Differential Equations in Banach Spaces II. Theory of Cosine Operator Functions.

Differential Equations in Banach Spaces II. Theory of Cosine Operator Functions.
V. Vasiliev and S. Piskarev

More than 13 years have passed since the fundamental survey (Vasiliev V., Krein S. and Piskarev S.) was prepared, which, as the author intended, should be the first part of a large work devoted to abstract differential equations and methods for their solution. However, the troubles being in the Russian science during the whole this period has influenced on the authors, and instead of two years sopposed, the preparation of the second part occupies considerably more time. During the later 10 years, the work in the field of differential equation in abstracr spaces was very active (in foreign countries), and every year, several books and a heavy number of papers devoted to this direction appear in the world (of course, the most of them are not available for the Russian reader). At the same time, only two books (Ph. Clement and P. Hess) of such a type appears being translated by the authors of the present survey and (J. Goldstein), which were editted by Yu. A. Daletskii. Therefore, the work whose second part is proposed to the reader will be undoubtedly usefull for the Russian reader. Its style coincides with that of (Vasiliev V., Krein S. and Piskarev S.), i.e., the material is often presented without proofs, and the main attention is paid to the structure of presentation, although we present certain proofs from foreing sources that are almost inacessible. From our viewpoint, this allows us to clearly demonstrate the philosophy, to describe the result obtained, and to indicate the main directions of the development of the theory in the framework of a limited volume of the survey. r Moreover, the authors have prepared a separate edition of the bibliographical index Vasiliev V., Piskarev S. , which can surve as a sufficiently complete source of information about the theory of differential equations in abstract spaces during the recent years. The main object of study in this part are second-order differential equation that are presented very little in Russian literature since now. Here, we can only mention the book of J. Goldstein written in accordance with the own interests of the author, which does not pretend to the exhausting description of all aspects of the theory. Moreover, the material of the present survey includes the presentation of the abstract Cauchy problem for first- and second-order equations that is not considered in the book Vasiliev V., Piskarev S. . As was already mentioned in (Vasiliev V., Krein S. and Piskarev S.), the theory of cosine operator-valued functions is philosophically very close to the operator semigroup theory and often is developped in parallel to it. Therefore, the reader easily draws analogies between the material presented here and that presented in (Vasiliev V., Krein S. and Piskarev S.). At the same time, the theory of cosine operator-valued functions considerably differs from the operator semigroup theory. First of all, these distinctions concer with the properties inherit to the corresponding parabolic and hyperbolic partial differential equations.