CUBO A Mathematical Journal
Vol.15, No¯ 01, (159–169). March 2013

Generalized Ulam - Hyers Stability of Derivations of a AQ -
Functional Equation

M. Arunkumar

Government Arts College,

Department Of Mathematics,

Tiruvannamalai - 606 603, Tamilnadu, India.

annarun2002@yahoo.co.in

ABSTRACT

In this paper, the author established the generalized Ulam - Hyers stability of Deriva-

tions of additive and quadratic (AQ)- functional equation

f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y).

RESUMEN

En este art́ıculo el autor establece la estabilidad generalizada Ulam-Hyers de deriva-

ciones de la ecuación (AQ)-funcional cuadrática y aditiva

f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y).

Keywords and Phrases: Additive functional equations, Quadratic functional equation, Mixed

type functional equation, Additive derivations, Quadratic derivations, Ulam - Hyers stability

2010 AMS Mathematics Subject Classification: 39B52, 32B72, 32B82



160 M. Arunkumar CUBO
15, 1 (2013)

1 Introduction

The study of stability problems for functional equations is related to a question of Ulam [33]

concerning the stability of group homomorphisms and affirmatively answered for Banach spaces

by Hyers [13]. It was further generalized and excellent results obtained by number of authors

[2, 10, 24, 30, 32].

Over the last six or seven decades, the above problem was tackled by numerous authors

and its solutions via various forms of functional equations like additive, quadratic, cubic, quartic,

mixed type functional equations which involves only these types of functional equations were

discussed. We refer the interested readers for more information on such problems to the monographs

[1, 9, 14, 19, 22, 31].

C. Park [25] applied Gavruta’s result to Banach modules over a C∗−algebra. Many authors

have studied the structure of C∗− algebras for different types of functional equations in various

settings one can refer [6, 8, 26, 28]. It seems that approximate derivations was first investigated

by K.W. Jun and D.W. Park [16]. Recently, the stability of derivations have been investigated

in [7, 11, 12, 20, 27, 29] and references therein. The stability of cubic derivations was first time

introduced and investigated by M.E. Gordji et. al.,[12]. With the help of [12], the stability of

quadratic derivations was discussed by M. Arunkumar et. al., [3].

Very recently M. Arunkumar and J.M. Rassias [5], established the generalized Ulam - Hyers

stability of an additive and quadratic (AQ)-mixed type functional equation

f(x + y) + f(x − y) = 2f(x) + f(y) + f(−y) (1)

in Banach spaces. The solution and stability of several types of Mixed type additive and quadratic

type functional equations were discussed in [4, 17, 18, 21, 23]

In this paper, the author first time established the generalized Ulam - Hyers stability of mixed

derivations of a additive - quadratic (AQ)- functional equation (1).

Hereafter through out this paper, let us consider X and Y to be a normed Algebra and a

Banach Algebra, respectively.

2 Stability Results: Additive Derivations

In this section, the authors investigate the generalized Ulam-Hyers stability of additive derivations

of the AQ-functional equation (1).

Definition 2.1. A C−linear mapping A : X → X is called Additive Derivation on X if A

satisfies

A(xy) = A(x)y + xA(y) (1)

for all x, y ∈ X.



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Generalized Ulam - Hyers Stability Of Derivations Of A Aq - ... 161

Theorem 2.1. Let j = ±1. Let fa : X → Y be a odd mapping for which there exist a function

α, β : X2 → [0, ∞) with the condition

∞∑

n=0

α
(

2njx, 2njy
)

2nj
converges in R and lim

n→∞

α
(

2njx, 2njy
)

2nj
= 0 (2)

∞∑

n=0

β
(

2njx, 2njy
)

22nj
converges in R and lim

n→∞

α
(

2njx, 2njy
)

22nj
= 0 (3)

such that the functional inequalities

‖fa(x + y) + fa(x − y) − 2fa(x) − fa(y) − fa(−y)‖ ≤ α (x, y) (4)

and

‖fa(xy) − fa(x)y − xfa(y)‖ ≤ β (x, y) (5)

for all x, y ∈ X. Then there exists a unique Additive Derivation mapping A : X → Y satisfying the

functional equation (1) and

‖fa(x) − A(x)‖ ≤
1

2

∞∑

k=
1−j
2

α(2kjx, 2kjx)

2kj
(6)

for all x ∈ X. The mapping A(x) is defined by

A(x) = lim
n→∞

fa(2
njx)

2nj
(7)

for all x ∈ X.

