Fixed points and the stability of Jensen’s functional equation. (English) Zbl 1043.39010

Cǎdariu, Liviu; Radu, Viorel

JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4, 7 p. (2003).

The authors use fixed point theorems to prove generalizations of earlier results on the stability of the Jensen equation. They prove the following
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 \[ \| 2f((x+ y)/2)- f(x)- f(y)\|\leq\varphi(x, y), \] \(x,y\in E\), where \(\varphi: E\times E\to [0,\infty[\) with \(\psi(x):= \varphi(x,0)\) satisfies \(\psi(x)\leq Lq_i\psi(x/q_i)\) 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 \(0\leq L< 1\) and some \(i\in \{0,1\}\).
Then there is a unique additive mapping \(j: E\to F\) such that \(\| f(x)- j(x)\|\leq {L^{1-i}\over 1-L} \psi(x)\) for all \(x\in E\).

Reviewer: Jens Schwaiger (Graz)

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

Jensen’s equation; fixed point theorems; Banach space; additive mapping; stability