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Stability of an Additive-Cubic-Quartic Functional Equation
Advances in Difference Equations volume 2009, Article number: 395693 (2010)
Abstract
In this paper, we consider the additive-cubic-quartic functional equation and prove the generalized Hyers-Ulam stability of the additive-cubic-quartic functional equation in Banach spaces.
1. Introduction
The stability problem of functional equations is originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influence in the development of what we callgeneralized Hyers-Ulam stability or asHyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Gvruta [8] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach (see [2, 5–13]).
Jun and Kim [14] introduced and investigate the following functional equation:

and prove the generalized Hyers-Ulam stability for the functional equation (1.1). Obviously, the function satisfies the functional equation (1.1), which is called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic mapping. Jun and Kim proved that a mapping
between two real vector spaces
and
is a solution of (1.1) if and only if there exists a unique mapping
such that
for all
; moreover,
is symmetric for each fixed one variable and is additive for fixed two variables.
In [15], Park and Bae considered the following quartic functional equation:

In fact, they proved that a mapping between two real vector spaces
and
is a solution of (1.2) if and only if there exists a unique symmetric multi-additive mapping
such that
for all
(see [7, 11]). It is easy to show that the function
satisfies the functional equation (1.2), which is called a quartic functional equation. Every solution of the quartic functional equation is said to be a quartic mapping.
In this paper, we aim to deal with the next functional equation derived from additive, cubic, and quadric mappings,

It is easy to show that the function satisfies the functional equation (1.3). We establish the general solution and prove the generalized Hyers-Ulam stability for the functional equation (1.3).
2. An Additive-Cubic-Quartic Functional Equation
Throughout this section, and
will be real vector spaces. Before proceeding the proof of Theorem 2.4 which is the main result in this section, we shall need the following two lemmas.
Lemma 2.1.
If an even mapping satisfies (1.3), then
is quartic.
Proof.
Putting in (1.3), we get
. Setting
in (1.3), by the evenness of
, we obtain

for all . Hence (1.3) can be written as

for all . Replacing
by
in (1.3), we obtain

for all . By (2.1) and (2.3), we obtain

for all . According to (2.4), (2.2) can be written as

for all . This shows that
is quartic, which completes the proof of the lemma.
Lemma 2.2.
If an odd mapping satisfies (1.3), then f is cubic-additive.
Proof.
We show that the mappings and
, respectively, defined by
and
, are additive and cubic, respectively.
Since is odd,
. Letting
in (1.3), we obtain

for all . Hence (1.3) can be written as

for all . Replacing
by
and
in (2.7), respectively, we get

for all . Replacing
by
in (2.7), we obtain

for all . Replacing
by
in (2.9), we get

for all . Replacing
by
and
by
in (2.9), we get

for all . Replacing
by
in (2.11), we get

for all .
Subtracting (2.12) from (2.10), we obtain

for all . By (2.8) and (2.13), we obtain

for all .
Replacing by
in (2.14), we get

for all .
By (2.14) and (2.15), we obtain

for all .
By (2.7) and (2.16), we have

for all . Replacing
by
in (2.17), we get

for all . Replacing
by
in (2.18), respectively, we get

for all .
By (2.18) and (2.19), we obtain

for all . Replacing
by
in (2.17), respectively, we get

for all . Thus it follows from (2.20) and (2.21) that

for all . Replacing
by
in (2.22), we obtain

for all . Replacing
by
in (2.23), respectively, we get

for all . By (2.23) and (2.24), we obtain

for all . Adding (2.22) to (2.25) and using (2.17), we get

for all . The last equality means that

for all Thus the mapping
is additive.
Replacing by
in (2.17), respectively, we get

for all . Since
for all

for all Hence it follows from (2.17) and (2.28) that

for all Thus the mapping
is cubic.
On the other hand, we have for all
This means that
is cubic-additive. This completes the proof of the lemma.
The following is suggested by an anonymous referee.
Remark 2.3.
The functional equation (1.3) is equivalent to the functional equation

