Approximate additive mappings related to a Cauchy additive functional inequality (Q2913891)

Let \(f:\mathcal X \to \mathcal Y\), \(\mathcal X\) be a normed space and \(\mathcal Y\) be a Banach space. After showing that \(f\) is additive if and only if it satisfies the inequality NEWLINE\[NEWLINE \| f(x)+f(y)+f(-z) \| \leq \| f(x+y)-f(z) \|, NEWLINE\]NEWLINE the authors prove the following stability result:NEWLINENEWLINETheorem: Let \(f:\mathcal X \to \mathcal Y\), \(\mathcal X\) be a mapping. If there exists a function \(\phi: \mathcal X^3 \to [0,\infty)\) satisfying NEWLINE\[NEWLINE \| f(x)+f(y)-f(z) \| \leq \| f(x+y)-f(z) \|+\phi(x,y,z) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \psi(x,y,z):=\sum_{j=0}^{\infty} \frac{1}{2^j} \phi((-2)^jx,(-2^j)y,(-2^j)z) < \infty NEWLINE\]NEWLINE for all \(x, y, z \in \mathcal X\), then there exists a unique additive mapping \(A:\mathcal X \to \mathcal Y\) such that NEWLINE\[NEWLINE \| f(x)-A(x)\| \leq \frac{1}{2}\psi(x,x,2x) NEWLINE\]NEWLINE for all \(x \in \mathcal X\).NEWLINENEWLINEThe proof is based on the standard direct method.NEWLINENEWLINEAn analogous stability theorem is proved when \(\mathcal X\) and \(\mathcal Y\) are non-Archimedean.NEWLINENEWLINE\vskip 4mmNEWLINENEWLINEReviewers's remark: The quoted generalization of Rassias' theorem due to Gǎvruta, is a special case of a result published in 1980 by the reviewer [\textit{G. L. Forti}, Stochastica 4, No. 1, 23--30 (1980; Zbl 0442.39005)].