A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation. (English) Zbl 0933.39053

The following generalization of the stability of the Jensen’s equation in the spirit of Hyers-Ulam-Rassias is proved: Let \(V\) be a normed space, \(X\) – a Banach space, \(p<1\) and \(0\leq a<3\). If \(f:V\rightarrow X\) satisfies \[ \left\|2f\left({x+y\over 2}\right)-f(x)-f(y)\right\|\leq \|x\|^p+\|y\|^p \] for all \(x,y\in V\) with \(\|x\|,\|y\|>a\), then there exists a unique additive mapping \(T:V\rightarrow X\) such that \[ \left\|f(x)-T(x)-f(0)\right\|\leq {3+3^p\over 3-3^p}\|x\|^p \] for all \(x\in V\) with \(\|x\|>a\). For the case \(p>1\) a corresponding result is obtained.


39B82 Stability, separation, extension, and related topics for functional equations
39B62 Functional inequalities, including subadditivity, convexity, etc.
39B52 Functional equations for functions with more general domains and/or ranges
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