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A fixed point approach to stability of a quadratic equation. (English) Zbl 1118.39015

The authors prove the orthogonal stability of the quadratic functional equation of Pexider type by using the fixed point alternative theorem. The following theorem is the main result of this paper: Suppose that $X$ is a real orthogonality space with a symmetric orthogonal relation $\perp$ and $Y$ is a Banach space. Let the mappings $f,g,h,k:X\to Y$ satisfy

$\parallel f\left(x+y\right)+g\left(x-y\right)-h\left(x\right)-k\left(y\right)\parallel \le \epsilon$

for all $x,y\in X$ with $x\perp y$. There exist an orthogonally additive mapping $T:X\to Y$ and a constant ${C}_{1}\ge 0$ such that

$\parallel f\left(x\right)-T\left(x\right)\parallel \le {C}_{1}\epsilon \phantom{\rule{1.em}{0ex}}\left(\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in X\right)$

if and only if there exists a constant ${C}_{2}\ge 0$ with

$\parallel f\left(2x\right)-f\left(-2x\right)-4f\left(x\right)-4f\left(-x\right)\parallel \le {C}_{2}\epsilon \phantom{\rule{1.em}{0ex}}\left(\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in X\right)·$

Indeed, if $f:X\to Y$ satisfies

$\parallel f\left(2x\right)-f\left(-2x\right)-4f\left(x\right)-4f\left(-x\right)\parallel \le \epsilon \phantom{\rule{1.em}{0ex}}\left(\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}x\in X\right),$

then there exist orthogonally additive mappings ${T}_{1},{T}_{2},{T}_{3}:X\to Y$ such that

$\begin{array}{cc}\hfill \parallel f\left(x\right)-f\left(0\right)-{T}_{1}\left(x\right)\parallel & \le \frac{140}{3}\epsilon ,\hfill \\ \hfill \parallel g\left(x\right)-g\left(0\right)-{T}_{2}\left(x\right)\parallel & \le \frac{98}{3}\epsilon ,\hfill \\ \hfill \parallel h\left(x\right)+k\left(x\right)-h\left(0\right)-k\left(0\right)-{T}_{3}\left(x\right)\parallel & \le \frac{256}{3}\epsilon \hfill \end{array}$

for all $x\in X$.

In the Introduction, quoting the stability of quadratic equations of Pexider type, the authors missed citing a paper “Stability of the quadratic equation of Pexider type” [Abh. Math. Semin. Univ. Hamb. 70, 175–190 (2000; Zbl 0991.39018)] by S.-M. Jung, which contains the first result about the stability of quadratic equations of Pexider type (see Theorem 5 and Corollary 6 in the paper).

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B55 Orthogonal additivity and other conditional functional equations 39B52 Functional equations for functions with more general domains and/or ranges