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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)].

34K30 Functional-differential equations in abstract spaces
34K05 General theory of functional-differential equations
Full Text: DOI
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