  We prove that whenever a sequence of invertible and bounded operators
$(A_m)_{m\in \mathbb{Z}}$ acting on a Banach space $X$ admits an exponential
dichotomy and a sequence of differentiable maps $f_m \colon X\to X$, $m\in
\mathbb{Z}$, has bounded and H\"{o}lder derivatives, the nonautonomous dynamics
given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \mathbb{Z}$ has various shadowing
properties. Hence, we extend recent results of Bernardes Jr. et al. in several
directions. As a nontrivial application of our results, we give a new proof of
the nonautonomous Grobman-Hartman theorem.
