Lakshmikantham V., Ladas G.E. Differential Equations in Abstract Spaces

New York: Academic Press, 1972. — 231 p.The theory of differential equations in abstract spaces is a fascinating field with important applications to a number of areas of analysis and
other branches of mathematics. At the present time, there is no single book that is self-contained and simple enough to appeal to the beginner.
Furthermore, if one desires to give a course so as to expose the student to this branch of research, such a book becomes handy. This being the
motivation, the aim of our book is, in fact, to introduce the nonspecialist to this elegant theory and powerful techniques. But for some familiarity
with the elements of functional analysis, all the important results used in this book are carefully stated in the appendixes so that, for the most part,
no other references are needed. The required theory, from the calculus of abstract functions and the theory of semigroups of operators, used in
connection with differential equations in Banach spaces is treated in detail. We have tried to present the fundamental theory of differential equations in Banach spaces: the first three chapters form an integrated whole together with, perhaps, Sections 6.1 and 6.3 of Chapter 6. Chapter 4 is devoted to the study of differential inequalities, mostly, in Hilbert spaces. The theory developed in Chapter 5 is interesting in itself and could be read independently. This also applies to Chapter 4. Throughout the book we give a number of examples and applications to functional and partial
differential equations which help to illustrate the abstract results developed. In most sections there are several problems with hints directly related to
the material in the text. The notes at the end of each chapter indicate the sources which have been consulted and those whose ideas are developed.
Several references are also included for further study on the subject. We hope that the reader who is familiar with the contents of this book will be
fully equipped to contribute to this field as well as read with ease the current literature.