Given a nonautonomous and nonlinear differential equation(0.1)$x^{\prime }=A(t)x+f(t,x)t\ge 0,$ on an arbitrary Banach space X, we formulate very general conditions for the associated linear equation $x^{\prime }=A(t)x$ 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 $x^{\prime }=A(t)x$ admits a very general type of dichotomy, which includes the classical hyperbolic behavior 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 discrete time versions of our results.

给定一个非自治和非线性微分方程（0.1）$X^{\prime }=\mathrm{一种}（Ť）X+F（Ť，X）Ť\ge 0，$在任意Banach空间X上，我们为关联的线性方程式制定了非常一般的条件$X^{\prime }=\mathrm{一种}（Ť）X$ 对于非线性项 $F：[0，+\infty ）\times X\to X$在这种情况下，上述系统可以满足阴影属性的适当版本。更确切地说，我们要求$X^{\prime }=\mathrm{一种}（Ť）X$承认非常普通的二分法类型，其中包括经典的双曲线行为作为非常特殊的情况。另外，我们要求f是第二变量中的Lipschitz，具有足够小的Lipschitz常数。我们的通用框架使我们能够处理以前未获得阴影结果的各种设置。此外，我们能够恢复和完善一些已知的结果。我们还展示了如何将我们的主要结果应用于研究高阶微分方程的遮蔽特性。最后，我们通过呈现结果的离散时间版本来结束本文。