Fuzzy stability of quartic mappings (Q2881398)

Let \(Df(x,y):=f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)\). The author proves the stability of the quartic equation \(Df(x,y)=0\) for mappings \(f: X\to Y\) from a linear space \(X\) into a fuzzy Banach space \((Y,N)\) with a control mapping \(\varphi: X\times X\to Z\) where \((Z,N')\) is a fuzzy normed space. Namely, assuming that NEWLINE\[NEWLINE N(Df(x,y),t)\geq N'(\varphi(x,y),t) NEWLINE\]NEWLINE and under some additional assumptions on \(\varphi\), there exists a unique quartic mapping \(Q: X\to Y\) in some sense close to \(f\).NEWLINENEWLINEAs an application, the classical Hyers-Ulam stability of the quartic equation in Banach spaces is proved.