# zbMATH — the first resource for mathematics

The stability of the equation $$f(xy)-f(x)-f(y)=0$$ on groups. (English) Zbl 1021.39013
Summary: Let $$G$$ be a group and let $$E$$ be a Banach space. Suppose that a mapping $$f:G\to E$$ satisfies the relation $$\|f(xy)-f(x)-f(y) \|\leq c$$ for some $$c>0$$ and any $$x,y\in G$$. The problem of existence of mappings $$l:G\to E$$ such that the following relations hold 1) $$l(xy)-l(x)-l(y) =0$$ for any $$x,y\in G$$; 2) the set $$\{\|l(x)-f(x) \|$$; $$\forall x,y\in G\}$$ is bounded, is considered.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 39B62 Functional inequalities, including subadditivity, convexity, etc. 39B72 Systems of functional equations and inequalities
Full Text: