Oscillations of advanced difference equations with variable arguments

Abstract

Consider the first-order advanced difference equation of the form \begin{equation*} \nabla x(n)-p(n)x(\mu (n))=0\text{, }\ n\geq 1\, [\Delta x(n)-p(n)x(\nu (n))=0, n\geq 0], \end{equation*} where $\nabla $ denotes the backward difference operator $\nabla x(n)=x(n)-x(n-1)$, $\Delta $ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$, $\left\{ p(n)\right\} $ is a sequence of nonnegative real numbers, and $\left\{ \mu (n)\right\} $ $\ \left[ \left\{ \nu (n)\right\} \right] $ is a sequence of positive integers such that \begin{equation*} \mu (n)\geq n+1\ \text{ for all }n\geq 1\, \left[ \nu (n)\geq n+2 \ \text{ for all }n\geq 0\right] \text{.} \end{equation*} Sufficient conditions which guarantee that all solutions oscillate are established. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.

Keywords

• advanced difference equation
• variable argument
• oscillatory solution
• nonoscillatory solution