Stability of an additive functional inequality (Q2913874)

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 an odd 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\[NEWLINE \psi(x,y,z):=\sum_{j=0}^{\infty} \frac{1}{2^j} \phi(2^{j+1}x,-2^jy,-2^jz) < \infty NEWLINE\]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\[NEWLINE \| f(x)-A(x)\| \leq \frac{1}{2}\psi(x,x,x) NEWLINE\]NEWLINE for all \(x \in \mathcal X\).NEWLINENEWLINEThe proof is based on the standard direct method.