Stability of generalized \(n\)-dimensional cubic functional equation in fuzzy normed spaces (Q2913886)

The authors consider the fuzzy stability of the \(n\)-dimensional cubic functional equation: NEWLINE\[NEWLINE\begin{multlined} f\left(ax_1+b\sum_{i=2}^{n}x_i\right)+f\left(ax_1-b\sum_{i=2}^{n}x_i\right)+2a(b^2-a^2)f(x_1) \\ =ab^2\left[f\left(\sum_{i=1}^{n}x_i\right)+f\left(x_1-\sum_{i=2}^{n}x_i\right)\right].\end{multlined} \tag{1}NEWLINE\]NEWLINENEWLINENEWLINELet \(X\) be a linear space, \((Z,N')\) a fuzzy normed space, \((Y,N)\) a fuzzy Banach space. Let \(Df(x_1,\dots,x_n)\) denote the difference between the left and right hand sides of (1). Let \(\alpha: X^n\to Z\) be a control mapping (satisfying several conditions). The main result of the paper states that if \(f: X\to Y\) satisfies NEWLINE\[NEWLINE N(Df(x_1,\dots,x_n),r)\geq N'(\alpha(x_1,\dots,x_n),r)\qquad \text{for all}\;x_1,\dots,x_n\in X,~r>0, NEWLINE\]NEWLINE then there exists a unique mapping \(C: X\to Y\) satisfying (1) and close to \(f\), namely such that NEWLINE\[NEWLINE N(f(x)-C(x),r)\geq N'(\alpha(x,0,\dots,0),r|a^3-d|)\qquad \text{for all}\;x\in X,~r>0 NEWLINE\]NEWLINE (\(d\) is some constant related to \(\alpha\)).