In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

在本文中，我们将 Gühring 和 Räbiger 首次获得的非自治和非齐次柯西问题的常数公式的变体扩展到非 Hille-Yosida 情况。通过使用这个常数的变化公式，我们推导出了相关非自治齐次问题生成的进化族指数二分法存在的充分必要条件。我们还证明了小扰动的指数二分法的持久性结果。最后，我们通过考虑两个例子来说明我们的结果。第一个例子是一个具有非局部和非自治边界条件的抛物线方程，第二个例子是一个年龄结构模型，它是一个双曲线方程。