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Uniqueness and nonuniqueness criteria for ordinary differential equations. (English) Zbl 0785.34003

Series in Real Analysis. 6. Singapore: World Scientific. xi, 312 p. (1993).
The book is devoted to a branch of the theory of differential equations being classical on the one hand but still alive and developing on the other hand. The topic investigated in the book may be sketched as follows: Consider the initial value problem (1) \(x'=f(t,x)\), \(x(t_ 0)=x_ 0\), where \(f\) is defined in a certain neighbourhood of the point \((t_ 0,x_ 0) \in R \times E\) and \(E\) denotes the Euclidean space \(R^ n\) or even an arbitrary Banach space. One can find conditions imposed on \(f\) which guarantee that the problem (1) has at most one solution (uniqueness criteria) or conditions ensuring that (1) has at least two solutions (nonuniqueness criteria). It is well-known that the most classical condition guaranteeing the uniqueness is the Lipschitz continuity of the function \(f\). More general uniqueness criteria are connected with the names of Nagumo, Osgood, Perron, Kamke etc. The authors give a very interesting and exhaustive survey of uniqueness and nonuniqueness criteria obtained by several mathematicians. They discuss various aspects of the theory of differential equations which are associated with those criteria (continuation of solutions, dependence on the right hand sides and initial data, stability and so on). Several carefully constructed examples illustrate the theoretical material presented in this book.
The book is divided into ten chapters discussing consecutively uniqueness and nonuniqueness criteria for first order differential equations, first order differential systems, higher order differential equations, differential equations in abstract spaces, complex differential equations, functional differential equations, impulsive differential equations and differential equations with hysteresis. Moreover, the book contains also an additional chapter (Notes) describing the bibliographical sources of the results presented. The bibliography contains 184 items which present efforts in the fascinating theory. The book is very interesting and well written. It may be warmly recommended to any student in analysis and to any specialist in the theory of differential equations.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34K05 General theory of functional-differential equations
34M99 Ordinary differential equations in the complex domain
34G20 Nonlinear differential equations in abstract spaces
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