In this paper, we solve the additive
ρ
functional equations
where
ρ
is a fixed number with
ρ
≠ 1, 2, and
where
ρ
is a fixed number with
ρ
≠ 1.
Using the fixed point method, we prove the HyersUlam stability of the additive
ρ
functional equations (0.1) and (0.2) in Banach spaces.
1. INTRODUCTION AND PRELIMINARIES
The stability problem of functional equations originated from a question of Ulam
[14]
concerning the stability of group homomorphisms.
The functional equation
is called the
Cauchy equation
. In particular, every solution of the Cauchy equation is said to be an
additive mapping
. Hyers
[7]
gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki
[1]
for additive mappings and by Rassias
[13]
for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta
[6]
by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.
We recall a fundamental result in fixed point theory.
Theorem 1.1
(
[2
,
5]
)
.
Let
(
X, d
)
be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α
< 1.
Then for each given element x
∈
X, either
for all nonnegative integers n or there exists a positive integer n_{0} such that
(1)
d
(
J^{n}x
,
J
^{n+1}
x
) < ∞, ∀
n
≥
n
_{0}
;
(2)
the sequence
{
J^{n}x
}
converges to a fixed point y
*
of J
;
(3)
y
^{∗}
is the unique fixed point of J in the set Y
= {
y
∈
X

d
(
J
^{n0}
x, y
) < ∞};
(4)
for all y
∈
Y
.
In 1996, G. Isac and Th.M. Rassias
[8]
were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see
[3
,
4
,
10
,
11
,
12]
).
In Section 2, we solve the additive functional equation (0.1) and prove the HyersUlam stability of the additive functional equation (0.1) in Banach spaces.
In Section 3, we solve the additive functional equation (0.2) and prove the HyersUlam stability of the additive functional equation (0.2) in Banach spaces.
Throughout this paper, assume that
X
is a normed space and that
Y
is a Banach space.
2. ADDITIVEρFUNCTIONAL EQUATION (0.1)
Let
ρ
be a number with
ρ
≠ 1, 2.
We solve and investigate the additive
ρ
functional equation (0.1) in normed spaces.
Lemma 2.1.
If a mapping f
:
X
→
Y satisfies
for all x, y, z
∈
X, then f : X → Y is additive
.
Proof
. Assume that
f
:
X → Y
satisfies (2.1).
Letting
x
=
y
=
z
= 0 in (2.1), we get −2
f
(0) = −
ρf
(0). So
f
(0) = 0.
Letting
y = x
and
z
= 0 in (2.1), we get
f
(2
x
) − 2
f
(
x
) = 0 and so
f
(2
x
) = 2
f
(
x
) for all
x
∈
X
. Thus
for all
x
∈
X
.
It follows from (2.1) and (2.2) that
and so
f
(
x + y + z
) =
f
(
x
) +
f
(
y
) +
f
(
z
) for all
x, y, z
∈
X
. Since
f
(0) = 0,
for all
x, y
∈
X
. ☐
We prove the HyersUlam stability of the additive
ρ
functional equation (2.1) in Banach spaces.
Theorem 2.2.
Let φ
:
X
^{3}
→ [0,∞)
be a function such that there exists an L
< 1
with
for all x
,
y
,
z
∈
X. and let f
:
X → Y be a mapping satisfying f
(0) = 0
and
for all x
,
y
,
z
∈
X. Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. Letting
y = x
and
z
= 0 in (2.4), we get
for all
x
∈
X
. So
for all
x
∈
X
.
Consider the set
and introduce the generalized metric on
S
:
where, as usual, inf
ϕ
= +∞. It is easy to show that (
S
,
d
) is complete (see
[9]
).
Now we consider the linear mapping
J : S → S
such that
for all
x
∈
X
.
Let
g
,
h
∈
S
be given such that
d
(
g
,
h
) =
ε
. Then
for all
x
∈
X
. Hence
for all
x
∈
X
. So
d
(
g
,
h
) =
ε
implies that
d
(
Jg
,
Jh
) ≤
Lε
. This means that
for all
g
,
h
∈
S
.
It follows from (2.6) that
for all
By Theorem 1.1, there exists a mapping
A
:
X → Y
satisfying the following:
(1)
A
is a fixed point of
J
, i.e.,
for all
x
∈
X
. The mapping
A
is a unique fixed point of
J
in the set
This implies that
A
is a unique mapping satisfying (2.7) such that there exists a
µ
∈ (0, ∞) satisfying
for all
x
∈
X
;
(2)
d
(
J^{l}f
,
A
) → 0 as
l
→ ∞. This implies the equality
for all
x
∈
X
;
(3)
, which implies
for all
x
∈
X
.
It follows from (2.3) and (2.4) that
for all
x
,
y
,
z
∈
X
. So
for all
x
,
y
,
z
∈
X
. By Lemma 2.1, the mapping
A
:
X → Y
is additive. ☐
Corollary 2.3.
