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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-3-6,
author = {Jacek Tabor},
title = {Stability of the Cauchy functional equation in quasi-Banach spaces},
journal = {Annales Polonici Mathematici},
volume = {83},
year = {2004},
pages = {243-255},
zbl = {1101.39021},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-3-6}
}
Jacek Tabor. Stability of the Cauchy functional equation in quasi-Banach spaces. Annales Polonici Mathematici, Tome 83 (2004) pp. 243-255. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap83-3-6/