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 .

Theorem 3.1.

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 .

Proof.

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.

Remark 3.2.

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 [26] and called JMRassias stability by several authors [27–30].

Corollary 3.3.

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

Proof.

Define by

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, .

Definition 3.4.

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.

Theorem 3.5.

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 .

Proof.

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 ,

and

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 [46], the cubic function satisfies for every .

The following corollary is the Hyers-Ulam stability [7] of (1.5).

Corollary 3.6.

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 .

Proof.

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.