Let r
> 1
and θ be nonnegative real numbers, and let f
:
X → Y be a mapping satisfying f
(0) = 0
and
for all x
,
y
,
z
∈
X
.
Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. The proof follows from Theorem 2.2 by takig
φ
(
x
,
y
,
z
) =
θ
(∥
x
∥
^{r}
+∥
y
∥
^{r}
+∥
z
∥
^{r}
) for all
x
,
y
,
z
∈
X
. Then we can choose
L
= 2
^{1−r}
and we get the desired result. ☐
Theorem 2.4.
Let φ
:
X
^{3}
→ [0,∞)
be a function such that there exists an L
< 1
with
for all x
,
y
,
z
∈
X Let f : X → Y be a mapping satisfying f
(0) = 0
and
(2.4).
Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. It follows from (2.6) that
for all
x
∈
X
.
Let (
S
,
d
) be the generalized metric space defined in the proof of Theorem 2.2.
Now we consider the linear mapping
J
:
S → S
such that
for all
x
∈
X
.
It follows from (2.9) that
. So
for all
x
∈
X
.
The rest of the proof is similar to the proof of Theorem 2.2. ☐
Corollary 2.5.
Let r
< 1
and θ be positive real numbers, and let f
:
X → Y be a mapping satisfying f
(0) = 0
and
(2.8).
Then there exists a unique additive mapping A
:
X → Y such that
for all
x
∈
X
.
Proof
. The proof follows from Theorem 2.4 by takig
φ
(
x
,
y
,
z
) =
θ
(∥
x
∥
^{r}
+∥
y
∥
^{r}
+∥
z
∥
^{r}
) for all
x
,
y
,
z
∈
X
. Then we can choose
L
= 2
^{r−1}
and we get the desired result. ☐
3. ADDITIVEρFUNCTIONAL EQUATION (0.2)
Let
ρ
be a number with
ρ
≠ 1.
We solve and investigate the additive
ρ
functional equation (0.2) in normed spaces.
Lemma 3.1.
If a mapping f
:
X → Y satisfies
for all x
,
y
,
z
∈
X
,
then f
:
X → Y is additive
.
Proof
. Assume that
f
:
X → Y
satisfies (3.1).
Letting
x
=
y
=
z
= 0 in (3.1), we get −2
f
(0) = −2
ρf
(0). So
f
(0) = 0.
Letting
y
=
x
and
z
= 0 in (3.1), we get
f
(2
x
) − 2
f
(
x
) = 0 and so
f
(2
x
) = 2
f
(
x
) for all
x
∈
X
. Thus
for all
x
∈
X
.
It follows from (3.1) and (3.2) that
and so
f
(
x
+
y
) =
f
(
x
) +
f
(
y
) for all
x
,
y
∈
X
. ☐
We prove the HyersUlam stability of the additive
ρ
functional equation (3.1) in Banach spaces.
Theorem 3.2.
Let φ
:
X
^{3}
→ [0,∞)
be a function such that there exists an L
< 1
with
for all x
,
y
,
z
∈
X
.
and let f
:
X → Y be a mapping satisfying f
(0) = 0
and
for all x
,
y
,
z
∈
X
.
Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. Letting
y
=
x
and
z
= 0 in (3.3), we get
for all x
∈
X
.
The rest of the proof is similar to the proof of Theorem 2.2. ☐
Corollary 3.3.
Let r
> 1
and θ be nonnegative real numbers, and let f
:
X → Y be a mapping satisfying f
(0) = 0
and
for all x
,
y
,
z
∈
X. Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. The proof follows from Theorem 3.2 by takig
φ
(
x
,
y
,
z
) =
θ
(∥
x
∥
^{r}
+∥
y
∥
^{r}
+∥
z
∥
^{r}
) for all
x
,
y
,
z
∈
X
. Then we can choose
L
= 2
^{1−r}
and we get the desired result. ☐
Theorem 3.4.
Let
φ
:
X
^{3}
→ [0,∞)
be a function such that there exists an L
< 1
with
for all x
,
y
,
z
∈
X Let f
:
X → Y be a mapping satisfying f
(0) = 0
and
(3.3).
Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. It follows from (3.4) that
for all x
∈
X
.
The rest of the proof is similar to the proof of Theorem 2.2. ☐
Corollary 3.5.
Let r
< 1
and θ be positive real numbers, and let f
:
X → Y be a mapping satisfying f
(0) = 0
and
(3.5).
Then there exists a unique additive mapping A
:
X → Y such that
for all x
∈
X
.
Proof
. The proof follows from Theorem 3.4 by takig
φ
(
x
,
y
,
z
) =
θ
(∥
x
∥
^{r}
+∥
y
∥
^{r}
+∥
z
∥
^{r}
) for all
x
,
y
,
z
∈
X
. Then we can choose
L
= 2
^{r−1}
and we get the desired result. ☐
Acknowledgements
This research was supported by Hanshin University Research Grant.
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