The aim of this text is to deal with functional differential equations with infinite delay on an abstract phase space characterized by several axioms which are satisfied by different kinds of function spaces. The standard spaces which we have in mind are the one of continuous functions on ( - ∞,0] that are endowed with some restriction on their asymptotic behavior at - ∞, and the one of measurable functions on ( - ∞,0] that are integrable with respect to some Borel measure equipped with mild conditions.

This text consists of nine chapters. Chapter 1 contains the formulation of axioms of the phase space together with many examples. Chapter 2 is a presentation of basic theory of existence, uniqueness, continuous dependence, etc. of solutions. After a brief introduction to Stieltjes integrals in Chapter 3, the theory of linear equations is developed from Chapter 4 through Chapter 6. Chapter 7 is an introduction to fading memory spaces. In Chapter 8, the stability problem in functional differential equations on a fading memory space is studied in connection with limiting equations. In succession, the existence of periodic and almost periodic solutions of functional differential equations is discussed in Chapter 9.

--From the Preface

The text lives up to the authors’ description. It is theoretical, with few attempts to relate the theory to practice. The only practical examples are given in chapter 8. There the authors consider a second-order damped differential equation with a delay in the forcing term. They also analyze a system of high-order integrodifferential equations such as arise in n-species systems in mathematical ecology.