This paper deals with the well-posedness for the fully nonlinear functional differential equation $u^{\prime }(t)\in A(t)u_t$ in a general Banach space X, where $\{A(t);t\in [a,b)\}$ is a family of operators whose domains are subsets of the so-called initial-history space $\mathfrak{X}$ and whose ranges are subsets of the space X. The special case where $A(t)\varphi =B(t)\varphi (0)+G(t,\varphi )$ for $t\in [a,b)$ and $\varphi \in \mathfrak{X}$ with $\varphi (0)\in D(B(t))$ has been extensively studied so far, but there has not been a satisfactory solution to the flow invariance problem. This paper establishes the well-posedness for the fully nonlinear functional differential equations and solves the above-mentioned problem on flow invariance.

中文翻译：

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全非线性泛函微分方程的适定性

本文讨论了全非线性泛函微分方程的适定性 ${你}^{\prime }(吨)\in \mathrm{一种}(吨){你}_{吨}$在一般 Banach 空间X 中，其中$\{\mathrm{一种}(吨);吨\in [\mathrm{一种},乙)\}$ 是一个算子族，其域是所谓的初始历史空间的子集 $\mathfrak{X}$并且其范围是空间X 的子集。特殊情况下$\mathrm{一种}(吨)\phi =乙(吨)\phi (0)+G(吨,\phi )$ 为了 $吨\in [\mathrm{一种},乙)$ 和 $\phi \in \mathfrak{X}$ 和 $\phi (0)\in D(乙(吨))$到目前为止已经被广泛研究，但是对于流动不变性问题还没有一个令人满意的解决方案。本文建立了全非线性泛函微分方程的适定性，解决了上述关于流动不变性的问题。