Mixed stability of the d’Alembert functional equation - Przybyła, Maciej (Annales Mathematicae Silesianae, 2005)
In the present paper we will prove the theorem concerning the mixed stability of the d'Alembert functional equation, i.e. we will show that if ɛ > 0, s ≥ 1, δ = [2s + 22s+16ɛ+8]/4, X is a real normed space and f: X→ℂ satisfies the inequality |f(x+y) + f(x-y) - 2f(x)f(y)| ≤ ɛ(∥x∥s + ∥y∥s) for all x,y∈X, then |f(x)| ≤ δ∥x∥s for all x∈X such that ∥x∥ ≥ 1, or f(x+y) + fx-y) = 2f(x)f(y) for all x,y∈X.