Ordinary differential equations in Banach spaces / Klaus Deimling

In the preface, the author announces that "the aim of this book has been to show more or less recent connections between differential equations and functional analysis''. Special attention is given to the theory of countable systems of ordinary differential equations, because of the great importance of these systems, as pointed out in the introduction of the book. The material is divided into eight sections, with the following headings: (1) Lipschitz type conditions, (2) Compactness conditions, (3) Conditions of dissipative type, (4) Solutions in closed sets, (5) Flow invariance and differential inequalities, (6) Countable systems of ordinary differential equations, (7) Approximate solutions, (8) Related topics. Applications of the theory to some concrete problems are also given; for example, in Section 7, the cases of branching processes and degradation of polymers are studied with the aid of some countable systems of ordinary differential equations. The reader may also find here many modern topics such as: stochastic differential equations, accretive operators, semigroup theory, evolution equations, etc. This well-written and interesting book, which is an expanded version of the material prepared for a one-semester graduate level course, includes an up-to-date bibliography of 199 of the most important recent papers and books in the theory of ordinary differential equations. (MathSciNet)