On the stability of Cauchy additive mappings
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 \textit{T. Aoki} [J. Math. Soc. Japan 2, 64--66 (1950; Zbl 0040.35501)] (with further generalizations by \textit{Th. M. Rassias}, \textit{Z. Gajda} and others).
scientific article; zbMATH DE number 5363263