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Remarks on the stability of functional equations. (English) Zbl 0549.39006
Let \((G,+)\) be an abelian group and let X be a Banach space. If f:\(G\to X\) is a function such that \(\| f(x+y)+f(x-y)-2f(x)-2f(y)\|\leq \delta\) for every x,\(y\in G\) and some \(\delta >0\), then there exists a unique function g: \(G\to X\) satisfying the equation \(g(x+y)+g(x- y)=2g(x)+2g(y)\) for every x,\(y\in G\) such that \(\| f(x)-g(x)\|\leq \delta /2\) for every \(x\in G.\)
In the second part there is a short proof of a stability theorem of D. H. Hyers and S. M. Ulam [Proc. Am. Math. Soc. 3, 821-828 (1952; Zbl 0047.295)] for the inequality \(f(tx+(1-t)y)=tf(x)+(1-t)f(y).\) Finally, the author gives a counterexample for Jensen-convex functions.
Reviewer: A.Smajdor

39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
Full Text: DOI EuDML
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