Mixed stability of the d'Alembert functional equation
Abstract
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.
References
2. Gavruta P., An answer to a question of Th. M. Rassias and J. Tabor on mixed stability of mappings, Bul. Stiintific al Univ. Politehnica din Timisoara, 42 (1997), 1-6.
Instytut Matematyki, Uniwersytet Śląski w Katowicach & Instytut Badań Systemowych Polskiej Akademii Nauk Poland
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