The left hand side is even with respect to and the right hand side is odd by the assumption of Lemma 2.2. Thus

So we conclude that , as desired.
Theorem 2.4.
If a mapping satisfies (1.3) for all
, then there exist a unique additive mapping
a unique mapping
, and a unique symmetric multi-additive mapping
such that
for all
and that
is symmetric for each fixed one variable and is additive for fixed two variables.
Proof.
Let satisfy (1.3). We decompose
into the even part and the odd part by setting

for all By (1.3), we have

for all This means that
satisfies (1.3). Similarly we can show that
satisfies (1.3). By Lemmas 2.1 and 2.2,
and
are quartic and cubic-additive, respectively. Thus there exist a unique additive mapping
a unique mapping
, and a unique symmetric multi-additive mapping
such that
and that
for all
and
is symmetric for each fixed one variable and is additive for fixed two variables. Thus
for all
as desired.
3. Stability of an Additive-Cubic-Quartic Functional Equation
We now investigate the generalized Hyers-Ulam stability problem of the functional equation (1.3). From now on, let be a real vector space and let
be a Banach space. Now before taking up the main subject, given
, we define the difference operator
by

for all We consider the following functional inequality:

for an upper bound
Theorem 3.1.
Let be fixed. Suppose that an even mapping
satisfies
and

for all If the upper bound
is a function such that

and that for all
then the limit

exists for all and
is a unique quartic mapping satisfying (1.3) and

for all
Proof.
Putting in (3.3), we obtain

for all On the other hand, replacing
by
in (3.3), we get

for all By (3.7) and (3.8), we get

for all Replacing
by
in (3.9), we get

for all . It follows from (3.10) that

for all . It follows from (3.11) that

for all .
This shows that is a Cauchy sequence in
. Since
is complete, the sequence
converges. We now define
by

for all It is clear that (3.6) holds, and
for all
By (3.3), we have

for all Hence by Lemma 2.1,
is quartic.
It remains to show that is unique. Suppose that there exists a quartic mapping
which satisfies (1.3) and (3.6). Since
and
for all
we conclude that

for all . By taking
in this inequality, we have
for all
which gives the conclusion for the case
Let
Then by (3.9), we have

for all Replacing
by
in (3.16) and dividing by 16, we get

for all By (3.16) and (3.17), we obtain

for all It follows from (3.18) that

for all Dividing both sides of (3.19) by
and then replacing
by
, we get

for all By taking
in (3.20),
is a Cauchy sequence in
. Then
exists for all
It is easy to see that (3.6) holds for
The rest of the proof is similar to the case
Theorem 3.2.
Suppose that an odd mapping satisfies

for all If the upper bound
is a function such that

and that for all
then the limit

exists for all and
is a unique additive mapping satisfying (1.3) and

for all
Proof.
Set in (3.21). Then by the oddness of
, we have

for all Replacing
by
in (3.21), we obtain

for all . Combining (3.25) and (3.26) yields that

for all Putting
and
for all
we get

for all . It follows from (3.28) that

for all Multiplying both sides of (3.29) by
and then replacing
by
, we get

for all So
is a Cauchy sequence in
. Put
for all
Then we have

for all . On the other hand, it is easy to show that

for all Hence it follows that

for all This means that
satisfies (1.3). Then by Lemma 2.2,
is additive. Thus (3.31) implies that
is additive.
To prove the uniqueness of , suppose that
is an additive mapping satisfying (3.24). Then for every
we have
and
Hence it follows that

for all . This shows that
for all
Theorem 3.3.
Suppose that an odd mapping satisfies

for all If the upper bound
is a function such that

and that

for all then the limit

exists for all and
is a unique cubic mapping satisfying (1.3), and

for all
Proof.
It is easy to show that satisfies (3.27). Setting
and then putting
in (3.27), we obtain

for all It follows from (3.40) that

for all . Replacing
by
in (3.41) and then multiplying both sides of (3.41) by
we get

