On stability of the homogeneity condition. (English) Zbl 0829.39011

Kominek, Zygfryd; Matkowski, Janusz

Result. Math. 27, No. 3-4, 373-380 (1995).

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

Keywords:

Hyers-Ulam stability; homogeneity condition; sequentially complete locally convex linear topological Hausdorff space; homogeneous mapping

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