# zbMATH — the first resource for mathematics

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

##### MSC:
 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges
Full Text:
##### 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.