Stability of the generalized alternative cauchy equation

Abstract

Let G be a commutative semigroup and letL be a complete Archimedean Riesz Space. Suppose thatF: G → L satisfies for somee ∈ L ₊ the inequality

$$\left| {|F(x + y)| - |F(x) + F(y)|} \right| \leqslant e for x, y \in G.$$

Then there exists a unique additive mappingA : G → L such that

$$\left| {F(x) - A(x)} \right| \leqslant e for x \in G.$$

As the method of the proof we use the Johnson-Kist Representation Theorem.