New oscillation criteria for first order neutral delay difference equations

In this paper we deal with the oscillatory criteria for all solutions of first order neutral delay difference equation with positive and negative coefficients. Here we consider the neutral delay difference equation of the form, 1.1 where nεN(n₀), n₀ is a nonnegative integer and Δ is the forward difference operator.

Here we assume {P_n}, {q_n} are positive sequences and {r_n} is a real sequence. f and g are continuous functions such that uf(u) ≠ 0, and ug(u) ≠0, for u ≠ 0. Also k,l,m are positive integers. Here Δ is the forward difference operator defined by ΔY_n= Y_n+l-Y_n. By a solution of (1.1) we mean a real sequence {x_n} which satisfies the equation(1.1) for all nεN₀. A solution {x_n} of (1.1) is said to be oscillatory if the terms of the sequence are not eventually positive, or not eventually negative. Otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Following this trend, in this paper we find some sufficient conditions for oscillation of all solutions of equation(1.1).