New oscillation criteria for delay difference equations

This paper is concerned with the oscillation of all solutions of the delay difference equations x(n+1) - x(n) + p(n)x(n-k) = 0, n = 0, 1, 2.... (*) and x(n+1) - x(n) + Sigma (m)(i=1) pi(n)x(n-ki) = 0, n = 0, 1, 2,..., (**) where {p(n)} and {p(i)(n)} are sequences of nonnegative real numbers and k and k(i) are positive integers. New oscillation criteria of the forms lim(n --> infinity) sup p(n) > alpha + C(alpha) for equation (*) and lim(n --> infinity) sup Sigma (m)(i=1) Sigma (n+ki)(s=n) p(i) (s) > 1 for equation (**) are obtained, where alpha = lim inf(n --> infinity) (1/k) Sigma (n-1)(i=n-k) p(i) and C(alpha) is "the best possible" function of a in some sense. (C) 2001 Elsevier Science Ltd. All rights reserved.