Second order differential equations in Banach space

Abstract

The theory of second-order differential equations in Banach space is surveyed. An introduction of certain selected aspects of the subject is attempted, rather than a complete discussion. References to various proofs are given, but not the proofs themselves. The bulk of the material concerns linear second-order differential equations, a subject linked intimately with the theory of strongly continuous cosine families of bounded linear operators in Banach space. Results are also presented on nonlinear second-order equations having a special semilinear form. The following topics are included: the basic theory of strongly continuous cosine families in Banach space, and the fundamental generation theorem of Sova--Da Prato--Giusti--Fattorini; the problem of converting an abstract second-order linear differential equation to an abstract first-order differential system; perturbations of the infinitesimal generator of a strongly continuous cosine family, and the approximation theorem of Konishi--Goldstein; the special properties of compactness, uniform continuity, and inhomogeneous equations for strongly continuous cosine families; and abstract semilinear second-order initial-value problems in which the linear term is a cosine family generator. (RWR