We will investigate the generalized Hyers-Ulam type theorem of the functional equation (1.5) in fuzzy normed spaces. In the following theorem, we will show that under special circumstances on the control function , every -almost cubic mapping can be approximated by a cubic mapping .
Let . Let be a linear space, and let be a fuzzy normed space. Suppose that an even function satisfies for all and for all . Suppose that is a fuzzy Banach space. If a function satisfies
for all and , then there exists a unique cubic function which satisfies (1.5) and the inequality
holds for all and .
We have the following two cases.
Case 1 ().
Replacing by in (3.1) and summing the resulting inequality with (3.1), we get
Since (3.1) and (3.3) hold for any , let us fix for convenience. By (), we have
Replacing by in (3.4). By (), we have
Replacing by in (3.5), we get
It follows from
and last inequality that
In order to prove convergence of the sequence , we replace by to find that for ,
Replacing by in last inequality to get
For every and , we put
Replacing by in last inequality, we observe that
Let and be given. Since , there is some such that for every . The convergence of the series gives some such that for every and . For every and , we have
This shows that is a Cauchy sequence in the fuzzy Banach space , therefore this sequence converges to some point . Fix and put in (3.13) to obtain
For every ,
The first two terms on the right hand side of the above inequality tend to 1 as . Therefore we have
for large enough. By last inequality, we have
Now, we show that is cubic. Use inequality (3.1) with replaced by and by to find that
On the other hand , hence by ()
We conclude that fulfills (1.5). It remains to prove the uniqueness assertion. Let be another cubic mapping satisfying (3.17). Fix . Obviously
for all . For every , we can write
Since , we have
Therefore for all , whence .
Case 2 ().
We can state the proof in the same pattern as we did in the first case. Replace , by and , respectively in (3.4) to get
We replace and by and in last inequality, respectively, we find that
For each , one can deduce
where . It is easy to see that is a Cauchy sequence in (). Since is a fuzzy Banach space, this sequence converges to some point , that is,
Moreover, satisfies (1.5) and
The proof for uniqueness of for this case proceeds similarly to that in the previous case, hence it is omitted.
We note that need not be equal to 27. But we do not guarantee whether the cubic equation is stable in the sense of Hyers, Ulam and Rassias if is assumed in Theorem 3.1.
Let . Suppose that the mapping from into is right continuous. Then we get a fuzzy approximation better than (3.17) as follows.
For every , we have
for large enough . It follows that
Tending to zero we infer
From Theorem 3.1, we obtain the following corollary concerning the stability of (1.5) in the sense of the JMRassias stability of functional equations controlled by the mixed product-sum of powers of norms introduced by J. M. Rassias  and called JMRassias stability by several authors [27–30].
Let be a Banach space and let be a real number. Suppose that a function satisfies
for all where . Then there exists a unique cubic function which satisfying (1.5) and the inequality
for all . The function is given by for all
It is easy to see that is a fuzzy Banach space. Denote by the map sending each to . By assumption,
Note that given by
is a fuzzy norm on . By Theorem 3.1, there exists a unique cubic function satisfies (1.5) and inequality
for all and . Consequently, .
Let be a mapping where and are fuzzy normed spaces. is said to be sequentially fuzzy continuous at if for any satisfying implies . If is sequentially fuzzy continuous at each point of , then is said to be sequentially fuzzy continuous on .
For the various definitions of continuity and also defining a topology on a fuzzy normed space we refer the interested reader to [61, 62]. Now we examine some conditions under which the cubic mapping found in Theorem 3.1 to be continuous. In the following theorem, we investigate fuzzy sequentially continuity of cubic mappings in fuzzy normed spaces. Indeed, we will show that under some extra conditions on Theorem 3.1, the cubic mapping is fuzzy sequentially continuous.
Denote the fuzzy norm obtained as Corollary 3.3 on . Suppose that conditions of Theorem 3.1 hold. If for every the mappings (from into and (from into are sequentially fuzzy continuous, then the mapping is sequentially continuous and for all .
We have the following case.
Case 1 ().
Let be a sequence in that converges to some , and let . Let be given. Since ,
there is such that
It follows form (3.17) and (3.38) that
By the sequentially fuzzy continuity of maps and , we can find some such that for any ,
Hence by last inequality and (3.38), we get
On the other hand,
Hence by last inequality and (3.42), we obtain
Therefore it follows from (3.44), (3.40) and (3.39) that for every ,
Therefore for every choice , and , we can find some such that for every . This shows that .
The proof for proceeds similarly to that in the previous case.
It is not hard to see that for every rational number . Since is a fuzzy sequentially continuous map, by the same reasoning as the proof of , the cubic function satisfies for every .
The following corollary is the Hyers-Ulam stability  of (1.5).
Let be a Banach space, and let be a real number. Suppose that a function satisfies
for all . Then there exists a unique cubic function which satisfies (1.5) and the inequality
for all . Moreover, if for each fixed the mapping from to is fuzzy sequentially continuous, then for all .
Denote and the fuzzy norms obtained as Corollary 3.3 on and , respectively. This time we choose . By Theorem 3.1, there exists a unique cubic function which satisfies the inequality
for all . It follows that . The rest of proof is an immediate consequence of Theorem 3.5.