The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces
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 \(f\).
scientific article; zbMATH DE number 5793477
scientific article