### 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.

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