Stability of an additive functional inequality with the fixed point alternative
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\[NEWLINE \| f(x)+2f(y)+2f(z) \| \leq \| f(x)+f(2y+2z) \|, NEWLINE\]NEWLINE the authors prove the following stability result:NEWLINENEWLINETheorem: Suppose that an odd mapping \(f:\mathcal X \to \mathcal Y\), \(\mathcal X\) satisfies the inequality NEWLINE\[NEWLINE \| f(x)+2f(y)+2f(z) \| \leq \| f(x)+f(2y+2z) \|+\phi(x,y,z) NEWLINE\]NEWLINE where \(\phi: \mathcal X^3 \to [0,\infty)\) is a given function. If there exists \(L <1\) such that NEWLINE\[NEWLINE \phi(x,y,z) \leq \frac{1}{2} L\phi(2x,2y,2z) NEWLINE\]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\[NEWLINE \| f(x)-A(x)\| \leq \frac{L}{2-2L}\phi(2x,-x,0) NEWLINE\]NEWLINE for all \(x \in \mathcal X\).NEWLINENEWLINEThe proof is based on the fixed point method.NEWLINENEWLINEReviewer'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)].
scientific article; zbMATH DE number 6085204