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Functional differential equations with infinite delay. (English) Zbl 0732.34051
Lecture Notes in Mathematics, 1473. Berlin etc.: Springer-Verlag. x, 317 p. DM 53.00 (1991).
This book deals with functional differential equations \(\dot x(t)=f(t,x_ t)\), where \(\dot x(t)\) denotes the derivative of x at t and \(x_ t\) stands for the retarded argument of the unknown function x at time t, more precisely, \(x_ t\) stands for a function in a fixed function space, called the phase space. The function \(x_ t\) describe the history of x up to t. The phase space is characterized by some axioms. The crucial assumptions is that the motion of \(x_ t\) in the phase space is continuous for t. The standard phase spaces are the one of continuous function on \((-\infty,0]\) that are endowed with some restriction on their asymptotic behavior at \(-\infty\) and the one of measurable function on \((-\infty,0]\) that are integrable with respect to a Borel measure equipped with mild conditions. Note that in the case of functional differential equations with finite delay, the natural phase space is a space of continuous function on \([-h,0]\), \(h>0\). These spaces are carefully adopted as phase spaces for equations with delay so as to solve each problem which we encounter in applications. Nevertheless many fundamental properties of these equations hold independently of the choice of phase spaces. The axiomatic approach permits to summarize these properties as well as to investigate the general mathematical structure which reflects the effect of infinite delay.
The book 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 in Chapters 4, 5 and 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 the last Chapter the existence of periodic and almost periodic solutions is discussed.

34K05 General theory of functional-differential equations
34K10 Boundary value problems for functional-differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K20 Stability theory of functional-differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
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