A sharp oscillation criterion for a difference equation with constant delay

It is known that all solutions of the difference equation $$\Delta x(n)+p(n)x(n-k)=0, \quad n\geq0, $$ Δ x ( n ) + p ( n ) x ( n − k ) = 0 , n ≥ 0 , where $\{p(n)\}_{n=0}^{\infty}$ { p ( n ) } n = 0 ∞ is a nonnegative sequence of reals and k is a natural number, oscillate if $\liminf_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> ( \frac {k}{k+1} ) ^{k+1}$ lim inf n → ∞ ∑ i = n − k n − 1 p ( i ) > ( k k + 1 ) k + 1 . In the case that $\sum_{i=n-k}^{n-1}p(i)$ ∑ i = n − k n − 1 p ( i ) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with $\limsup_{n\rightarrow\infty}\sum_{i=n-k}^{n-1}p(i)> ( \frac{k}{k+1} ) ^{k+1}$ lim sup n → ∞ ∑ i = n − k n − 1 p ( i ) > ( k k + 1 ) k + 1 . An example illustrating the applicability and importance of the result is presented.