Solvability and Mann iterative approximations for a higher order nonlinear neutral delay differential equation

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.