## 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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### References:

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