
Theorem 3. Let \(n\) be even. If \[ \liminf_{k\to\infty}\, k^{n-1} \sum^\infty_{i=k} p(i)> {|M_{n0}|\over n-1},\tag{3} \] then the difference equation \(\Delta^n x(k)+ p(k) x(k)= 0\) is oscillatory, where \(M_{n0}\) is the minimum of \(P_n(\lambda)= \lambda(\lambda- 1)\cdots(\lambda- n+1)\), \(\lambda\in [0,1]\).
In particular, we get that the fourth-orer difference equation \(\Delta^4 x(k)+ p(k)x(k)= 0\) is oscillator if \(\liminf_{k\to\infty} k^4p(k)> 1\).