On the stability of Cauchy additive mappings. (English) Zbl 1156.39018

The following inequality and the two other of similar type is considered:
\[ \| f(x)+f(y)+f(z)\| \leq \left\| 2f\left(\frac{x+y+z}{2}\right)\right\| ,\qquad x,y,z\in X\tag{1} \]
where \(f\colon X\to Y\), \(X,Y\) are Banach spaces. It is easy to see that a solution of the above inequality has to be an additive mapping. The aim of the paper is to show the stability of inequality (1). It is proved that if \(f\) satisfies
\[ \| f(x)+f(y)+f(z)\| \leq \left\| 2f\left(\frac{x+y+z}{2}\right)\right\| +\varepsilon\left(\| x\| ^r+\| y\| ^r+\| z\| ^r\right),\qquad x,y,z\in X\tag{2} \]
with \(r\neq 1\) or
\[ \| f(x)+f(y)+f(z)\| \leq \left\| 2f\left(\frac{x+y+z}{2}\right)\right\| +\varepsilon\left(\| x\| ^r\cdot\| y\| ^r\cdot\| z\| ^r\right),\qquad x,y,z\in X\tag{3} \]
with \(r\neq \frac{1}{3}\), then \(f\) can be approximated by an additive mapping. Namely, there exists a unique additive mapping \(a\colon X\to Y\) such that \(\| f(x)-a(x)\| \) is bounded by \(M\| x\| ^r\) or \(N\| x\| ^{3r}\) in the case of (2) or (3), respectively (with some constants \(M,N\) depending on \(\varepsilon\) and \(r\)).
In the proofs, the authors use the standard technique involving the Hyers’ sequence. However, in the reviewers opinion, it would be much simpler to reduce the considered inequalities so that the classical results might be applied directly. For example, it is easy to see that (2) yields (with some \(k\)) \[ \| f(x+y)-f(x)-f(y)\| \leq k\varepsilon (\| x\| ^r+\| y\| ^r),\qquad x,y\in X \] and the assertion follows immediately from the result of T. Aoki [J. Math. Soc. Japan 2, 64–66 (1950; Zbl 0040.35501)] (with further generalizations by Th. M. Rassias, Z. Gajda and others).


39B82 Stability, separation, extension, and related topics for functional equations


Zbl 0040.35501
Full Text: Euclid