Functional inequalities associated with additive mappings
Consider a Banach space \(X,\) an additive group \(G\) divisible by 2 and mappings \(\varphi:G\times G\times G\rightarrow[0,\infty)\) such that \(\varphi(2x,-x,0)=\varphi(0,x,-x)\) and \(\lim_{n\rightarrow\infty} (1/2^{n})\varphi(2^{n+1}x,2^{n}y,2^{n}z)=0\) (or \(\lim_{n\rightarrow\infty }2^{n}\varphi(x/2^{n-1},y/2^{n},z/2^{n})=0)\) for all \(x,y,z\in G.\) Under these hypotheses it is proved that any mapping \(f:G\rightarrow X\) that verifies the functional inequality \(\left\| f(x)+2f(y)+2f(z)\right\| \leq\left\| 2f(x/2+y+z)\right\| +\varphi(x,y,z)\) for all \(x,y,z\in G\) is additive. This result is related to the paper of \textit{Gy. Maksa} and \textit{P. Volkmann} [Publ. Math. 56, No. 1--2, 197--200 (2000; Zbl 0991.39015)].
scientific article; zbMATH DE number 5352107
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