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Volterra and integral equations of vector functions. (English) Zbl 0940.45002

Pure and Applied Mathematics, Marcel Dekker. 224. New York, NY: Marcel Dekker. vi, 349 p. (2000).
This book treats the theory of abstract Volterra equations, that is equations of the form \[ x(t) = \int_0^t f(t,s,x(s)) ds, \] in, e.g., Banach spaces, and in addition it treats a large class of generalizations of this equation. Chapter 1 contains a detailed and carefully written exposition of the mathematical tools used in the book, i.e., compactness and fixed-point theory, measurable functions and ideal spaces (if \(|y(s)|\leq |x(s)|\) a.e., then \(\|y\|\leq \|x\|\)). In Chapter 2 the general existence theory for abstract Volterra equations is presented. Here abstract Volterra operators are defined, using projections with nondecreasing range and a linearly ordered index set, in such a way that they describe systems where the current state does not depend on the “future”. In Chapter 3 the author considers Volterra integral equations in Banach spaces with focus on the existence and uniqueness of solutions. In this chapter a large number of general compactness results are derived. In the last chapter the dependence on parameters is studied in a way that makes it easy to obtain results on “averaging” problems as well.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45-02 Research exposition (monographs, survey articles) pertaining to integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
45G10 Other nonlinear integral equations
47G10 Integral operators
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