  A boundary value problem associated to the difference equation with advanced
argument \begin{equation} \label{*}\Delta\bigl (a_{n}\Phi(\Delta
x_{n})\bigr)+b_{n}\Phi(x_{n+p} )=0,\ \ n\geq1 \tag{$*$} \end{equation} is
presented, where $\Phi(u)=|u|^{\alpha}$sgn $u,$ $\alpha>0,p$ is a positive
integer and the sequences $a,b,$ are positive. We deal with a particular type
of decaying solutions of (\ref{*}), that is the so-called intermediate
solutions (see below for the definition) . In particular, we prove the
existence of these type of solutions for (\ref{*}) by reducing it to a suitable
boundary value problem associated to a difference equation without deviating
argument. Our approach is based on a fixed point result for difference
equations, which originates from existing ones stated in the continuous case.
Some examples and suggestions for future researches complete the paper.
