Second order linear differential equations in Banach spaces. (English) Zbl 0564.34063

Fattorini, H. O.

North-Holland Mathematics Studies, 108. Notas de Matemática, 99. Amsterdam - New York - Oxford: North-Holland. XIII, 314 p. (1985).

The work presents the theory and applications of the second order linear differential equations in Banach spaces. The method of the study gives the possibility to reduce an initial value or initial-boundary value problem with partial differential equations to an ordinary differential initial value problem, which is very useful not only in the theory but also in the applications. The monograph contains 8 chapters. The first three of them and the last deal with the theory and the other with the applications. Chapter I contains the study of the Cauchy problem for first order equations and the semigroup theory. Chapter II presents the Cauchy problem for second order equations and cosine function theory. Chapter III deals with the reduction of a second order equation to a first order system and with phase spaces. In chapters IV-VII the applications to partial differential equations e.g. to wave equations with different boundary conditions, to equations with almost periodic and periodic solutions, to parabolic singular perturbation problem with applications to diverse physical situations etc., are treated. Chapter VIII deals with the complete second order equation theory. The monograph is written on a high scientific and methodological level, with many exercises throughout the book, which complete the width of the theory.

Reviewer: M.Greguš

MSC:

34G10	Linear differential equations in abstract spaces
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A30	Linear ordinary differential equations and systems
34C20	Transformation and reduction of ordinary differential equations and systems, normal forms

Keywords:

second order linear differential equations in Banach spaces; Cauchy problem; reduction; phase spaces; wave equations; parabolic singular perturbation problem; applications