We will show the general solution of the functional equation
f
(
x
+
ay
) +
f
(
x
−
ay
) + 2(
a
^{2}
− 1)
f
(
x
) =
a
^{2}
f
(
x
+
y
) +
a
^{2}
f
(
x
−
y
) + 2
a
^{2}
(
a
^{2}
− 1)
f
(
y
) and investigate the stability of quartic Lie *derivations associated with the given functional equation.
1. INTRODUCTION
The stability problem of functional equations originated from a question of Ulam
[17]
concerning the stability of group homomorphisms. Hyers
[7]
gave a first affirmative partial answer to the question of Ulam. Afterwards, the result of Hyers was generalized by Aoki
[1]
for additive mapping and by Rassias
[14]
for linear mappings by considering a unbounded Cauchy difference. Later, the result of Rassias has provided a lot of in°uence in the development of what we call HyersUlam stability or HyersUlamRassias stability of functional equations. For further information about the topic, we also refer the reader to
[10]
,
[8]
,
[2]
and
[3]
.
Recall that a Banach *algebra is a Banach algebra (complete normed algebra) which has an isometric involution. Jang and Park
[9]
investigated the stability of *derivations and of quadratic *derivations with Cauchy functional equation and the Jensen functional equation on Banach *algebra. The stability of *derivations on Banach *algebra by using fixed point alternative was proved by Park and Bodaghi and also Yang et al.; see
[12]
and
[19]
, respectively. Also, the stability of cubic Lie derivations was introduced by Fošner and Fošner; see
[6]
.
Rassias
[13]
investigated stability properties of the following quartic functional equation
It is easy to see that
f
(
x
) =
x
^{4}
is a solution of (1.1) by virtue of the identity
For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo
[4]
determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function
f
: ℝ → ℝ is a solution of (1.1) if and only if
f
(
x
) =
A
(
x
,
x
,
x
,
x
) , where the function
A
: ℝ
^{4}
→ ℝ is symmetric and additive in each variable.
In this paper, we deal with the following functional equation:
for all
x
,
y
∈
X
and an integer
a
(
a
≠ 0 , ±1) . We will show the general solution of the functional equation (1.3), define a quartic Lie *derivation related to equation (1.3) and investigate the HyersUlam stability of the quartic Lie *derivations associated with the given functional equation.
2. A QUARTIC FUNCTIONAL EQUATION
In this section let
X
and
Y
be real vector spaces and we investigate the general solution of the functional equation (1.3). Before we proceed, we would like to introduce some basic definitions concerning
n
additive symmetric mappings and key concepts which are found in
[16]
and
[18]
. A function
A
:
X
→
Y
is said to be
additive
if
A
(
x
+
y
) =
A
(
x
) +
A
(
y
) for all
x
,
y
∈
X
: Let
n
be a positive integer. A function
A_{n}
:
X^{n}
→
Y
is called
n

