Consider the following system of difference equations in Banach space \(\bf X\times Y\):
$$
\label{1}
\left\{\begin{array}{lclcl}
x(t+1)&=&A(t)x(t)+f(t,x(t),y(t)),\\
y(t+1)&=&B(t)y(t)+g(t,x(t),y(t)),\\
\end{array}
\right.
$$
satisfying the conditions of separation
\[\nu=\max\left(\sup_{t\in\bf Z}\sum_{s=-\infty}^{t-1}|Y(t,s+1)| \, |X(s,t)|, \sup_{t\in\bf Z}\sum^{+\infty}_{s=t}|X(t,s+1)| \, |Y(s,t)|\right)<+\infty\]
and \(f(t,\cdot)\), \(g(t,\cdot)\) are \(\varepsilon\)-Lipshitz, \(f(t,0,0)=0\), \(g(t,0,0)=0\).
We find a simpler system of difference equations that is conjugated and asymptotic equivalent to the
given one. Using this result we obtain sufficient conditions that invertible and noninvertible system (\ref{1}) is asymptotic equivalent to the linear system
\begin{eqnarray}\label{2}
\left\{\begin{array}{lclcl}
x(t+1)&=&A(t)x(t),\\
y(t+1)&=&0,\\
\end{array}
\right.
\end{eqnarray}
in the case when \(\varepsilon\) depends on \(t\) and tends to zero as \(t\to+\infty\) sufficiently rapidly.
This work was partially supported by the grant Nr. 345/2012 of the Latvian Council of Science.
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