## 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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