additive
if it is additive in each of its variables. A function
A_{n}
is said to be
symmetric
if
A_{n}
(
x
_{1}
, ⋯ ,
x_{n}
) =
A_{n}
(
x
_{σ(1)}
, ⋯ ,
x
_{σ(n)}
) for every permutation {
σ
(1) , ⋯ ,
σ
(
n
)} of {1 , 2 , ⋯ ,
n
} . If
A_{n}
(
x
_{1}
,
x
_{2}
, ⋯ ,
x_{n}
) is an
n
additive symmetric map, then
A^{n}
(
x
) will denote the diagonal
A_{n}
(
x
,
x
, ⋯ ,
x
) and
A^{n}
(
rx
) =
r^{n}A^{n}
(
x
) for all
x
∈
X
and all
r
∈ ℚ . such a function
A^{n}
(
x
) will be called a
monomial function
of degree
n
(assuming
A^{n}
≢ 0). Furthermore the resulting function after substitution
x
_{1}
=
x
_{2}
= ⋯ =
x_{s}
=
x
and
x
_{s+1}
=
x
_{s+2}
= ⋯ =
x_{n}
=
y
in
A_{n}
(
x
_{1}
,
x
_{2}
, ⋯ ,
x_{n}
) will be denoted by
A
^{s,n−s}
(
x
,
y
) .
Theorem 2.1.
A function f
:
X
→
Y is a solution of the functional equation (1.3) if and only if f is of the form f
(
x
) =
A
^{4}
(
x
)
for all x
∈
X
,
where
A
^{4}
(
x
)
is the diagonal of the
4
additive symmetric mapping
A
_{4}
:
X
^{4}
→
Y
.
Proof
. Assume that
f
satisfies the functional equation (1.3). Letting
x
=
y
= 0 in the equation (1.3), we have
that is,
f
(0) = 0 . Putting
x
= 0 in the equation (1.3), we get
for all
y
∈
X
. Replacing
y
by −
y
in the equation (2.1),we obtain
for all
y
∈
X
. Combining two equations (2.1) and (2.2), we have
f
(
y
) =
f
(−
y
) , for all
y
∈
X
. That is,
f
is even. We can rewrite the functional equation (1.3) in the form
for all
x
,
y
∈
X
and an integer
a
(
a
≠ 0 , ±1) . By Theorem 3.5 and 3.6 in
[18]
,
f
is a generalized polynomial function of degree at most 4, that is,
f
is of the form
for all
x
∈
X
, where
A
^{0}
(
x
) =
A
^{0}
is an arbitrary element of
Y
, and
A^{i}
(
x
) is the diagonal of the
i
additive symmetric mapping
A_{i}
:
X^{i}
→
Y
for
i
= 1, 2, 3, 4 . By
f
(0) = 0 and
f
(−
x
) =
f
(
x
) for all
x
∈
X
; we get
A
^{0}
(
x
) =
A
^{0}
= 0 : Substituting (2.3) into the equation (1.3) we have
for all
x
,
y
∈
X
. Note that
Since
a
≠ 0, ±1 , we have
for all
y
∈
X
. Thus
for all
x
∈
X
.
Conversely, assume that
f
(
x
) =
A
^{4}
(
x
) for all
x
∈
X
, where
A
^{4}
(
x
) is the diagonal of a 4additive symmetric mapping
A
_{4}
:
X
^{4}
→
Y
. Note that
where 1 ≤
s
,
t
≤ 3 and
c
∈ ℚ . Thus we may conclude that
f
satisfies the equation (1.3).
3. QUARTIC LIE *DERIVATIONS
Throughout this section, we assume that
A
is a complex normed *algebra and
M
is a Banach
A
bimodule. We will use the same symbol  ·  as norms on a normed algebra
A
and a normed
A
bimodule
M
. A mapping
f
:
A
→
M
is a
quartic homogeneous mapping
if
f
(
μa
) =
μ
^{4}
f
(
a
) ; for all
a
∈
A
and
μ
∈ ℂ . A quartic homogeneous mapping
f
:
A
→
M
is called a
quartic derivation
if
holds for all
x
,
y
∈
A
. For all
x
,
y
∈
A
, the symbol [
x
,
y
] will denote the commutator
xy
−
yx
. We say that a quartic homogeneous mapping
f
:
A
→
M
is a quartic Lie derivation if
for all
x
,
y
∈
A
. In addition, if
f
satisfies in condition
f
(
x
*) =
f
(
x
)* for all
x
∈
A
, then it is called the
quartic Lie
*
derivation
.
Example 3.1.
Let
A
= ℂ be a complex field endowed with the map
z
↦
z
* =
(where
is the complex conjugate of
z
). We define
f
:
A
→
A
by
f
(
a
) =
a
^{4}
for all
a
∈
A
. Then
f
is quartic and
for all
a
∈
A
. Also,
for all
a
∈
A
. Thus
f
is a quartic Lie *derivation.
In the following, 𝕋
^{1}
will stand for the set of all complex units, that is,
For the given mapping
f
:
A
→
M
, we consider
for all
x
,
y
∈
A
,
μ
∈ ℂ and
s
∈ ℤ (
s
≠ 0 , ±1) .
Theorem 3.2.
Suppose that f
:
A
→
M is an even mapping with f
(0) = 0
for which there exists a function ϕ
:
A
^{5}
→ [0, ∞)
such that
for all
and all a
,
b
,
x
,
y
,
z
∈
A in which
n
_{0}
∈ ℕ .
Also, if for each fixed b
∈
A the mapping r
↦
f
(
rb
)
from
ℝ
to M is continuous, then there exists a unique quartic Lie
*
derivation L
:
A
→
M satisfying
Proof
. Let
a
= 0 and
μ
= 1 in the inequality (3.3), we have
for all
b
∈
A
. Using the induction, it is easy to show that
for
t
>
k
≥ 0 and
b
∈
A
. The inequalities (3.2) and (3.7) imply that the sequence
is a Cauchy sequence. Since
M
is complete, the sequence is convergent. Hence we can define a mapping
L
:
A
→
M
as
for
b
∈
A
. By letting
t
=
n
and
k
= 0 in the inequality (3.7), we have
for
n
> 0 and
b
∈
A
. By taking
n
→ ∞ in the inequality (3.9), the inequalities (3.2) implies that the inequality (3.5) holds.
Now, we will show that the mapping
L
is a unique quartic Lie *derivation such that the inequality (3.5) holds for all
b
∈
A
. We note that
for all
a
,
b
∈
A
and
. By taking
μ
= 1 in the inequality (3.10), it follows that the mapping
L
is a quartic mapping. Also, the inequality (3.10) implies that Δ
_{μ}
L
(0,
b
) = 0 . Hence
for all
b
∈
A
and
. Let
μ
∈ 𝕋
^{1}
= {λ ∈ ℂ  λ = 1} . Then
μ
=
e^{iθ}
, where 0 ≤
θ
≤ 2π . Let
. Hence we have 이미지 . Then
for all
μ
∈ 𝕋
^{1}
and
a
∈
A
. Suppose that
ρ
is any continuous linear functional on
A
and
b
is a fixed element in
A
. Then we can define a function
g
: ℝ → ℝ by
for all
r
∈ ℝ . It is easy to check that
g
is cubic. Let
for all
k
∈ ℕ and
r
∈ ℝ .
Note that
g
as the pointwise limit of the sequence of measurable functions
g_{k}
is measurable. Hence
g
as a measurable quartic function is continuous (see
[5]
) and
for all
r
∈ ℝ . Thus
for all
r
∈ ℝ . Since
ρ
was an arbitrary continuous linear functional on
A
we may conclude that
for all
r
∈ ℝ . Let
μ
∈ ℂ (
μ
≠ 0) . Then
. Hence
for all
b
∈
A
and
μ
∈ ℂ (
μ
≠ 0) . Since
b
was an arbitrary element in
A
, we may conclude that
L
is quartic homogeneous.
Next, replacing
x
,
y
by
s^{k}x
,
s^{k}y
, respectively, and
z
= 0 in the inequality (3.4), we have
for all
x
,
y
∈
A
. Hence we have Δ
L
(
x
,
y
) = 0 for all
x
,
y
∈
A
. That is,
L
is a quartic Lie derivation. Letting
x
=
y
= 0 and replacing
z
by
s^{k}z
in the inequality (3.4), we get
for all
z
∈
A
. As
n
→ ∞ in the inequality (3.11), we have
for all
z
∈
A
. This means that
L
is a quartic Lie *derivation. Now, assume
L'
:
A
→
A
is another quartic *derivation satisfying the inequality (3.5). Then
which tends to zero as
k
→ ∞ , for all
b
∈
A
. Thus
L
(
b
) =
L'
(
b
) for all
b
∈
A
. This proves the uniqueness of
L
.
Corollary 3.3.
Let θ
,
r be positive real numbers with r
< 4
and let f
:
A
→
M be an even mapping with f
(0) = 0
such that
for all
and a
,
b
,
x
,
y
,
z
∈
A
.
Then there exists a unique quartic Lie *derivation L
:
A
→
M satisfying
for all b
∈
A
.
Proof
. The proof follows from Theorem 3.2 by taking
ϕ
(
a
,
b
,
x
,
y
,
z
) =
θ
(
a

