Stability of Functional Equations in Several Variables

Donald H. Hyers, George Isac, Themistocles M. Rassias (auth.)9783764340247, 376434024X, 081764024X

The notion of stability of functional equations of several variables in the sense used here had its origins more than half a century ago when S. Ulam posed the fundamental problem and Donald H. Hyers gave the first significant partial solution in 1941. The subject has been revised and de­ veloped by an increasing number of mathematicians, particularly during the last two decades. Three survey articles have been written on the subject by D. H. Hyers (1983), D. H. Hyers and Th. M. Rassias (1992), and most recently by G. L. Forti (1995). None of these works included proofs of the results which were discussed. Furthermore, it should be mentioned that wider interest in this subject area has increased substantially over the last years, yet the pre­ sentation of research has been confined mainly to journal articles. The time seems ripe for a comprehensive introduction to this subject, which is the purpose of the present work. This book is the first to cover the classical results along with current research in the subject. An attempt has been made to present the material in an integrated and self-contained fashion. In addition to the main topic of the stability of certain functional equa­ tions, some other related problems are discussed, including the stability of the convex functional inequality and the stability of minimum points. A sad note. During the final stages of the manuscript our beloved co­ author and friend Professor Donald H. Hyers passed away.

---

Table of contents :
Front Matter….Pages i-vii
Prologue….Pages 1-10
Introduction….Pages 11-14
Approximately Additive and Approximately Linear Mappings….Pages 15-44
Stability of the Quadratic Functional Equation….Pages 45-77
Generalizations. The Method of Invariant Means….Pages 78-101
Approximately Multiplicative Mappings. Superstability….Pages 102-131
The Stability of Functional Equations for Trigonometric and Similar Functions….Pages 132-154
Functions with Bounded n th Differences….Pages 155-165
Approximately Convex Functions….Pages 166-179
Stability of the Generalized Orthogonality Functional Equation….Pages 180-203
Stability and Set-Valued Functions….Pages 204-231
Stability of Stationary and Minimum Points….Pages 232-245
Functional Congruences….Pages 246-276
Quasi-Additive Functions and Related Topics….Pages 277-289
Back Matter….Pages 290-318