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Stability of a functional equation of Whitehead on semigroups. (English) Zbl 1252.39034
Summary: Let $$S$$ be a semigroup and $$X$$ a Banach space. The functional equation $$\varphi (xyz) + \varphi (x) + \varphi (y) + \varphi (z) = \varphi (xy) + \varphi (yz) + \varphi (xz)$$ is said to be stable for the pair (X, S) if and only if $$f : S \rightarrow X$$ satisfying $$f(xyz)+f(x)+f (y) + f (z) - f (xy) - f (yz) - f (xz) \leq \delta$$for some positive real number $$\delta$$ and all $$x, y, z \in S$$, there is a solution $$\varphi$$: S $$\rightarrow X$$ such that $$f-\varphi$$ is bounded.
In this paper, among others, we prove the following results: 1) this functional equation, in general, is not stable on an arbitrary semigroup; 2) this equation is stable on periodic semigroups; 3) this equation is stable on abelian semigroups; 4) any semigroup with left (or right) law of reduction can be embedded into a semigroup with left (or right) law of reduction where this equation is stable.
The main results of this paper generalize the works of S.-M. Jung [J. Math. Anal. Appl. 222, No. 1, 126–137 (1998; Zbl 0928.39013)], Pl. Kannappan [Result. Math. 27, No. 3–4, 368–372 (1995; Zbl 0836.39006)] and W. Fechner [J. Math. Anal. Appl. 322, No. 2, 774–786 (2006; Zbl 1101.39017)].
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.)
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