An Oscillation Criterion for First Order Difference Equations
This paper is concerned with the oscillatory behavior of first order difference equation with general argument

 $$\Delta x(n) + p(n)x\left( \tau (n)\right) = 0,\quad n = 0,1,\ldots \qquad\qquad (\star)$$
 where $${{(p(n))_{n\geq 0}}}$$
 is a sequence of nonnegative real numbers and $${{(\tau (n))_{n\geq 0}}}$$
 is a sequence of integers. Let the numbers k and L be defined by

 $$k = \liminf\limits_{n \rightarrow \infty} \sum \limits_{j=\tau (n)}^{n-1}p(j)$$
 and

 $$L=\limsup\limits_{n\rightarrow \infty} \sum\limits_{j=\tau(n)}^{n}p(j).$$
 It is proved that, when L < 1 and $${{0 < k \leq \frac{1}{e},}}$$
 all solutions of Equation (
 $${{\star}}$$
 ) oscillate if the condition

 $$L > 2k+\frac{2}{{\rm \lambda} _{1}}-1$$
 where $${{{\rm \lambda} _{1}\in \lbrack 1,e]}}$$
 is the unique root of the equation $${{{\rm \lambda} =e^{k\lambda },}}$$
 is satisfied.