for all . Since the right hand side of the inequality (3.42) tends to 0 as
the sequence
is Cauchy. Now we define

for all Then we have

for all Let

for all . Then we have

for all . Since
is an odd mapping,
satisfies (2.6). By (3.44), we conclude that
for all
Then
is cubic.
We have to show that is unique. Suppose that there exists another cubic mapping
which satisfies (1.3) and (3.39). Since
and
for all
we have

for all By letting
in the above inequality, we get
for all
which gives the conclusion.
Theorem 3.4.
Suppose that an odd mapping satisfies

for all If the upper bound
is a function such that

and that for all
then there exist a unique cubic mapping
, and a unique additive mapping
such that

for all
Proof.
By Theorems 3.2 and 3.3, there exist an additive mapping and a cubic mapping
such that

for all Combining two equations in (3.51) yields that

for all So we get (3.50) by letting
and
for all
To prove the uniqueness of and
let
be other additive and cubic mappings satisfying (3.50). Let
Then

for all Since

for all Hence (3.53) implies that

for all Since
, by (3.55), we obtain that
for all
Again by (3.55), we have
for all
Now we prove the generalized Hyers-Ulam stability of the functional equation (1.3).
Theorem 3.5.
Suppose that a mapping satisfies
and
for all
If the upper bound
is a function such that

and that for all
then there exist a unique additive mapping
a unique cubic mapping
, and a unique quartic mapping
such that

for all .
Proof.
Let for all
Then
and

for all Hence in view of Theorem 3.1, there exists a unique quartic mapping
satisfying (3.6). Let
for all
Then
, and
for all
From Theorem 3.4, it follows that there exist a unique cubic mapping
and a unique additive mapping
satisfying (3.44). Now it is obvious that (3.57) holds for all
and the proof of the theorem is complete.
Corollary 3.6.
Let and let
be a positive real number. Suppose that a mapping
satisfies
and

for all Then there exist a unique additive mapping
a unique cubic mapping
, and a unique quartic mapping
satisfying

for all .
Proof.
It follows from Theorem 3.5 by taking for all
.
Theorem 3.7.
Suppose that an odd mapping satisfies

for all If the upper bound
is a function such that

and that for all
then the limit

exists for all and
is a unique additive mapping satisfying (1.3) and

for all
Proof.
The proof is similar to the proof of Theorem 3.2.
Employing a similar way to the proof of Theorem 3.3, we get the following theorem.
Theorem 3.8.
Suppose that an odd mapping satisfies

for all If the upper bound
is a function such that

and that for all
then the limit

exists for all and
is a unique cubic mapping satisfying (1.3), and

for all
Theorem 3.9.
Suppose that an odd mapping satisfies

for all If the upper bound
is a function such that

and that for all
then there exist a unique additive mapping
, and a unique cubic mapping
such that

for all
Proof.
The proof is similar to the proof of Theorem 3.4.
Theorem 3.10.
Suppose that satisfies
and

for all If the upper bound
is a function such that

and that for all
then there exist a unique additive mapping
a unique cubic mapping
, and a unique quartic mapping
such that

for all .
Proof.
The proof is similar to the proof of Theorem 3.5.
Corollary 3.11.
Let and let
be a positive real number. Suppose that
satisfies
and

for all Then there exist a unique additive mapping
a unique cubic mapping
, and a unique quartic mapping
satisfying

for all .
Corollary 3.12.
Let be a positive real number. Suppose that a mapping
satisfies
and
for all
Then there exist a unique additive mapping
a unique cubic mapping
, and a unique quartic mapping
such that

for all .
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Acknowledgments
The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper. The third and corresponding author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).
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Eshaghi-Gordji, M., Kaboli-Gharetapeh, S., Park, C. et al. Stability of an Additive-Cubic-Quartic Functional Equation. Adv Differ Equ 2009, 395693 (2010). https://doi.org/10.1155/2009/395693
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DOI: https://doi.org/10.1155/2009/395693