Convergence rates for solutions of inhomogeneous ill-posed problems in Banach space with sufficiently smooth data
The subject of this paper is the ill-posed Cauchy problem \[ u^\prime(t) = Au(t) + h(t), \ 0 < t < T, \quad u(0) = \varphi, \tag{1}\] where \( A \) is a closed linear operator on a Banach space \( X \) such that \( -A \) is the infinitesimal generator of a holomorphic semigroup \( e^{-tA} \) of angle \( \theta \in (0,\frac{\pi}{2}) \) on \( X \). In addition, \( \varphi \in X \) as well as the function \( h:[0,T] \to X \) are given, and \( 0 \) belongs to the resolvent set of \( A \). For the regularization of the given ill-posed Cauchy problem, the author considers the regularized Cauchy problem \[v_\beta^\prime(t) = f_\beta(A)v_\beta(t) + h(t), \ 0 < t < T, \quad u(0) = \varphi, \tag{2}\] where, for each parameter \( 0 < \beta < 1 \), the operator \(f_\beta(A) \) is the infinitesimal generator of a \( C_0 \) semigroup on \( X \). It is assumed that \( f_\beta(A) \) satisfies condition (A\(^+\)) which by definition means that the following are satisfied: (i) \( \textup{Dom}(f_\beta(A)) \supseteq \textup{Dom}(A^2) \), and for some constant \( R \) independent of \( \beta \), there holds \( \Vert (-A + f_\beta(A)) x \Vert \le \beta R \Vert A^2 x \Vert \) for each \( x \in \textup{Dom}(A^2) \), and (ii) there exist \( \frac{\pi}{2} - \theta < \nu < \frac{\pi}{2} \) and some constant \( C \ge 0 \) independent of \( \beta \) such that the inequality \( \textup{Re} (-w + f_\beta(w) ) \le C \vert w \vert \) holds for each \( w \in S_\nu := \{0 \neq w \in \mathbb{C} : \vert \textup{arg} \, w \vert < \nu \} \). It is shown that under those conditions, the estimate \[ \Vert u(t) - v_\beta(t) \Vert \le c\beta^{1-\omega(t)} M^{\omega(t)}, \quad 0 < t < T, \] holds, where \( c \) and \( M \) denote some constants which depend on the smoothness both of the initial data \( \varphi \in X \) and the inhomogeneity \( h:[0,T] \to X \). In addition, \( \omega \) is some function satisfying \( 0 = \omega(0) \le \omega(t) < \omega(T) = 1 \) for each \( 0 \le t < T \), and \( u \) and \( v_\beta \) denote strong solutions of (1) and (2), respectively. Subsequently, four examples of operators \( f_\beta(A) \) are given which illustrate condition~(A\(^+\)): \( A (I+\beta A)^{-1}, \, A -\beta A^\sigma, \, A e^{-\beta A} \), and \( (\ln 2)^{-1} A\hspace{1pt}\textup{Log}\hspace{1pt}(1+e^{-\beta A}) \). The paper concludes with some comments on regularization and applications. For the entire collection see [Zbl 1471.47002].
scientific article; zbMATH DE number 7393032
scientific article