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Stability of generalized quadratic functional equation on a set of measure zero

Youssef Aribou, Hajira Dimou, Abdellatif Chahbi, Samir Kabbaj (2015)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

In this paper we prove the Hyers-Ulam stability of the following K-quadratic functional equation [...] where E is a real (or complex) vector space. This result was used to demonstrate the Hyers-Ulam stability on a set of Lebesgue measure zero for the same functional equation.

Stability of the Cauchy functional equation in quasi-Banach spaces

Jacek Tabor (2004)

Annales Polonici Mathematici

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 θ ( k ) : = w ( 2 k ) / 2 k for k ∈ ℤ and extended in a piecewise linear way over the rest of ℝ.

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