  Given a nonautonomous and nonlinear differential equation
\begin{equation}\label{DE} x'=A(t)x+f(t,x) \quad t\geq 0, \end{equation} on an
arbitrary Banach space $X$, we formulate very general conditions for the
associated linear equation $x'=A(t)x$ and for the nonlinear term
$f:[0,+\infty)\times X\to X$ under which the above system satisfies an
appropriate version of the shadowing property. More precisely, we require that
$x'=A(t)x$ admits a very general type of dichotomy, which includes the
classical hyperbolic behaviour as a very particular case. In addition, we
require that $f$ is Lipschitz in the second variable with a sufficiently small
Lipschitz constant. Our general framework enables us to treat various settings
in which no shadowing result has been previously obtained. Moreover, we are
able to recover and refine several known results.
  We also show how our main results can be applied to the study of the
shadowing property for higher order differential equations. Finally, we
conclude the paper by presenting a discrete time versions of our results.
