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

Abstract

More than 13 years have passed since the fundamental survey [16] was prepared, which, as the author intended, should be the first part of a large work devoted to abstract differential equations and methods for solving them. However, the troubles being in the Russian science during the whole this period have influenced also on the authors, and instead of two years supposed, the preparation of the second part has occupied considerably more time. During the last 10 years, the work in the field of differential equations in abstract 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 [33, 75] of such a type appeared being translated by the authors of the present survey and [20], which were edited by Yu. A. Daletskii. Therefore, the work whose second part is proposed to the reader will be undoubtedly useful for the Russian reader. Its style coincides with that of [16], 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 foreign sources that are almost inaccessible for Russian readers. From our viewpoint, this allows us to demonstrate clearly the philosophy, to describe the results obtained, and to indicate the main directions of the development of the theory in the framework of a limited volume of the survey.