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This paper investigates the second order nonlinear neutral delay difference equation Δ[anΔ(xn+bxn−τ−dn)]+Δf(n,xf1(n),xf2(n),…,xfk(n))+g(n,xg1(n),xg2(n),…,xgk(n))=cn,n≥n0. $$\begin{aligned}& \Delta \bigl[a_{n}\Delta (x_{n}+bx_{n-\tau}-d_{n} ) \bigr] +\Delta f (n,x_{f_{1}(n)},x_{f_{2}(n)},\ldots,x_{f_{k}(n)} ) \\& \quad{} +g (n,x_{g_{1}(n)},x_{g_{2}(n)},\ldots,x_{g_{k}(n)} )=c_{n},\quad n\ge n_{0}. \end{aligned}$$ By using the Banach fixed point theorem and some new techniques, we establish the existence results of uncountably many bounded nonoscillatory solutions for the above equation, propose a few Mann type iterative approximation schemes with errors and obtain several errors estimates between the iterative approximations and the nonoscillatory solutions. Examples that cannot be solved by known results are given to illustrate our theorems.

Existence and iterative approximations of nonoscillatory solutions for second order nonlinear neutral delay difference equations