Some sufficient conditions guaranteeing the topological and strong topological equivalence of the two perturbed difference systems of the form \[ x_{n+1}=A_nx_n+f(n,x_n),\tag{1} \]
\[ y_{n+1}=A_ny_n+g(n,y_n)\tag{2} \] are obtained whose linear parts \[ \begin{aligned} & x_{n+1}=A_nx_n,\\ & y_{n+1}=A_ny_n\end{aligned} \] have the generalized exponential dichotomy property. There are two innovation in this work, first of which is considering generalized exponential dichotomy and second is proving the continuity of the map \(x\mapsto H(n,x)\) in details, where the map \(H:\mathbb{Z}\times\mathbb{R}^d\rightarrow\mathbb{R}^d\) satisfies the properties:

(i)

\(u\mapsto H(n,u)\) is an homeomorphism of \(\mathbb{R}^d\) for each fixed \(n\in\mathbb{Z}\);

(ii)

\(H(n,u)-u\) is bounded \(\mathbb{Z}\times\mathbb{R}^d\);

(iii)

If \(x_n\) is a solution (1), then \(H(n,x_n)\) is a solution of (2);

(iv)

\(u\mapsto H^{-1}(n,u)\) has Properties (i)–(iii).

To obtain the stronger results like uniform continuity of the map \(H(n,x)\), the authors overcome the technical difficulties by imposing some conditions on the nonlinear functions \(f\) and \(g\), one of which is \[ |f(n,x)|\leq F_n\quad\text{and}\quad|g(n,x)|\leq G_n \] for some non-negative sequences \(F_n\) and \(G_n\) satisfying some properties given in the paper with details.