EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5719
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Quadratic f-hom-ders in Banach Algebra Related to
System of Quadratic Functional Equations

Choonkil Park1, Siriluk Donganont2,∗, Se Won Min3,∗

1 Research Institute for Convergence of Basic Sciences, Hanyang University, Seoul 04763,
Korea
2 School of Science, University of Phayao, Phayao 56000, Thailand
3 Department of Mathematics, Hanyang University, Seoul 04763, Korea

Abstract. Mirzavaziri and Moslehian [13] introduced the concept of f -derivations and Sripattanet
et al. [18] introduced a quadratic hom-der in Banach algebras. In this paper, we solve the system
of quadratic functional equations{

f(x+ y) + f(x− y) = g(x) + g(y),
g
(
x+y
2

)
+ g

(
x−y
2

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

Using Mirzavaziri and Moslehian’s idea and Sripattanet et al.’s idea, we define a quadratic f -hom-
der in Banach algebras, and we investigate the Hyers-Ulam stability of quadratic f -hom-ders in
Banach algebras.

2020 Mathematics Subject Classifications: 47H10, 47B47, 17B40, 39B72, 39B52

Key Words and Phrases: quadratic f -hom-der; fixed point method; Hyers-Ulam stability;
system of quadratic functional equations

1. Introduction

Let B be a complex Banach algebra and f : B → B be a C-linear mapping. Mirzavaziri
and Moslehian [13] introduced the concept of f -derivation g : B → B as follows:

g(xy) = f(x)g(y) + g(x)f(y) (1)

for all x, y ∈ B.

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5719

Email addresses: baak@hanyang.ac.kr (C. Park),
siriluk.pa@up.ac.th (S. Donganont), blavely526@hanyang.ac.kr (S. W. Min)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 2 of 10

Park et al. [16] introduced the concept of hom-derivation on B, i.e., g : B → B is
a homomorphism and f satisfies (1) for all x, y ∈ B. Dehghanian et al. introduced the
concept of hom-der g : B → B as follows:

g(x)g(y) = xg(y) + g(x)y

for all x, y ∈ B. Dehghanian et al. [6] introduced and investigated ternary hom-ders in
ternary Banach algebras and Kheawborisuk et al. [10] defined and studied hom-ders in
fuzzy Banach algebras. Recently, Sripattanet et al. [18] introduced a quadratic hom-der
in Banach algebras B as follows: A quadratic mapping D : B → B is said to be a quadratic
hom-der if it satisfies

D(x)D(y) = x2D(y) +D(x)y2

for all x, y ∈ B.
In this papere, we introduce the concept of quadartic hom-der in Banach algebras.

Definition 1. Let B be a complex Banach algebra and f : B → B be a quadratic mapping.
A quadratic mapping g : B → B is called a quadratic f -hom-der if it satisfies

g(x)g(y) = f(x)g(y) + g(x)f(y)

for all x, y ∈ B.

Example 1. Let C0(X) be the complex Banach algebra of complex valued continuous
functions on a locally compact Hausdorff space X and g : C0(X) → C0(X) be defined by
g(M) = 2M2 and f : C0(X) → C0(X) be defined by f(M) = M2. Then f is a quadratic
mapping and g is a quadratic f -hom-der.

We say that an equation is stable if any function satisfying the equation approximately
is near to an exact solution of the equation.

The stability analysis of functional equations emanated from a question of Ulam [19],
was raised in 1940, about the stability of group homomorphisms and then was extended
by Hyers [9]. Recently, results on the so-called Hyers-Ulam stability have comfortabled
the stability conditions. Dehghanian and Modarres [3] studied ternary γ-homomorphisms
and ternary γ-derivations on ternary semigroups, Dehghanian et al. [4] studied ternary
3-derivations on C∗-ternary algebras and Dehghanian and Park [5] studied C∗-ternary 3-
homomorphisms on C∗-ternary algebras. Moreover, Senthil Kumar et al. [11] investigated
modular stabilities of a reciprocal second power functional equation and Bowniya et al.
[1, 2] obtained the Hyers-Ulam stability results of linear differential equations.

The method provided by Hyers [9] which produces the additive function will be called
a direct method. This method is the most significant and strong tool to concerning the
stability of different functional equations. That is, the exact solution of the functional
equation is explicitly constructed as a limit of a sequence, starting from the given approx-
imate solution [17]. The other significant method is fixed point theorem, that is, the exact
solution of the functional equation is explicitly created as a fixed point of some certain
mapping [8, 14, 15].

We remember a fixed point alternative theorem.



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 3 of 10

Theorem 1. [7] If (B, d) is a complete generalized metric space and I : B → B a strictly
contractive mapping, that is,

d(Iu, Iv) ≤ Ld(u, v)

for all u, v ∈ B and a Lipschitz constant L < 1. Then for each given element u ∈ B, either

d(Inu, In+1u) = +∞, ∀n ≥ 0,

or

d(Inu, In+1u) < +∞, ∀n ≥ n0,

for some positive integer n0. Furthermore, if the second alternative holds, then

(i) the sequence (Inu) is convergent to a fixed point v∗ of I;

(ii) v∗ is the unique fixed point of I in the set V := {v ∈ B, d(In0u, v) < +∞};
(iii) d(v, v∗) ≤ 1

1−Ld(v, Iv) for all u, v ∈ V .

