Bounded Positive Solutions for a Third Order Discrete Equation

Abstract

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 ${\lim \hspace{0.17em}}_{n\to \infty }(n-d_{in})=+\infty$ for $i\in \{\mathrm{1,2},\dots ,l\}$ and $f\in C({\mathbb{N}}_{n_0}\times {ℝ}^l,ℝ)$. 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.