Stability by fixed point theory for functional differential equations. (English) Zbl 1160.34001

Mineola, NY: Dover Publications (ISBN 0-486-45330-8/pbk). xiv, 348 p. (2006).
The central idea presented in the book is to ‘reconstruct’ solutions to differential equations (existence and uniqueness of which is already known) in some special function space, typically of functions decaying to zero. This construction uses fixed point theorems – mostly Banach’s contraction theorem, but also, for instance, Schauder’s fixed point theorem. If the solution for all initial values is obtained, say, in a space of functions that converge to zero, then clearly the zero solution is globally asymptotically stable. Versions of this method are extensively applied to examples, and compared with Liapunov’s direct method.
The last chapter, contributed by J.A.D. Appleby, extends the methods to stochastic perturbations of the equations treated so far.
The promise of the cover text that the material is accessible to advanced undergraduates is really kept by the book.
Students and researchers can also profit from the extensive list of references which includes classical work, e.g., of Volterra and Liapunov. The author dedicated the volume to V. Lakshmikantham.


34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34K50 Stochastic functional-differential equations
47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations