On the completeness of fuzzy normed spaces

Abstract

In \cite{mmm, mm}, the fuzzy stability problems for the Cauchy additive functional equation and the Jensen additive functional equation in fuzzy Banach spaces have been investigated.

We introduce the notion of the following Jensen type functional equation \begin{eqnarray*} f(px+ qy) = p f(x) + q f(y) \end{eqnarray*} for positive integers $p, q$ with $p \ne 1$. We prove that if every approximately Jensen type additive mapping from the set $\mathbb N$ of positive integers into a fuzzy normed vector space can be approximated by a Jensen type additive mapping, then the fuzzy normed vector space is complete. And we also get the same result for approximately quadratic mapping and quadratic mapping.