Proof. Assume j = 1. Replacing y by x in (4) and using oddness of f, we get

∥

∥

∥

∥

fa(x) −
fa(2x)

2

∥

∥

∥

∥

≤
α (x, x)

2
(8)

for all x ∈ X. Now replacing x by 2x and dividing by 2 in (8), we get

∥

∥

∥

∥

fa(2x)

2
−

fa(2
2x)

22

∥

∥

∥

∥

≤
α (2x, 2x)

22
(9)

for all x ∈ X. From (8) and (9), we obtain

∥

∥

∥

∥

fa(x) −
fa(2

2x)

22

∥

∥

∥

∥

≤

∥

∥

∥

∥

fa(x) −
fa(2x)

2

∥

∥

∥

∥

+

∥

∥

∥

∥

fa(2x)

2
−

fa(2
2x)

22

∥

∥

∥

∥

≤
1

2

[

α(x, x) +
α(2x, 2x)

2

]

(10)



162 M. Arunkumar CUBO
15, 1 (2013)

for all x ∈ X. In general for any positive integer n , we get

∥

∥

∥

∥

fa(x) −
fa(2

nx)

2n

∥

∥

∥

∥

≤
1

2

n−1∑

k=0

α(2kx, 2kx)

2k
(11)

≤
1

2

∞∑

k=0

α(2kx, 2kx)

2k

for all x ∈ X. In order to prove the convergence of the sequence
{
fa(2

nx)

2n

}

,

replace x by 2mx and dividing by 2m in (11), for any m, n > 0 , we deduce
∥

∥

∥

∥

fa(2
mx)

2m
−

fa(2
n+mx)

2(n+m)

∥

∥

∥

∥

=
1

2m

∥

∥

∥

∥

fa(2
mx) −

fa(2
n · 2mx)

2n

∥

∥

∥

∥

≤
1

2

n−1∑

k=0

α(2k+mx, 2k+mx)

2k+m

≤
1

2

∞∑

k=0

α(2k+mx, 2k+mx)

2k+m

→ 0 as m → ∞

for all x ∈ X. Hence the sequence

{
fa(2

nx)

2n

}

is Cauchy sequence. Since Y is complete, there exists

a mapping A : X → Y such that

A(x) = lim
n→∞

fa(2
nx)

2n
∀ x ∈ X.

Letting n → ∞ in (11) we see that (6) holds for all x ∈ X. To prove that A satisfies (1), replacing

(x, y) by (2nx, 2ny) and dividing by 2n in (4), we obtain

1

2n

∥

∥

∥
fa(2

nx + 2ny) + fa(2
nx − 2ny) − 2fa(2

nx) + fa(2
ny) + fa(−2

ny)
∥

∥

∥
≤

1

2n
α(2nx, 2ny)

for all x, y ∈ X. Letting n → ∞ in the above inequality and using the definition of A(x), we see

that

A(x + y) + A(x − y) = 2A(x) + A(y) + A(−y).

Hence A satisfies (1) for all x, y ∈ X. It follows from (5) that

‖A(xy) − A(x)y − xA(y)‖

=
1

22n
‖fa(2

n(xy)) − fa(2
nx)(2ny) − (2nx)fa(2

ny)‖

≤
1

2n
β (2nx, 2ny)

→ 0 as n → ∞



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Generalized Ulam - Hyers Stability Of Derivations Of A Aq - ... 163

for all x, y ∈ X. To prove that A is unique, let B(x) be another mapping satisfying (1) and (6),

then

‖A(x) − B(x)‖ =
1

2n
‖A(2nx) − B(2nx)‖

≤
1

2n
{‖A(2nx) − fa(2

nx)‖ + ‖fa(2
nx) − B(2nx)‖}

≤
∞∑

k=0

α(2k+nx, 2k+nx)

2(k+n)

→ 0 as n → ∞

for all x ∈ X. Hence A is unique. Thus the mapping A : X → Y is a unique Additive Derivation

mapping satisfying (6).

For j = −1, we can prove a similar stability result. This completes the proof of the theorem.

The following Corollary is an immediate consequence of Theorem 2.1 concerning the stability

of (1).

