The implicit function theorem : history, theory and applications / Steven G. Krantz and Harold R. Parks

This book is devoted to basic results with the generally accepted name Implicit Function Theorem (sometimes Inverse Function Theorem) and which appear everywhere, beginning with elementary courses of Calculus and ending up with fundamental monographs on Functional Analysis, Differential and Integral Equations, Calculus of Variations, and so on. It seems that the book is concerned with all problems related to implicit function theorems. However, this book really does not cover all fields with the name “Implicit Function Theorem”. So, the authors do not recall G. Peano, E. Goursat, G. A. Bliss, Young, A. Lamson, S. M. Lozinskii, and others, who made a significant contribution in this field; the authors do not consider different variants of the Implicit Function Theorem in Banach spaces for operators differentiable only in separate points, or variants of the Implicit Function Theorem based on fixed point principles different from Banach-Caccioppoli; the authors also omitted variants of the Implicit Function Theorem for operators in locally convex spaces or for operators in spaces of formal power series. One can find some information in this direction in [J. Appell, A. Vignoli and P. P. Zabreiko, Expo. Math. 14, No. 5, 385-424 (1996; Zbl 0869.26003)]. However, in general, this book is the first that is especially devoted to the Implicit Function Theorem. The authors offer a useful and fascinating manuscript which should be of interest and useful to graduate and postgraduate students, professional mathematicians, teachers as well as researchers. Most of them will be able to find all fundamental ideas in the field and simple and transparent proofs of all main implicit function theorems and theorems closely related to these. (Zentralblatt)

Bibliogr. p. [151]-159. Index