Oscillation Criteria for Higher-Order Nonlinear Delay Dynamic Equations on Time Scales

Abstract

In this paper, oscillation criteria are obtained for higher-order half-linear delay difference equations involving generalized difference operator of the form Delta(b) (p(n) (Delta(m-1)(b) x(n))(alpha)) + q(n)x(n-sigma)(beta) = 0, n >= n(0), where Delta(b) is defined by Delta(b)y(n) = y(n+1) - by(n), b is an element of R - {0}, p: N -> R+, alpha, beta are the ratio of odd positive integers with beta <= alpha; m, n, n(0), sigma are non-negative integers, q : N -> R. The cases of b negative and positive and q(n) >= 0, which has important role for oscillation of this equation, are considered. Also we provide some examples to illustrate our main results. (C) 2016 Mathematical Institute Slovak Academy of Sciences