Hyers-Ulam stability of Flett's points

K e y w o r d s H y e r s U l a m stability, Flett mean value theorem, Flett's point. 1. I N T R O D U C T I O N In 1968, Ulam [1] proposed the general problem: "When is it t rue tha t by changing a l i t t le the hypotheses of a theorem one can still assert t ha t the thesis of the theorem remains t rue or approx imate ly t rue?" In 1978, Gruber [2] proposed the following Ulam type problem: "Suppose a ma themat i ca l ob jec t satisfies a cer tain p roper ty approximately. Is it then possible to approximate this objec t by objects , sat isfying the p roper ty exactly?" According to Gruber , this kind of s tabi l i ty problem is of par t icu lar interest in probabi l i ty theory. Given an opera tor T and a solution class {u} with the p roper ty tha t T(u) = 0, when does HT(v)I[ _< E for an e > 0 imply tha t I[u vlt <~ 5(c) for some u and for some ~ > 0? This problem is called the s tabi l i ty of the functional t ransformat ion. A great deal of work has been done in connection with the ord inary and par t ia l differential equations. In 1940, Ulam [3] asked the following problem: "Give condit ions in order for a linear mapping near an approx imate ly l inear mapping to exist." If f is a function from a normed vector space into a Banach space and satisfies IIf(x + y) f ( x ) f(Y)l[ ~< ¢, Hyers [4] proved tha t there exists an addi t ive function A such tha t [If(x) A(x)ll <_ ~. If f ( x ) is a real continuous function of x over R, and I f ( x + y) f ( x ) f(y)] < ¢, it was shown by Hyers and Ulam [5] tha t there exists a constant k such tha t I f (x) kx[ < 2c. The interested reader could refer to the recent book [6] for an account on Hyers-Ulam type s tabi l i ty problems. In the paper [7], Hyers and Ulam consider the s tabi l i ty of differential expressions and proved following theorem. THEOREM 1. Let f : • -~ R be nt imes differentiable in a neighborhood N o f the point "q. Suppose that f(n)(~) = 0 and f (n ) (x ) changes sign at ~. Then, for all ~ > O, there exists a ~ > 0 such that for every function g : R --~ • which is nt imes differentiable in N and 0893-9659/03/$ see front matter (~) 2003 Elsevier Science Ltd. All rights reserved. Typeset by A~-TEX PII: S0893-9659 (02)00190-8