The Hyers theorem via the Markov–Kakutani fixed point theorem

Barbara Przebieracz

doi:10.1007/s11784-013-0102-y

Abstract

We present an application of the Markov–Kakutani common fixed point theorem to the theory of stability of functional equation by proving some version of the Hyers theorem concerning approximate homomorphisms.

Highlights

One of the most celebrated results of the theory of common fixed points is a theorem proved independently by Markov [8] and Kakutani [7] (see [9] and [10])
One of the most celebrated results of the theory of common fixed points is a theorem proved independently by Markov [8] and Kakutani [7].Theorem 1.1 (Markov–Kakutani fixed point theorem)
The theorem of Hyers [5] was a partial answer to the problem posed by Ulam: does there exist for an approximate homomorphism φ a homomorphism which approximates φ? The result of Hyers initiated the works of many authors on the stability of functional equations; it suffices to mention that his paper was cited several hundred times

Summary

Introduction

One of the most celebrated results of the theory of common fixed points is a theorem proved independently by Markov [8] and Kakutani [7] (see [9] and [10]). Let F be a family of affine continuous self-mappings of K such that F ◦ G = G ◦ F for F, G ∈ F. The result of Hyers initiated the works of many authors on the stability of functional equations (see [6]); it suffices to mention that his paper was cited several hundred times.

Results

Conclusion