The stability of d’Alembert’s functional equation

Summary.

In the present paper we study the stability problem of d’Alembert’s functional equation for vector-valued mappings. In particular, for a function f from an abelian group (G,+) to the normed algebra $M_{n} (\mathbb{C})$ of complex n × n-matrices satisfying the inequality

$$ ||f(x + y) + f(x - y) - 2f(x)f(y)|| \leq \delta $$

for some δ > 0 and all x, y ∈ G we show that there exists a function $h:G \to M_{n} (\mathbb{C})$ such that

• (i) $||f(x) - h(x)|| \leq\epsilon$ for some $\epsilon \geq 0$ and all x ∈ G,

• (ii) (h(x + y) + h(x − y) − 2h(x)h(y))² = 0 for all x, y ∈ G.