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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.

MSC:

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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References:

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