Sharp conditions for the oscillation of delay difference equations

Suppose that {pn} is a nonnegative sequence of real numbers and let k be a positive integer. We prove that liminf−rân⇒•â[k’-¹Σ−rn-1âi=n-kâpi]>k^k/(k+1)^k’+¹ is a sufficient condition for the oscillation of all solutions of the delay difference equation AnÅ+1-An+pnAnÅ-k=0, n=0,1,2,.... This result is sharp in that the lower bound k^k/(k+1)^k’+¹ in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.