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On nonautonomous functional differential equations. (English) Zbl 0949.34063
The author investigates the existence of an evolution family for the nonautonomous Cauchy problem $x'(t)= A(t) x(t),\quad 0\leq s\leq t\leq T,\quad x(s)= x,$ in a Banach space $$X$$. Each $$A(t)$$ is a linear operator on $$X$$. The following result is obtained:
Let $$X$$, $$Y$$, and $$D$$ be Banach spaces, $$D$$ densely and continuously imbedded in $$X$$. Let $$A(t)\in L(D,X)$$, $$L(t)\in L(D, Y)$$, $$\Phi(t)\in L(X,Y)$$ be such that
(i) $$t\to A(t)x$$, $$L(t)x$$ are $$C^1$$ for all $$x\in D$$,
(ii) $$\{A_0(t)\}_{0\leq t\leq T}$$ with $$A_0(t)$$ defined as $$A(t)$$ restricted to $$\text{ker }L(t)$$ is stable,
(iii) $$L(t)$$ is surjective for every $$t\in [0,T]$$,
(iv) $$t\to \Phi(t)x$$ is $$C^1$$ for all $$x\in E$$,
(v) there exist constants $$\gamma> 0$$, $$v\in\mathbb{R}$$ such that $$\|L(t)x\|\geq \gamma(\lambda- v)\|x\|$$ for all $$x\in \text{ker}(\lambda- A(t))$$, and all $$\lambda> v$$.
Then there is an evolution family $$\{U_\Phi(t, s)\}_{0\leq s\leq t\leq T}$$ generated by $$\{A_\Phi(t)\}_{0\leq t\leq T}$$ where $$A_\Phi$$ is $$A$$ restricted to $$\text{ker}(L- \Phi)$$, and ${\partial\over\partial t} U_\Phi(t, s)x= A_\Phi(t) U_\Phi(t, s)x$ for every $$x\in D(A_\Phi(s))$$.
Some applications are given. The proof is based on a generalization of an idea by G. Greiner [Houston J. Math. 13, 213-229 (1987; Zbl 0639.47034)].

##### MSC:
 34K30 Functional-differential equations in abstract spaces 34K05 General theory of functional-differential equations
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