Remarks on the stability of functional equations. (English) Zbl 0549.39006

Cholewa, Piotr W.

Aequationes Math. 27, 76-86 (1984).

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

MSC:

39B72	Systems of functional equations and inequalities
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

stability of functional equations; quadratic functionals; approximately convex functions; Jensen’s inequality; Banach space; counterexample; Jensen-convex functions

Citations:

Zbl 0047.295

References:

[1]	Hyers, D. H.,On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A.27 (1941), 411–416. · Zbl 0061.26403
[2]	Hyers, D. H.,Transformations with bounded m-th differences. Pacific J. Math.11 (1961), 591–602. · Zbl 0099.10501
[3]	Hyers, D. H. andUlam, M.,Approximately convex functions. Proc. Amer. Math. Soc.3 (1952), 821–828. · Zbl 0047.29505 · doi:10.1090/S0002-9939-1952-0049962-5
[4]	Rockafellar, R. T.,Convex analysis. Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
[5]	Ulam, S. M.,A collection of mathematical problems. Interscience Publishers, Inc., New York, 1960. · Zbl 0086.24101