## 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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