Nonlinear $\mathcal{L}$--Fuzzy stability of k-cubic functional equation

Abstract

We establish the stability result for the k-cubic functional
equation
$$ 2 [ k f ( x + k y ) + f ( k x - y ) ] = k ( k ^2 + 1 ) [ f ( x + y ) + f ( x - y ) ]
+ 2 ( k ^ 4 - 1 ) f ( y )
,$$
where $k $ is a real number different from $0$ and $1$,
in the setting of various $\mathcal{L}$--fuzzy normed spaces that
in turn generalize a Hyers--Ulam stability result in the framework
of classical normed spaces. First we shall prove the stability
of k-cubic functional equations in the $\mathcal{L}$-fuzzy
normed space under arbitrary $t$-norm which generalizes previous works. Then, we shall prove the stability of k-cubic functional
equations in the non-Archimedean $\mathcal{L}$-fuzzy normed space.
We therefore provide a link among different disciplines: fuzzy
set theory, lattice theory, non-Archimedean spaces, and mathematical
analysis.