Corollary 2.2. Let fa : X → Y be a odd mapping and there exists real numbers λ and s such that

‖fa(x + y) + fa(x − y) − 2fa(x) − fa(y) − fa(−y)‖

≤






λ,

λ {||x||s + ||y||s} , s < 1 or s > 1;

λ ||x||s||y||s, s < 1
2

or s > 1
2
;

λ
{
||x||s||y||s +

{
||x||2s + ||y||2s

}}
, s < 1

2
or s > 1

2
;

(12)

‖fa(xy) − fa(x)y − xfa(y)‖

≤






λ,

λ {||x||s + ||y||s} , s < 1 or s > 1;

λ ||x||s||y||s, s < 1
2

or s > 1
2
;

λ
{
||x||s||y||s +

{
||x||2s + ||y||2s

}}
, s < 1

2
or s > 1

2
;

(13)

for all x, y ∈ X. Then there exists a unique Additive Derivation function A : X → Y such that

‖fa(x) − A(x)‖ ≤






λ,

2λ||x||s

|2 − 2s|
,

λ||x||2s

|2 − 22s|
3λ||x||2s

|2 − 22s|

(14)

for all x ∈ X.



164 M. Arunkumar CUBO
15, 1 (2013)

3 Stability Results: Quadratic Derivations

In this section, the author establish the generalized Ulam-Hyers stability of quadratic derivations

of the AQ-functional equation (1).

Definition 3.1. Quadratic Derivation. A C−linear mapping Q : X → X is called Quadratic

Derivation on X if Q satisfies

Q(xy) = Q(x)y2 + x2Q(y) (1)

for all x, y ∈ X.

Theorem 3.1. Let j = ±1. Let fq : X → Y be a even mapping for which there exist a function

α, β : X2 → [0, ∞) with the condition

∞∑

n=0

α
(

2njx, 2njy
)

22nj
converges in R and lim

n→∞

α
(

2njx, 2njy
)

22nj
= 0 (2)

∞∑

n=0

β
(

2njx, 2njy
)

24nj
converges in R and lim

n→∞

β
(

2njx, 2njy
)

24nj
= 0 (3)

such that the functional inequalities

‖fq(x + y) + fq(x − y) − 2fq(x) − fq(y) − fq(−y)‖ ≤ α (x, y) (4)

and
∥

∥fq(xy) − x
2fq(y) − fq(x)y

2
∥

∥ ≤ β (x, y) (5)

for all x, y ∈ X. Then there exists a unique Quadratic Derivation mapping Q : X → Y satisfying

the functional equation (1) and

‖fq(x) − Q(x)‖ ≤
1

4

∞∑

k=
1−j
2

α(2kjx, 2kjx)

22kj
(6)

for all x ∈ X. The mapping Q(x) is defined by

Q(x) = lim
n→∞

fq(2
njx)

22nj
(7)

for all x ∈ X.

Proof. It follows from (5) that

∥

∥Q(xy) − x2Q(y) − Q(x)y2
∥

∥

=
1

24n

∥

∥fq(2
n(xy)) − (2nx)2fq(2

ny) − fq(2
nx)(2ny)2

∥

∥

≤
1

24n
β (2nx, 2ny)

→ 0 as n → ∞



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Generalized Ulam - Hyers Stability Of Derivations Of A Aq - ... 165

for all x, y ∈ X. The rest of the proof is similar tracing to that of Theorem 2.1. Thus the mapping

Q : X → Y is a unique Quadratic Derivation mapping satisfying (6).

The following Corollary is an immediate consequence of Theorem 3.1 concerning the stability

of (1).

Corollary 3.2. Let fq : X → Y be a even mapping and there exists real numbers λ and s such that

‖fq(x + y) + fq(x − y) − 2fq(x) − fq(y) − fq(−y)‖

≤






λ,

λ {||x||s + ||y||s} , s < 2 or s > 2;

λ ||x||s||y||s, s < 1 or s > 1;

λ
{
||x||s||y||s +

{
||x||2s + ||y||2s

}}
, s < 1 or s > 1;

(8)

∥

∥fq(xy) − x
2fq(y) − fq(x)y

2
∥

∥

≤






λ,

λ {||x||s + ||y||s} , s < 2 or s > 2;

λ ||x||s||y||s, s < 1 or s > 1;

λ
{
||x||s||y||s +

{
||x||2s + ||y||2s

}}
, s < 1 or s > 1;

(9)

for all x, y ∈ X. Then there exists a unique quadratic Deviation function Q : X → Y such that

‖fq(x) − Q(x)‖ ≤






λ

3
,

2λ||x||s

|4 − 2s|
,

λ||x||2s

|4 − 22s|
3λ||x||2s

|4 − 22s|

(10)

for all x ∈ X.

