Integrated semigroups and \(C\)-semigroups and their applications
The survey is devoted to recent advances in integrated semigroups and $C$-semigroups of operators in Banach space and their applications to the regularization of ill-posed problems. Typically all theorems, propositions, etc., are given with relevant references and without proofs. \par Chapter 1 concerns $n$-times integrated semigroups on Banach spaces. General properties of such type of operator semigroups and the relationship to abstract Cauchy problems are discussed. The generator of an integrated semigroup is defined and its (spectral) properties are described. It is emphasized that in the general case the generator may be not densely defined. \par Special attention is paid to the case of exponentially bounded integrated semigroups. The authors review results on relations between the semigroup, the resolvent of the generator, and the spectral properties of the generator. Interpolation results are given as well. It is considered the particular case of local integrated semigroups and their properties. \par Chapter 2 concerns $C$-semigroups. The first part of Chapter~2 is devoted to exponentially bounded semigroups. The authors give the definitions of the generator of a semigroup and the generators in the sense of Miyadera and in the sense of Da Prato. They discuss the relations between different notions of the semigroup generators and their spectral properties. The relations between semigroups and resolvents are discussed as well. Some interpolation results for the semigroup generator are given. The case of analytic semigroups is considered. The relations with abstract Cauchy problems and basic perturbation results are considered as well. \par The second part of Chapter~2 concerns local $C$-semigroups and $C$-cosine operator functions. Basic notions and results on local $C$-semigroups (i.e., generator, asymptotic $C$-resolvent, etc.)\ are discussed. General results on local $C$-cosine families and the relations with abstract second-order Cauchy problems are given. \par The third part of Chapter~2 is devoted to applications to inhomogeneous abstract Cauchy problems in Banach spaces. \par Chapter 3 concerns applications to the regularization of ill-posed problems. In the first part, the authors consider basic notions of the regularization problem. The second part provides a brief survey of approximations of well-posed evolution problems. \par The third part of Chapter~3 concerns the regularization of the linear problem $Bx=y$, provided that $B^{-1}$ is unbounded. Iterative procedures for this problem are discussed. \par The fourth part of Chapter~3 is devoted to the regularization of the backward Cauchy problem in the form \[ v'(t)=Av(t),\quad v(T)=v_T,\quad 0\leq t\leq T \] (strongly ill-posed evolution equation). Different approximation schemes are reviewed, including stochastic approximations. \par The last part of Chapter~3 concerns the Cauchy problem in the form \[ u'(t)=Au(t)+f(t),\quad u(0)=u_0,\quad 0\leq t\leq T, \] where the operator $A$ is the generator of a $k$-times integrated semigroup. Different approximation techniques are considered, including discretization, semidiscrete regularization, Lavrent'iev method for $C_0$- and for integrated semigroups, and other methods.
scientific article; zbMATH DE number 6892358
scientific article