On the Jensen's equation in Banach modules.
\textit{Y.-H. Lee} and \textit{K.-W. Jun} [J. Math. Anal. Appl. 238, 305--315 (1999; Zbl 0933.39053)] proved the following Theorem: Let \(X\) be a Banach space, \(G\) be an Abelian group and~\(E\) be a~subset of \(G\) such that \(nx\in E\) for any integer~\(n\) and for all \(x\in E\) and \(2x\neq 0\), \(3x\neq 0\) for all \(x\in E\setminus\{0\}\). Let \(\varphi :E\setminus\{0\}\times E\setminus\{0\}\to[0,\infty)\) be a mapping such that \[ \widetilde\varphi(x,y)= \sum_{k=0}^{\infty} 3^{-k} \varphi(3^k x, 3^ky)< \infty\tag{1} \] for all \(x,y\in E\setminus \{0 \}\). Let \(f:E\to X\) be a mapping such that \(\| 2f(\frac {x+y}{2})- f(x) -f(y)\|\leq \varphi(x,y)\) for all \(x,y\in E\setminus\{0\}\) with \(\frac{x+y}{2}\in E\). Then there exists a unique mapping \(T:E\to X\) such that \(T(x+y)= T(x)+T(y)\) for all \(x,y\in E\) with \(x+y\in E\) and \[ \| f(x)-T(x)-f(0) \|\leq \tfrac{1}{3} (\widetilde\varphi(x,-x)+ \widetilde\varphi(-x,3x)) \tag{2} \] for all \(x\in E\setminus\{0\}\). In the present paper the Hyers-Ulam-Rassias stability of the Jensen's equation in Banach modules over a~Banach algebra is proved. Let~\(B\) be a unital Banach algebra and \(B_1\) the set of all elements of \(B\) having norm 1. Let \(_B{\mathbb B}_1\), \({}_B{\mathbb B}_2\) be left Banach \(B \)-modules. Using the theorem above the following result is obtained. Theorem: Let \(f:{}_B{\mathbb B}_1\to{}_B{\mathbb B}_2\) be a mapping for which there exists a function \(\varphi:{}_B{\mathbb B}_1\setminus \{0\} \times{}_B {\mathbb B}_1\setminus\{0\}\to[0,\infty)\) satisfying~(1) for all \(x,y\in{}_B {\mathbb B}_1\setminus\{0\}\). Assume that \[ \biggl\| 2f \Bigl(\frac{ax+ ay} {2}\Bigr)-af(x)-af(y)\biggr\|\leq\varphi(x,y) \] for all \(a\in B_1\cup {\mathbb R}^+\), \(x,y\in{}_B{\mathbb B}_1\setminus\{0\}\). Then there exists a unique \(B\)-linear mapping \(T:{}_B{\mathbb B}_1\to{}_B {\mathbb B}_2\) such that~(2) holds for all \(x\in{}_B{\mathbb B}_1\setminus \{0\}\). The corresponding result is proved under assumption \(\widetilde\varphi (x,y)= \sum_{k=0}^{\infty}3^k \varphi(3^{-k}x, 3^{-k} y) <\infty\). Some special results obtained by an appropriate specification of \(\varphi\) are discussed. Similar results are proved for a unital Banach \(\ast\)-algebra over \({\mathbb C}\) [see also \textit{W.-G. Park}, Nonlinear Funct. Anal. Appl. 7, 405--412 (2002; Zbl 1023.39023)].
scientific article; zbMATH DE number 1931472