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Zbl 1043.39010
Fixed points and the stability of Jensen's functional equation.
(English)
[J] JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4, 7 p., electronic only (2003). ISSN 1443-5756/e

The authors use fixed point theorems to prove generalizations of earlier results on the stability of the Jensen equation. They prove the following\par Theorem: Let $E$ be a (real or complex) linear space and let $F$ be a Banach space; and let $q_0:= 2$, $q_1:= 1/2$. Suppose that $f: E\to F$ satisfies $f(0)= 0$ and $$\Vert 2f((x+ y)/2)- f(x)- f(y)\Vert\le\varphi(x, y),$$ $x,y\in E$, where $\varphi: E\times E\to [0,\infty[$ with $\psi(x):= \varphi(x,0)$ satisfies $\psi(x)\le Lq_i\psi(x/q_i)$ and $\lim_{n\to\infty} {\varphi(2q^n_i x,2q^n_i y)\over 2q^n_i}= 0$ for all $x,y\in E$ and some $0\le L< 1$ and some $i\in \{0,1\}$.\par Then there is a unique additive mapping $j: E\to F$ such that $\Vert f(x)- j(x)\Vert\le {L^{1-i}\over 1-L} \psi(x)$ for all $x\in E$.
[Jens Schwaiger (Graz)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: Jensen's equation; fixed point theorems; Banach space; additive mapping; stability

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