Stability of the Cauchy functional equation in quasi-Banach spaces
Jacek Tabor (2004)
Annales Polonici Mathematici
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Let X be a quasi-Banach space. We prove that there exists K > 0 such that for every function w:ℝ → X satisfying ||w(s+t)-w(s)-w(t)|| ≤ ε(|s|+|t|) for s,t ∈ ℝ, there exists a unique additive function a:ℝ → X such that a(1)=0 and ||w(s)-a(s)-sθ(log₂|s|)|| ≤ Kε|s| for s ∈ ℝ, where θ: ℝ → X is defined by for k ∈ ℤ and extended in a piecewise linear way over the rest of ℝ.