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Zbl 1221.39036
Eshaghi Gordji, M.; Khodaei, H.
The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces.
(English)
[J] Discrete Dyn. Nat. Soc. 2010, Article ID 140767, 15 p. (2010). ISSN 1026-0226; ISSN 1607-887X/e

Suppose that $X$ is a linear space, $(Z,N')$ a fuzzy normed space, $(Y,N)$ a fuzzy Banach space, $f: X\to Y$, $n\geq 2$ a fixed integer. The authors consider the stability of the functional equation $$\Delta f(x_1,\dots, x_n)=0$$ where $$\Delta f(x_1,\dots, x_n)=\sum_{i=1}^{n}f\left(x_i-\frac{1}{n}\sum_{j=1}^{n}x_j\right)-\sum_{i=1}^{n}f(x_i)+nf\left(\frac{1}{n}\sum_{i=1}^{n}x_i\right).$$ The main result reads, roughly, as follows. Suppose that $f: X\to Y$ satisfy $f(0)=0$ and $$N(\Delta f(x_1,\dots,x_n),t_1+\cdots+t_n)\geq \min\{N'(\phi(x_1),t_1),\dots,N'(\phi(x_n),t_n)\}$$ for all $x_1,\dots,x_n\in X$, $t_1,\dots,t_n>0$ where $\phi: X\to (Z,N')$ is a control mapping satisfying $\phi(2x)=\alpha \phi(x)$, $x\in X$ with some $|\alpha|<2$. Then, there exists a unique quadratic function $Q: X\to Y$ and a unique additive function $A: X\to Y$ such that the mapping $Q+A$ approximates $f$ (in terms of the fuzzy norm $N$). Moreover, if $f$ is odd, it can be approximated by an additive function $A$ and if $f$ is even, a quadratic mapping $Q$ approximates $f$.
[Jacek Chmieliński (Kraków)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
46S40 Fuzzy functional analysis
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
39B52 Functional equations for functions with more general domains

Keywords: stability of functional equations; fuzzy norm; fixed point theorem; fixed point method; fuzzy approximation; inner product spaces; fuzzy normed space; fuzzy Banach space; quadratic function; additive function

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