×

Stability of the generalized alternative Cauchy equation. (English) Zbl 0951.39007

Let \(G\) be a commutative semigroup and let \(L\) be a complete Archimedean Riesz space. Suppose that \(F: G\to L\) satisfies for some \(e\in L_+\) the inequality \[ \biggl||F(x+ y)|-|F(x)+ F(y)|\biggr|\leq e\quad\text{for }x,y\in G. \] Then there exists a unique additive mapping \(A: G\to L\) such that \[ |F(x)- A(x)|\leq e\quad\text{for }x\in G. \]

MSC:

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

References:

[1] Batko, B.; Tabor, J., Stability of an alternative Cauchy equation on a restricted domain, Aequat. Math., 57, 221-232 (1999) · Zbl 0935.39011 · doi:10.1007/s000100050079
[2] Ger, R., On a characterization of strictly convex spaces, Atti della Acad. delle Sci. di Torino, 127, 131-138 (1993) · Zbl 0843.46010
[3] W. A. J. Luxemburg andA. C. Zaanen,Riesz Spaces. North-Holland, 1971. · Zbl 0231.46014
[4] F. Skof, On the stability of functional equations on a restricted domain and a related topic. In: J. Tabor, T. M. Rassias (ed.)Stability of mappings of Hyers-Ulam type, Hadronic Press (1994, 141-151. · Zbl 0844.39006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.