# Oscillation Criteria for Delay and Advanced Difference Equations with Variable Arguments

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 47)

## Abstract

Consider the first-order delay difference equation
$$\displaystyle{\Delta x(n) + p(n)x(\tau (n)) = 0\text{, }n \geq 0,}$$
and the first-order advanced difference equation
$$\displaystyle{\nabla x(n) - p(n)x(\mu (n)) = 0\text{,}n \geq 1\text{,}[\Delta x(n) - p(n)x(\nu (n)) = 0\text{,}n \geq 0],}$$
where $$\Delta$$ denotes the forward difference operator $$\ \Delta x(n) = x(n + 1) - x(n)$$, ∇ denotes the backward difference operator $$\nabla x(n) = x(n) - x(n - 1)$$, $$\left \{p(n)\right \}$$ is a sequence of nonnegative real numbers, $$\left \{\tau (n)\right \}$$ is a sequence of positive integers such that τ(n) ≤ n − 1, for all n ≥ 0, and $$\left \{\mu (n)\right \}$$$$\left [\left \{\nu (n)\right \}\right ]$$ is a sequence of positive integers such that
$$\displaystyle{\mu (n) \geq n + 1\text{ for all }n \geq 1\text{,}\left [\nu (n) \geq n + 2\text{ for all }n \geq 0\right ].}$$
The state of the art on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

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