Oscillation of linear difference equations with deviating arguments

The following difference equation with deviating arguments: Delta (2)u(k) + Sigma (m)(j=1) p(j)(k)u(sigma (j)(k)) = 0, is considered, where Deltau(k) = u(k + 1) - u(k), Delta (2) = Delta circle Delta, p(j) (j = 1,..., m) is a sequence of nonnegative numbers, sigma (j) : N --> N and lim(k -->+infinity) sigma (j)(k) = +infinity (j = 1,..., m). In the paper, sufficient conditions are established for all proper solutions of the above equation to be oscillatory. (C) 2001 Elsevier Science Ltd. All rights reserved.