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Fixed points and the stability of Jensen’s functional equation. (English) Zbl 1043.39010
The authors use fixed point theorems to prove generalizations of earlier results on the stability of the Jensen equation. They prove the following
Theorem: Let \(E\) be a (real or complex) linear space and let \(F\) be a Banach space; and let \(q_0:= 2\), \(q_1:= 1/2\). Suppose that \(f: E\to F\) satisfies \(f(0)= 0\) and \[ \| 2f((x+ y)/2)- f(x)- f(y)\|\leq\varphi(x, y), \] \(x,y\in E\), where \(\varphi: E\times E\to [0,\infty[\) with \(\psi(x):= \varphi(x,0)\) satisfies \(\psi(x)\leq Lq_i\psi(x/q_i)\) and \(\lim_{n\to\infty} {\varphi(2q^n_i x,2q^n_i y)\over 2q^n_i}= 0\) for all \(x,y\in E\) and some \(0\leq L< 1\) and some \(i\in \{0,1\}\).
Then there is a unique additive mapping \(j: E\to F\) such that \(\| f(x)- j(x)\|\leq {L^{1-i}\over 1-L} \psi(x)\) for all \(x\in E\).

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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