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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:
39B82Stability, separation, extension, and related topics
39B62Functional inequalities, including subadditivity, convexity, etc. (functional equations)
39B52Functional equations for functions with more general domains and/or ranges
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References:
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