ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2015.
Nov,
22(4):
375-382

- Received : October 01, 2015
- Accepted : October 12, 2015
- Published : November 30, 2015

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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 direct method, we prove the Hyers-Ulam stability of the additive
ρ
-functional equations (0.1) and (0.2) in Banach spaces.
is called the
Cauchy equation
. In particular, every solution of the Cauchy equation is said to be an
additive mapping
. Hyers
[3]
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
[4]
for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Gǎvruta
[2]
by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.
In Section 2, we solve the additive functional equation (0.1) and prove the Hyers-Ulam 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 Hyers-Ulam 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.
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 Hyers-Ulam stability of the additive
ρ
-functional equation (2.1) in Banach spaces.
Theorem 2.2.
Let φ
:
X
^{3}
→ [0, ∞)
be a function 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
. Hence
for all nonnegative integers
m
and
l
with
m
>
l
and all
x
∈
X
. It follows from (2.7) that the sequence
is Cauchy for all
x
∈
X
. Since
Y
is a Banach space, the sequence
converges. So one can define the mapping
A
:
X
→
Y
by
for all
x
∈
X
. Moreover, letting
l
= 0 and passing the limit
m
→ ∞ in (2.7), we get (2.5).
Now, let
T
:
X
→
Y
be another additive mapping satisfying (2.5). Then we have
which tends to zero as
q
→ ∞ for all
x
∈
X
. So we can conclude that
A
(
x
) =
T
(
x
) for all
x
∈
X
. This proves the uniqueness of
A
.
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
. Letting
φ
(
x
,
y
,
z
) :=
in Theorem 2.2, we get the desired result. ☐
Theorem 2.4.
Let φ
:
X
^{3}
→ [0, ∞)
be a function and let f
:
X
→
Y be a mapping satisfying f
(0) = 0, (2.4)
and
for all x
,
y
,
z
∈
X
.
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
. Hence
for all nonnegative integers
m
and
l
with
m
>
l
and all
x
∈
X
. It follows from (2.10) that the sequence
is a Cauchy sequence for all
x
∈
X
. Since
Y
is complete, the sequence
converges. So one can define the mapping
A
:
X
→
Y
by
for all
x
∈
X
. Moreover, letting
l
= 0 and passing the limit
m
→ ∞ in (2.10), we get (2.9).
The rest of the proof is similar to the proof of Theorem 2.2. ☐
Corollary 2.5.
Let r
< 1
and θ be nonnegative 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
. Letting
φ
(
x
,
y
,
z
) :=
in Theorem 2.4, we get the desired result. ☐
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 (2.1), we get -2
f
(0) = -2
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 (3.1) and (3.2) that
and so
for all
x
,
y
∈
X
. ☐
We prove the Hyers-Ulam stability of the additive
ρ
-functional equation (3.1) in Banach spaces.
Theorem 3.2.
Let φ
:
X
^{3}
→ [0, ∞)
be a function 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
. Letting
φ
(
x
,
y
,
z
) :=
in Theorem 3.2, we get the desired result. ☐
Theorem 3.4.
Let φ
:
X
^{3}
→ [0, ∞)
be a function and let f
:
X
→
Y be a mapping satisfying f
(0) = 0, (3.3)
and
for all x
,
y
,
z
∈
X
.
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 proofs of Theorems 2.2 and 2.4. ☐
Corollary 3.5.
Let r
< 1
and θ be nonnegative 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
. Letting
φ
(
x
,
y
,
z
) :=
in Theorem 3.4, we get the desired result. ☐

1. INTRODUCTION

The stability problem of functional equations originated from a question of Ulam
[5]
concerning the stability of group homomorphisms.
The functional equation
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2. ADDITIVEρ-FUNCTIONAL EQUATION (0.1)

Let
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3. ADDITIVEρ-FUNCTIONAL EQUATION (0.2)

Let
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Aoki T.
(1950)
On the stability of the linear transformation in Banach spaces
J. Math. Soc. Japan
2
64 -
66
** DOI : 10.2969/jmsj/00210064**

Gǎvruta P.
(1994)
A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings
J. Math. Anal. Appl.
184
431 -
443
** DOI : 10.1006/jmaa.1994.1211**

Hyers D.H.
(1941)
On the stability of the linear functional equation
Proc. Natl. Acad. Sci. U.S.A.
27
222 -
224
** DOI : 10.1073/pnas.27.4.222**

Rassias Th.M.
(1978)
On the stability of the linear mapping in Banach spaces
Proc. Amer. Math. Soc.
72
297 -
300
** DOI : 10.1090/S0002-9939-1978-0507327-1**

Ulam S.M.
1960
A Collection of the Mathematical Problems
Interscience Publ.
New York

Citing 'ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES
'

@article{ SHGHCX_2015_v22n4_375}
,title={ADDITIVE ρ-FUNCTIONAL EQUATIONS IN BANACH SPACES}
,volume={4}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.4.375}, DOI={10.7468/jksmeb.2015.22.4.375}
, number= {4}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={JUN, IN WHAN
and
SEO, JEONG PIL
and
LEE, SUNGJIN}
, year={2015}
, month={Nov}