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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, $\left(Z,{N}^{\text{'}}\right)$ a fuzzy normed space, $\left(Y,N\right)$ a fuzzy Banach space, $f:X\to Y$, $n\ge 2$ a fixed integer.

The authors consider the stability of the functional equation

${\Delta }f\left({x}_{1},\cdots ,{x}_{n}\right)=0$

where

${\Delta }f\left({x}_{1},\cdots ,{x}_{n}\right)=\sum _{i=1}^{n}f\left({x}_{i}-\frac{1}{n}\sum _{j=1}^{n}{x}_{j}\right)-\sum _{i=1}^{n}f\left({x}_{i}\right)+nf\left(\frac{1}{n}\sum _{i=1}^{n}{x}_{i}\right)·$

The main result reads, roughly, as follows. Suppose that $f:X\to Y$ satisfy $f\left(0\right)=0$ and

$N\left({\Delta }f\left({x}_{1},\cdots ,{x}_{n}\right),{t}_{1}+\cdots +{t}_{n}\right)\ge min\left\{{N}^{\text{'}}\left(\phi \left({x}_{1}\right),{t}_{1}\right),\cdots ,{N}^{\text{'}}\left(\phi \left({x}_{n}\right),{t}_{n}\right)\right\}$

for all ${x}_{1},\cdots ,{x}_{n}\in X$, ${t}_{1},\cdots ,{t}_{n}>0$ where $\phi :X\to \left(Z,{N}^{\text{'}}\right)$ is a control mapping satisfying $\phi \left(2x\right)=\alpha \phi \left(x\right)$, $x\in X$ 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 $f$.

##### MSC:
 39B82 Stability, separation, extension, and related topics 46S40 Fuzzy functional analysis 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 39B52 Functional equations for functions with more general domains and/or ranges