^{r}
+ 
b

^{r}
+ 
x

^{r}
+ 
y

^{r}
+ 
z

^{r}
) for all
a
,
b
,
x
,
y
,
z
∈
A
.
In the following corollaries, we show the hyperstability for the quartic Lie *derivations.
Corollary 3.4.
Let r be positive real numbers with r
< 4
and let f
:
A
→
M be an even mapping with f
(0) = 0
such that
for all
and
a
,
b
,
x
,
y
,
z
∈
A
.
Then f is a quartic Lie *derivation on A
.
Proof
. By taking
ϕ
(
a
,
b
,
x
,
y
,
z
) = (
a

^{r}
+ 
x

^{r}
)(
b

^{r}
+ 
y

^{r}

z

^{r}
) in Theorem 3.2 for all
a
,
b
,
x
,
y
,
z
∈
A
, we have
. Hence the inequality (3.5) implies that
f
=
L
, that is,
f
is a quartic Lie *derivation on
A
.
Corollary 3.5.
Let r be positive real numbers with r
< 4
and let f
:
A
→
M be an even mapping with f
(0) = 0
such that
for all
and
a
,
b
,
x
,
y
,
z
∈
A
.
Then f is a quartic Lie *derivation on A
.
Proof
. By taking
ϕ
(
a
,
b
,
x
,
y
,
z
) = (
a

^{r}
+ 
x

^{r}
)(
b

^{r}
+ 
y

^{r}
+ 
z

^{r}
) in Theorem 3.2 for all
a
,
b
,
x
,
y
,
z
∈
A
, we have
. Hence the inequality (3.5) implies that
f
=
L
, that is,
f
is a quartic Lie *derivation on
A
.
Now, we will investigate the stability of the given functional equation (3.1) using the alternative fixed point method. Before proceeding the proof, we will state the theorem, the alternative of fixed point; see
[11]
and
[15]
.
Definition 3.6.
Let
X
be a set. A function
d
:
X
×
X
→ [0, ∞] is called a
generalized metric
on
X
if
d
satisfies

(1)d(x,y) = 0 if and only ifx=y;

(2)d(x,y) =d(y,x) for allx,y∈X;

