Dover is mostly known for reprinting older books, so this is a comparative rarity: a Dover original. In his preface, the author expresses his admiration for the way Dover makes mathematics books available at affordable prices. Clearly this is why he decided to publish Stability by Fixed Point Theory with them.
The classical way to study the stability of solutions to differential equations is Lyapunov's "direct method." This has many theoretical advantages, but sometimes it can be hard to use. In particular, constructing the right Lyapunov function or functional can be a challenge, and verifying the required conditions can be difficult or impossible.
Burton proposes to attack the problem in a different way, via fixed-point theory for transformations of metric spaces. This is quite elegant, since it manages to yield results on existence, uniqueness, and stability, all in one blow.
Of course, there are issues with the alternative method as well, mostly centering on finding the right way to express the problem as a fixed-point problem. Throughout the book, Burton not only develops the theory using fixed-point methods but also compares it with the Lyapunov approach.
Writing in Mathematical Reviews, A. F. Ivanov describes the book as "a useful addition to the libraries of all those interested in the theory and applications of stability of functional differential equations." (MR2281958) That seems about right.
Fernando Q. Gouvêa is the editor of MAA Reviews.