An Oscillation Criterion for First Order Difference Equations

Abstract

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.