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On stability of the homogeneity condition. (English) Zbl 0829.39011
The paper deals with the stability (in the sense of Hyers-Ulam) of the homogeneity condition. The main result is the following Theorem: Let $$S$$ be a cone in a real vector space $$X$$ and $$f$$ maps $$S$$ into a sequentially complete locally convex linear topological Hausdorff space $$Y$$. If there exist $$A \subset (1,+\infty)$$, $$\text{int }A\neq \emptyset$$, and a bounded subset $$V$$ of $$Y$$ such that $$\alpha^{-1} f(\alpha x)-f(x) \in V$$, $$\alpha \in A$$, $$x \in S$$, then there exists a unique positively homogeneous mapping $$F : S \to Y$$ such that $$F(x)-f(x) \in c(c-1)^{-1} \text{seq cl conv} (V \cup \{0\})$$, $$x \in S$$, where $$c = \text{sup} A$$.
The same conclusion holds if $$X$$ is a topological vector space, $$f$$ is continuous and $$A$$ contains $$\alpha, \beta$$ with $$\log \alpha/ \log \beta$$ irrational. From this result the authors obtain also a stability theorem for linear functions.
Reviewer: G.L.Forti (Milano)

##### MSC:
 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
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##### References:
 [1] Z. Gajda, On stability of the Cauchy equation on semigroups, Aequationes Math. 3 (1988) 76–79. · Zbl 0658.39006 · doi:10.1007/BF01837972 [2] J. Matkowski, Cauchy functional equations on a restricted domain and commuting functions, In Iteration Theory and its Functional Equations, (Proc. Schloss Hoffen, 1984), Lecture Notes in Math. Vol. 1163, Springer, Berlin-Heidelberg-New York-Tokyo, 1985, 101–106. [3] J. Tabor, On approximate by linear mappings, (submitted). · Zbl 0844.39011 [4] J. Tabor, Jr., J. Tabor, Homogeneity is superstable, (to appear). · Zbl 0823.39008
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