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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