(3)d(x,z) ≤d(x,y) +d(y,z) for allx,y,z∈X.
Theorem 3.7
(The alternative of fixed point
[11]
,
[15]
).
Suppose that we are given a complete generalized metric space
(Ω,
d
)
and a strictly contractive mapping T
: Ω → Ω
with Lipschitz constant l
.
Then for each given x
∈ Ω ,
either
or there exists a natural number
n
_{0}
such that

(1)d(Tnx,Tn+1x) < ∞for all n≥n0;

(2)The sequence(Tnx)is convergent to a fixed point y* of T;

(3)y* is the unique fixed point of T in the set


(4)for all y∈ △ .
Theorem 3.8.
Let f
:
A
→
M be a continuous even mapping with f
(0) = 0
and let
ϕ
:
A
^{5}
→ [0, ∞)
be a continuous mapping such that
for all
and a
,
b
,
x
,
y
,
z
∈
A
.
If there exists a constant l
∈ (0, 1)
such that
for all a
,
b
,
x
,
y
,
z
∈
A
,
then there exists a quartic Lie *derivation L
:
A
→
M satisfying
for all b
∈
A
.
Proof
. Consider the set
and introduce the generalized metric on Ω,
It is easy to show that (Ω,
d
) is complete. Now we define a function
T
: Ω → Ω by
for all
b
∈
A
. Note that for all
g
,
h
∈ Ω , let
c
∈ (0, ∞) be an arbitrary constant with
d
(
g
,
h
) ≤
c
. Then
for all
b
∈
A
. Letting
b
=
sb
in the inequality (3.17) and using (3.14) and (3.16), we have
that is,
Hence we have that
for all
g
,
h
∈ Ω , that is,
T
is a strictly selfmapping of Ω with the Lipschitz constant
l
. Letting
μ
= 1 ,
a
= 0 in the inequality (3.12), we get
for all
b
∈
A
. This means that
We can apply the alternative of fixed point and since lim
_{n→∞}
d
(
T^{n}
f
,
L
) = 0 , there exists a fixed point
L
of
T
in Ω such that
for all
b
∈
A
. Hence
This implies that the inequality (3.15) holds for all
b
∈
A
. Since
l
∈ (0, 1) , the inequality (3.14) shows that
Replacing
a
,
b
by
s^{n}a
,
s^{n}b
, respectively, in the inequality (3.12), we have
Taking the limit as
k
tend to infinity, we have Δ
_{μ}
f
(
a
,
b
) = 0 for all
a
,
b
∈
A
and all
. The remains are similar to the proof of Theorem 3.2.
Corollary 3.9.
Let θ
,
r be positive real numbers with r
< 4
and let f
:
A
→
M be a mapping with f
(0) = 0
such that
for all
and a
,
b
,
x
,
y
,
z
∈
A
.
Then there exists a unique quartic Lie *derivation L
:
A
→
M satisfying
for all b
∈
A
.
Proof
. The proof follows from Theorem 3.8 by taking
ϕ
(
a
,
b
,
x
,
y
,
z
) =
θ
(
a

^{r}
+ 
b

^{r}
+ 
x

^{r}
+ 
y

^{r}
+ 
z

^{r}
) for all
a
,
b
,
x
,
y
,
z
∈
A
.
In the following corollaries, we show the hyperstability for the quartic Lie *derivations.
Corollary 3.10.
Let r be positive real numbers with r
< 4
and let f
:
A
→
M be an even mapping with f
(0) = 0
such that
for all
and a
,
b
,
x
,
y
,
z
∈
A
.
Then f is a quartic Lie *derivation on A
Proof
. By taking
ϕ
(
a
,
b
,
x
,
y
,
z
) = (
a

^{r}
+ 
x

^{r}
)(
b

^{r}
+ 
y

^{r}

z

^{r}
) in Theorem 3.8 for all
a
,
b
,
x
,
y
,
z
∈
A
, we have
. Hence the inequality (3.15) implies that
f
=
L
, that is,
f
is a quartic Lie *derivation on
A
.
Corollary 3.11.
Let r be positive real numbers with r
< 4
and let f
:
A
→
M be an even mapping with f
(0) = 0
such that
for all
and a
,
b
,
x
,
y
,
z
∈
A
.
Then f is a quartic Lie *derivation on A
Proof
. By taking
ϕ
(
a
,
b
,
x
,
y
,
z
) = (
a

^{r}
+ 
x

^{r}
)(
b

^{r}
+ 
y

^{r}
+ 
z

^{r}
) in Theorem 3.8 for all
a
,
b
,
x
,
y
,
z
∈
A
, we have
. Hence the inequality (3.15) implies that
f
=
L
, that is,
f
is a quartic Lie *derivation on
A
.
Acknowledgements
The present research was conducted by the research fund of Dankook University in 2014.
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