Shadowing for Nonautonomous Dynamics

We prove that whenever a sequence of bounded operators (Am)m∈ℤ{(A_{m})_{m\in\mathbb{Z}}} acting on a Banach space X admits an exponential dichotomy and a sequence of differentiable maps fm:X→X{f_{m}\colon X\to X}, m∈ℤ{m\in\mathbb{Z}}, has bounded and Hölder derivatives, the nonautonomous dynamics given by xm+1=Am⁢xm+fm⁢(xm){x_{m+1}=A_{m}x_{m}+f_{m}(x_{m})}, m∈ℤ{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.