Cauchy′s functional equation in restricted complex domains

Watcharapon Pimsert,Sajee Pianskool,Vichian Laohakosol

doi:10.1155/ijmms/2006/69368

Abstract

Using a method modified from that used by Pisot and Schoenberg in 1964‐1965, a Cauchy′s functional equation with restricted domains in the complex field is solved for uniformly continuous solutions.

Highlights

Consider the Cauchy’s functional equation, hereby called CFE, ⎛ ⎞A f ⎝ ujαj⎠ = f ujαj . j=1 j=1 (1.1)Under various assumptions on A and αjs, Pisot and Schoenberg [8] solved for monotone solutions of (1.1)
Using a method modified from that used by Pisot and Schoenberg in 1964-1965, a Cauchy’s functional equation with restricted domains in the complex field is solved for uniformly continuous solutions
In a subsequent paper [9], they treated the case where the domain of solutions is a subset of Rn with the following main result

Summary

Introduction

If f : S+ → C is a uniformly continuous solution of the functional equation We define an R-linear function λ : C → C and show that λ is uniformly continuous on S+.

Results

Conclusion