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This paper deals with the third order nonlinear neutral delay difference equation with a forced term $$\begin{aligned}& \Delta^{2} \bigl(a(n)\Delta \bigl(x(n)+c(n)x(n-\tau) \bigr) \bigr)+f \bigl(n,x \bigl(n-b_{1}(n) \bigr),x \bigl(n-b_{2}(n) \bigr), \ldots,x \bigl(n-b_{k}(n) \bigr) \bigr) \\& \quad =d(n),\quad n\geq n_{0}. \end{aligned}$$ Using the Banach fixed point theorem, we prove the existence of uncountably many bounded positive solutions for the equation, suggest some Mann iterative schemes and obtain the error estimates between these bounded positive solutions and the sequences generated by the iterative schemes. Five nontrivial examples are also included.

This paper investigates the second order nonlinear neutral delay difference equation $$\begin{aligned}& \Delta \bigl[a_{n}\Delta (x_{n}+bx_{n-\tau}-d_{n} ) \bigr] +\Delta f (n,x_{f_{1}(n)},x_{f_{2}(n)},\ldots,x_{f_{k}(n)} ) \\& \quad{} +g (n,x_{g_{1}(n)},x_{g_{2}(n)},\ldots,x_{g_{k}(n)} )=c_{n},\quad n\ge n_{0}. \end{aligned}$$ 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.

In this paper, fixed point theorems for contractive and expansive maps are established, some of which extend a few results of Das and Debata, Edelstein, Fisher, Leader, Shih and Yeh, and Jungck.

The purpose of this paper is to study solvability of the higher order nonlinear neutral delay differential equation $$\begin{aligned}& \frac{d^{n}}{dt^{n}}\bigl[x(t)+c(t)x(t-\tau)\bigr]+(-1)^{n+1}f\bigl(t,x \bigl(\sigma _{1}(t)\bigr),x\bigl(\sigma_{2}(t)\bigr), \ldots,x\bigl(\sigma_{k}(t)\bigr)\bigr) \\ & \quad =g(t),\quad t\geq t_{0}, \end{aligned}$$ where n and k are positive integers, $\tau>0 $ , $t_{0}\in{\mathbb{R}}$ , $f\in C ([t_{0},+\infty)\times {\mathbb{R}}^{k},{\mathbb{R}} ) $ , $c,g,\sigma_{i}\in C([t_{0},+\infty),{\mathbb{R}})$ and $\lim_{t\rightarrow +\infty}\sigma_{i}(t)=+ \infty$ for $i \in\{1,2,\ldots,k\}$ . Under suitable conditions, several existence results of uncountably many nonoscillatory solutions and convergence of Mann iterative approximations for the above equation are shown. Three nontrivial examples are given to demonstrate the advantage of our results over the existing ones in the literature.