This paper studies the following third order neutral delay discrete equation $\mathrm{\Delta}({a}_{n}{\mathrm{\Delta}}^{2}({x}_{n}+{p}_{n}{x}_{n-\tau}))+f(n,{x}_{n-{d}_{1n}},\dots ,{x}_{n-{d}_{ln}})={g}_{n},n\ge {n}_{0}$, where $\tau ,l\in \mathbb{N}$, ${n}_{0}\in \mathbb{N}\cup \{0\}$, $\{{a}_{n}{\}}_{n\in {\mathbb{N}}_{{n}_{0}}}$, $\{{p}_{n}{\}}_{n\in {\mathbb{N}}_{{n}_{0}}}$, $\{{g}_{n}{\}}_{n\in {\mathbb{N}}_{{n}_{0}}}$ are real sequences with ${a}_{n}\ne 0$ for $n\ge {n}_{0}$, $\{{d}_{in}{\}}_{n\in {\mathbb{N}}_{{n}_{0}}}\subseteq \mathbb{Z}$ with ${\mathrm{lim}\hspace{0.17em}}_{n\to \infty}(n-{d}_{in})=+\infty $ for $i\in \{\mathrm{1,2},\dots ,l\}$ and $f\in C({\mathbb{N}}_{{n}_{0}}\times {\mathbb{R}}^{l},\mathbb{R})$. By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

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