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 + \sqrt{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) + f{x-y) = 2f(x)f(y) for all x,y∈X.
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Vol. 19 (2005)
Published: 2005-09-30
10.1515/amsil
English