On Ulam's Type Stability of the Cauchy Additive Equation

Janusz Brzdęk

doi:10.1155/2014/540164

Abstract

We prove a general result on Ulam's type stability of the functional equation f(x + y) = f(x) + f(y), in the class of functions mapping a commutative group into a commutative group. As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by Isaac and Rassias.

Highlights

The issue of stability of functional equations has been a very popular subject of investigations for the last nearly fifty years
In this paper we prove a quite general result that allows us to generalize and extend Theorem 2 in various directions
+ d (ξ (f2 (x)), μ (f2 (x))), (8) ξ, μ ∈ YZ, x ∈ Z, and Λ : RZ+ → RZ+ is an operator defined by Λδ (x) := δ (f1 (x)) + δ (f2 (x)) δ ∈ RZ+, x ∈ Z. (9)

Summary

Introduction

The issue of stability of functional equations has been a very popular subject of investigations for the last nearly fifty years (see, e.g., [1,2,3,4,5,6,7,8]). We can introduce the following definition, which somehow describes the main ideas of such stability notion for equations in two variables (R+ stands for the set of nonnegative reals). Let f : E1 → E2 be an operator such that. Note that Theorem 2 with p = 0 yields the result of Hyers [9] and it is known (see [17]; cf [18, 19]) that for p = 1 an analogous result is not valid. In this paper we prove a quite general result that allows us to generalize and extend Theorem 2 in various directions

An Auxiliary Result

The Main Theorem

Some Consequences