QUALITATIVE ANALYSIS ON EXISTENCE OF FIRST ORDER NEUTRAL DELAY DIFFERENCE EQUATIONS

In this paper, some oscillatory criteria for the solutions of first order neutral delay difference equation with negative coefficient are obtained. Here, I consider the neutral delay difference equation of the form, ∆(xn+rnxn-k)-qng(xn-m) =0, (1.1) Where, nεN(n0), n0 is a nonnegative integer and ∆ is the forward difference operator.{qn} is the positive sequences and {rn} is a real sequence, g is the continuous functions such that ug(u)≠0, for u≠0 . Also k, m are positive integers. Here, ∆ is the forward difference operator defined by, ∆yn= yn+1-yn. By a solution of (1.1), we mean a real sequence {xn} which satisfies the equation(1.1) for all nεN0. A solution {xn} 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 non-oscillatory. Equation (1.1) is said to be oscillatory, if all its solutions are oscillatory. Following this trend, in this paper, some sufficient conditions for oscillation of all solutions of equation (1.1) is obtained.