In contemplating the third edition, I have had multiple objectives to
achieve. The first and foremost important objective is to maintain the accessibility
and readability of the book to a broad readership with varying
mathematical backgrounds and sophistication. More proofs, more graphs,
more explanations, and more applications are provided in this edition.
The second objective is to update the contents of the book so that the
reader stays abreast of new developments in this vital area of mathematics.
Recent results on local and global stability of one-dimensional maps are
included in Chapters 1, 4, and Appendices A and C. An extension of the
Hartman–Grobman Theorem to noninvertible maps is stated in Appendix
D. A whole new section on various notions of the asymptoticity of solutions
and a recent extension of Perron’s Second Theorem are added to Chapter
8. In Appendix E a detailed proof of the Levin–May Theorem is presented.
In Chapters 4 and 5, the reader will find the latest results on the larval–
pupal–adult flour beetle model.
The third and final objective is to better serve the broad readership of
this book by including most, but certainly not all, of the research areas
in difference equations. As more work is being published in the Journal of
Difference Equations and Applications and elsewhere, it became apparent
that a whole chapter needed to be dedicated to this enterprise. With the
prodding and encouragement of Gerry Ladas, the new Chapter 5 was born.
Major revisions of this chapter were made by Fozi Dannan, who diligently
and painstakingly rewrote part of the material and caught several errors
and typos. His impact on this edition, particularly in Chapters 1, 4, and
Chapter 8 is immeasurable and I am greatly indebted to him. My thanks
go to Shandelle Henson, who wrote a thorough review of the book and
suggested the inclusion of an extension of the Hartman–Groman Theorem,
and to Julio Lopez and his student Alex Sepulveda for their comments and
discussions about the second edition.