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Stability of an alternative functional equation. (English) Zbl 1133.39025

The author considers the alternative functional equation
\[ f(x+y)+f(x)+f(y) \neq 0 \implies f(x+y)=f(x)+f(y), \]
where \(f:S \to X\), \((S,+)\) is an abelian semigroup and \((X,\|\cdot\|)\) a Banach space. The problem to be solved is the following: assume that \(f:S \to X\) satisfies
\[ \| f(x+y)+f(x)+f(y)\|>\Phi_1(x,y) \implies \| f(x+y)-f(x)-f(y)\| \leq \Phi_2(x,y), \]
where \(\Phi_1, \Phi_2:S \times S \to \mathbb R_+\) are given functions. Does there exist an additive function \(a:S \to X\) such that \[ \| f(x)-a(x)\| \leq \Psi(x), \] where \(\Psi:S \to \mathbb R_+\) is a function which an be explicitly computed starting from \(\Phi_1\) and \(\Phi_2\)? The answer is positive if the functions \(\Phi_1\) and \(\Phi_2\) satisfy one of the following conditions.
1. Each of the series \[ \sum_k^{\infty} 2^{-k}\Phi_i(2^kjx,2^kx), \] where \(j\in \{1,2,3\}\), \(i\in \{1,2\}\), converges for every \(x \in S\) and
\[ \lim_{k \to +\infty} 2^{-k}\Phi_i(2^kx,2^ky)=0 \]
for \(x,y \in S\);
2. \(S\) is uniquely \(2\)-divisible, each of the series \[ \sum_k^{\infty} 2^k \Phi_i(2^{-k}jx,2^{-k}x), \] where \(j\in \{1,2,3\}\), \(i\in \{1,2\}\), converges for every \(x \in S\) and
\[ \lim_{k \to +\infty} 2^k \Phi_i(2^{-k}x,2^{-k}y)=0 \] for \(x,y \in S\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
39B55 Orthogonal additivity and other conditional functional equations
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