Necessary and sufficient conditions for oscillations of partial difference equations with continuous variables

This paper is concerned with the delay partial difference equation A(χ + 1, y) + A(χ, y +) − A(χ,y) + pA(χ − γ, y − τ) = 0, where p is a real number, σ and τ are nonnegative real numbers. Sufficient and necessary conditions for all continuous solutions of this equation to be oscillatory are obtained. Explicit condition for oscillation in terms of p, σ, and τ is also given.