Fuzzy stability for the functional equation stemming from quadratic-additive mapping (Q2913917)

Let \((X,N)\) be a fuzzy normed space and \((Y,N')\) a fuzzy Banach space. Let \(q\in (0,\infty)\setminus\{\frac{1}{2},1\}\). For a mapping \(f: X\to Y\) denote NEWLINE\[NEWLINE Df(x,y):=f(x-y)-f(-x+y)-4f(x)+f(2x)-f(-y)+f(y). NEWLINE\]NEWLINE It is proved that if \(f\) satisfies \(f(0)=0\) and NEWLINE\[NEWLINE N'(Df(x,y),t+s)\geq \min\{N(x,s^q),N(y,t^q)\}\qquad \text{for all}\;x,y\in X,\;s,t>0, NEWLINE\]NEWLINE then there exists a unique mapping \(F: X\to Y\) satisfying \(Df(x,y)=0\) for all \(x,y\in X\) and close to \(f\); namely such that NEWLINE\[NEWLINE N'(f(x)-F(x),t)\geq \varphi(x,t),\qquad x\in X,\;t>0 NEWLINE\]NEWLINE where \(\varphi (x,t)\) is explicitly given in the paper. As a corollary it is shown that for a normed space \(X\) and for a Banach space \(Y\), each mapping \(f: X\to Y\) satisfying NEWLINE\[NEWLINE \|Df(x,y)\|\leq \|x\|^p+\|y\|^p,\qquad x,y\in X, NEWLINE\]NEWLINE with some \(p\neq 1,2\), admits a unique mapping \(F: X\to Y\) satisfying \(DF=0\) and close to \(f\). In particular, for \(p\in (0,1)\) the estimation is given by NEWLINE\[NEWLINE \|f(x)-F(x)\|\leq\frac{2\|x\|^p}{2-2^p},\qquad x\in X\setminus \{0\}. NEWLINE\]