## A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation.(English)Zbl 0957.39008

Let $$V$$ be a normed vector space and $$X$$ a Banach space, and let $$f,g,h: V\to X$$. The authors prove that the Pexider equation $f(x+y)= g(x)+h(y)$ is stable in the following sense: If there exists a real number $$p\neq 1$$, such that $\bigl\|f(x+y)- g(x)-h(y) \bigr\|\leq\|x \|^p+ \|y\|^p$ for all $$x,y\in V\setminus \{0\}$$, then there exists exactly one additive map $$T:V\to X$$ such that $\bigl\|f(x)- T(x)-f(0) \bigr\|\leq C(p)\|x\|^p$ for all $$x\in V$$. Here $$C(p)$$ is a certain specified constant.

### MSC:

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

### Keywords:

Hyers-Ulam stability; Banach space; Pexider equation
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### References:

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