Oscillation of Higher-order Delay Difference Equations

where {pi(n)} are sequences of nonnegative real numbers and not identically equal to zero, and ki is positive integer, i = 1,2, . . . , and is the first-order forward difference operator, xn = xn+1− xn, and xn = l−1( xn) for l ≥ 2. By a solution of (1.1) or inequality (1.2), we mean a nontrival real sequence {xn} satisfying (1.1) or inequality (1.2) for n ≥ 0. A solution {xn} is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise. An equation is said to be oscillatory if its every solution is oscillatory. The oscillatory behavior of difference equations has been intensively studied in recent years. Most of the literature has been concerned with equations of type (1.1) with l = 1 (see [1–10] and references cited therein). But very little is known regarding the oscillation of higher-order difference equation similar to (1.1). The purpose of this paper is to study the oscillatory properties of (1.1).