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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
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