Spectral representation theory and stability of the multiplicative Dhombres functional equation in f-algebras

Bogdan Batko

doi:10.1007/s00010-013-0234-6

Abstract

We describe a method of extending certain stability results valid for real-valued functions to the class of functions with range in an f-algebra. The method is based on the Spectral Representation Theory for Riesz spaces. Details will be presented for the multiplicative Dhombres functional equation $$(F(x) + F(y))(F(x + y) - F(x) - F(y)) = 0.$$In this note we use the Ogasawara–Maeda Spectral Representation Theorem for Riesz spaces which will be firstly adapted to the f-algebras reality.

Highlights

In this paper we investigate the possibility of applying the Spectral Representation Theory for Riesz spaces (SRT) with the purpose to develop a method of extending some results in the stability theory of functional equations for realvalued functions to the class of functions taking values in f-algebras
The SRT for Riesz spaces provides us with a representation of vectors of a given Riesz space L by extended real continuous functions on a certain topological space X which are finite on a dense subset of X; we denote this class of functions by C∞(X)
Speaking Theorem 6 states that the Dhombres equation in the multiplicative form (1) in f-algebras is stable in the Ulam–Hyers sense

Summary

Introduction

In this paper we investigate the possibility of applying the Spectral Representation Theory for Riesz spaces (SRT) with the purpose to develop a method of extending some results in the stability theory of functional equations for realvalued functions to the class of functions taking values in f-algebras. The stability of the Dhombres equation in the conditional form (2) in the class of functions mapping an Abelian group into a Riesz space was investigated in [4] with the use. In [10] Moszner proved the superstability of the multiplicative Dhombres equation in the class of functions mapping a groupoid into a finite-dimensional normed algebra without zero divisors (cf [10, Theorem 2.5.2]). It raises the natural question if a similar result holds true in a more general order setting. We prove that Eq (1) is stable in the Ulam–Hyers sense, i.e. any given f : G → L satisfying inequality (3) can be approximated by a unique additive function a : G → L in the sense that the set {|f (x) − a(x)| : x ∈ G} is bounded in L

Preliminaries

Stability results

Auxiliary results and proofs