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The authors consider the third order neutral delay difference equation with positive and negative coefficients ∆(an∆(bn∆(xn + pxn−m))) + pn f (xn−k) − qn1(xn−l) = 0, n ≥ n0, and give some new sufficient conditions for the existence of nonoscillatory solutions. Banach’s fixed point theorem plays a major role in the proofs. Examples are provided to illustrate their main results.

Existence of nonoscillatory solutions to third order nonlinear neutral difference equations