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.