A general approach to nonautonomous shadowing for nonlinear dynamics

Given a nonautonomous and nonlinear differential equation(0.1)x′=A(t)x+f(t,x)t≥0, on an arbitrary Banach space X, we formulate very general conditions for the associated linear equation x′=A(t)x term f:[0,+∞)×X→X under which above system satisfies appropriate version of shadowing property. More precisely, require that admits type dichotomy, includes classical hyperbolic behavior as particular case. In addition, f is Lipschitz in second variable with sufficiently small constant. Our framework enables us to treat various settings no result has been previously obtained. Moreover, are able recover refine several known results. We also show how our main results can be applied study property higher order equations. Finally, conclude paper by presenting discrete time versions