### Existence of Nonoscillatory Solutions to Third Order Nonlinear Neutral Difference Equations

#### Abstract

The authors consider the third order neutral delay difference equation with positive and negative coefficients

\begin{equation*}

\Delta(a_n\Delta(b_n\Delta (x_n+px_{n-m})))+p_nf(x_{n-k})-q_ng(x_{n-l})=0,\;n\geq n_0,

\end{equation*}

and give some new sufficient conditions for the existence of nonoscillatory solutions. Banach's fixed point theorem plays a major role in the proofs.

Examples are provided to illustrate their main results.

\begin{equation*}

\Delta(a_n\Delta(b_n\Delta (x_n+px_{n-m})))+p_nf(x_{n-k})-q_ng(x_{n-l})=0,\;n\geq n_0,

\end{equation*}

and give some new sufficient conditions for the existence of nonoscillatory solutions. Banach's fixed point theorem plays a major role in the proofs.

Examples are provided to illustrate their main results.

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