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The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces. (English) Zbl 1221.39036
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 $f$.

39B82Stability, separation, extension, and related topics
46S40Fuzzy functional analysis
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
39B52Functional equations for functions with more general domains and/or ranges
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