Abstract
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 $f: S\to X$ 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 $\varphi : S \to X$ 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 Jung [J. Math. Anal. Appl. 222 (1998), 126--137], Kannappan [Results Math. 27 (1995), 368--372] and Fechner [J. Math. Anal. Appl. 322 (2006), 774--786].
Citation
Valeriy A. Fa\u iziev. Prasanna K. Sahoo. "Stability of a functional equation of Whitehead on semigroups." Ann. Funct. Anal. 3 (2) 32 - 57, 2012. https://doi.org/10.15352/afa/1399899931
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