4 Stability Results: Mixed Derivations

In this section, the author present the generalized Ulam-Hyers stability of mixed derivations of the

AQ-functional equation (1).

Theorem 4.1. Let j = ±1. Let f : X → Y be a odd mapping for which there exist a function

α, β : X2 → [0, ∞) with the conditions (2), (3), (2) and (3) such that the functional inequalities

‖f(x + y) + f(x − y) − 2f(x) − f(y) − f(−y)‖ ≤ α (x, y) (1)



166 M. Arunkumar CUBO
15, 1 (2013)

(5) and (5) for all x, y ∈ X. Then there exists a unique Additive Derivation mapping A : X → Y

and a unique Quadratic Derivation mapping Q : X → Y satisfying the functional equation (1) and

‖f(x) − A(x) − Q(x)‖ ≤
1

2





1

2

∞∑

k=
1−j
2

(

α(2kjx, 2kjx)

2kj
+

α(−2kjx, −2kjx)

2kj

)

+
1

4

∞∑

k=
1−j
2

(

α(2kjx, 2kjx)

22kj
+

α(−2kjx, −2kjx)

22kj

)



 (2)

for all x ∈ X. The mapping A(x) and Q(x) are defined in (6) and (6) respectively for all x ∈ X.

Proof. Let fo(x) =
fa(x) − fa(−x)

2
for all x ∈ X. Then fo(0) = 0 and fo(−x) = −fo(x) for all

x ∈ X. Hence

‖fo(x + y) + fo(x − y) − 2fo(x) − fo(y) − fo(−y)‖ ≤
α(x, y)

2
+

α(−x, −y)

2
(3)

By Theorem 2.1, we have

‖fo(x) − A(x)‖ ≤
1

4

∞∑

k=
1−j
2

(

α(2kjx, 2kjx)

2kj
+

α(−2kjx, −2kjx)

2kj

)

(4)

for all x ∈ X. Also, let fe(x) =
fq(x) + fq(−x)

2
for all x ∈ X. Then fe(0) = 0 and fe(−x) = fe(x)

for all x ∈ X. Hence

‖fe(x + y) + fe(x − y) − 2fe(x) − fe(y) − fe(−y)‖ ≤
α(x, y)

2
+

α(−x, −y)

2
(5)

By Theorem 3.1, we have

‖fe(x) − Q(x)‖ ≤
1

8

∞∑

k=
1−j
2

(

α(2kjx, 2kjx)

22kj
+

α(−2kjx, −2kjx)

22kj

)

(6)

for all x ∈ X. Define

f(x) = fe(x) + fo(x) (7)

for all x ∈ X. From (4),(6) and (7), we arrive

‖f(x) − A(x) − Q(x)‖ = ‖fe(x) + fo(x) − A(x) − Q(x)‖

≤ ‖fo(x) − A(x)‖ + ‖fe(x) − Q(x)‖

≤
1

4

∞∑

k=
1−j
2

(

α(2kjx, 2kjx)

2kj
+

α(−2kjx, −2kjx)

2kj

)

+
1

8

∞∑

k=
1−j
2

(

α(2kjx, 2kjx)

22kj
+

α(−2kjx, −2kjx)

22kj

)

for all x ∈ X. Hence the theorem is proved.



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Generalized Ulam - Hyers Stability Of Derivations Of A Aq - ... 167

Using Corollaries 2.2 and 3.2 we have the following Corollary concerning the stability of (1).

Corollary 4.1. Let f : X → Y be a mapping and there exits real numbers λ and s such that

‖f(x + y) + f(x − y) − 2f(x) − f(y) − f(−y)‖

≤






λ,

λ {||x||s + ||y||s} , s < 1 or s > 1;

λ ||x||s||y||s, s < 1
2

or s > 1
2
;

λ
{
||x||s||y||s +

{
||x||2s + ||y||2s

}}
, s < 1

2
or s > 1

2
;

(8)

and (13), (9) for all x, y ∈ X. Then there exists a unique Additive Deviation function A : X → Y

and a unique quadratic Deviation function Q : X → Y such that

‖f(x) − A(x) − Q(x)‖ ≤






λ

(

1 +
1

3

)

,

2λ

(

1

|2 − 2s|
+

1

|4 − 2s|

)

||x||s,

λ

(

1

|2 − 22s|
+

1

|4 − 22s|

)

||x||2s,

3λ

(

1

|2 − 22s|
+

1

|4 − 22s|

)

||x||2s

(9)

for all x ∈ X.

Received: December 2012. Revised: February 2013.

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