Date: 29 Jul 2013

Oscillation Criteria for Delay and Advanced Difference Equations with Variable Arguments

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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.