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Regularization for a class of ill-posed Cauchy problems. (English) Zbl 1073.47016

The paper is concerned with the ill-posed Cauchy problem \(u^{\prime}(t)=Au(t)\) \((0<t \leq T)\), \(u(0)=x\) associated with a densely defined linear operator \(A\) in a Banach space. The main assumption is that operator \(-A\) generates an analytic semigroup of angle \(\alpha \in [\pi/4, \pi/2)\). Given \(\varepsilon>0\), the authors define the bounded operator \(A_{\varepsilon}=\varepsilon^{-1}(I-(I+\varepsilon A)^{-1})\) and the set \(S_{\varepsilon}(t)=\exp(-tA_{\varepsilon})\), \(t \in (-\infty,\infty)\).
The main result states that \(R_{\varepsilon,t}=S_{\varepsilon}(t)\) is a family of regularizing operators for the problem of finding \(u(t)\) when \(x\) is given approximately. Namely, there exists a coordination rule \(\varepsilon=\varepsilon(\delta)\) such that \(\lim_{\delta\to 0}\varepsilon(\delta)= 0\) and \(\lim_{\delta \to 0} \| R_{\varepsilon(\delta),t} x_{\delta}-u(t)\|=0\) for each \(t \in (0,T]\) under the condition that \(\| x_{\delta}-x \| \leq \delta\).

MSC:

47A52 Linear operators and ill-posed problems, regularization
47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
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