In this paper, we consider the following system of additive functional equations{
f(x+ y) + f(x− y) = g(x) + g(y),

g
(x+y

2

)
+ g

(x−y
2

)
= f(x) + f(y)

(2)

for all x, y ∈ B. The aim of the present paper is to solve the system of quadratic functional
equations and prove the Hyers-Ulam stability of quadratic f -hom-ders in complex Banach
algebras by using the fixed point method.

Throughout this paper, assume that B is a complex Banach algebra.

2. Stability of system of quadratic functional equations

We solve and investigate the system of quadratic functional equations (2) in complex
Banach algebras.

Lemma 1. Let f, g : B → B be mappings satisfying g(0) = 0 and (2) for all x, y ∈ B.
Then the mappings f, g : B → B are quadratic.

Proof. Letting x = y = 0 in (2), we get

f(0) = g(0) = 0.

Putting y = 0 in (2), we have

2f(x) = g(x),

2g
(x
2

)
= f(x) =

1

2
g(x) (3)



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 4 of 10

for all x ∈ B. So
f(x+ y) + f(x− y) = 2f(x) + 2f(y)

for all x, y ∈ B. Hence the mapping f : B → B is quadratic. Moreover, by (3),

g
(x
2

)
=

1

4
g(x)

and so

1

4
g(x+ y) +

1

4
g(x− y) = g

(
x+ y

2

)
+ g

(
x− y

2

)
= f(x) + f(y) =

1

2
g(x) +

1

2
g(y)

for all x, y ∈ B. Thus
g(x+ y) + g(x− y) = 2g(x) + 2g(y)

for all x, y ∈ B and so the mapping g : B → B is quadratic.

Using the fixed point technique, we prove the Hyers-Ulam stability of the system of
quadratic functional equations (2) in complex Banach algebras.

Theorem 2. Suppose that ∆ : B2 → [0,∞) is a function such that there exists an L < 1
with

∆(
x

2
,
y

2
) ≤ L

4
∆(x, y) (4)

for all x, y ∈ B. Let f, g : B → B be mappings satisfying g(0) = 0 and{
∥f(x+ y) + f(x− y)− g(x)− g(y)∥ ≤ ∆(x, y),

∥g
(x+y

2

)
+ g

(x−y
2

)
− f(x)− f(y)∥ ≤ ∆(x, y)

(5)

for all x, y ∈ B. Then there exist unique quadratic mappings F,G : B → B such that

∥F (x)− f(x)∥ ≤ 2 + L

4(1− L)
∆(x, x), (6)

∥G(x)− g(x)∥ ≤ 2 + L

2(1− L)
∆(x, x) (7)

for all x ∈ B.

Proof. Putting x = y = 0 in (5), we get{
∥2f(0)− 2g(0)∥ ≤ ∆(0, 0) = 0,
∥2g(0)− 2f(0)∥ ≤ ∆(0, 0) = 0

and so f(0) = g(0) = 0.



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 5 of 10

Letting y = x in (5), we obtain{
∥f(2x)− 2g(x)∥ ≤ ∆(x, x),
∥g(x)− 2f(x)∥ ≤ ∆(x, x)

and so { ∥∥g(x)− 4g
(
x
2

)∥∥ ≤ 2∆
(
x
2 ,

x
2

)
+∆(x, x) ≤ 2+L

2 ∆(x, x),∥∥f(x)− 4f
(
x
2

)∥∥ ≤ ∆
(
x
2 ,

x
2

)
+ 1

2∆(x, x) ≤ 2+L
4 ∆(x, x)

(8)

for all x ∈ B.
Let Γ = {γ : B → B : γ(0) = 0}. We define a generalized metric on Γ as follows:

d : Γ× Γ −→ [0,∞] by

d(δ, γ) = inf {µ ∈ R+ : ∥δ(x)− γ(x)∥ ≤ µ∆(x, x),∀x ∈ B} ,

and we consider inf ∅ = +∞. Then d is a complete generalized metric on Γ (see [12]).
Now, we define the mapping J : (Γ, d) → (Γ, d) such that

J δ(x) := 4δ
(x
2

)
for all x ∈ B.

Actually, let δ, γ ∈ (Γ, d) be given such that d(δ, γ) = µ. Then

∥δ(x)− γ(x)∥ ≤ µ∆(x, x)

for all x ∈ B. Hence

∥J δ(x)− J γ(x)∥ =
∥∥∥4δ (x

2

)
− 4γ

(x
2

)∥∥∥ ≤ 4µ∆
(x
2
,
x

2

)
≤ Lµ∆(x, x)

for all x ∈ B. It follows that d(J δ(x),J γ(x)) ≤ Lµ. So

d(J δ(x),J γ(x)) ≤ Ld(δ, γ)

for all x ∈ B and all δ, γ ∈ Γ.
It follows from (8) that d(f,J f) ≤ 2+L

4 and d(g,J g) ≤ 2+L
2 .

Using the fixed point alternative we deduce the existence of a unique fixed point of
J and a unique fixed point of J , that is, the existence of mappings F,G : B → B,
respectively, such that

F (x) = 4F
(x
2

)
, G(x) = 4G

(x
2

)
with the following property: there exist µ, η ∈ (0,∞) satisfying

∥f(x)− F (x)∥ ≤ µ∆(x, x), ∥g(x)−G(x)∥ ≤ η∆(x, x)

for all x ∈ B.



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 6 of 10

Since limn→+∞ d(J nf, F ) = 0 and limn→+∞ d(J ng,G) = 0,

lim
n→+∞

4nf
( x

2n

)
= F (x), lim

n→+∞
4ng

( x

2n

)
= G(x)

for all x ∈ B.
Next, d(f, F ) ≤ 1

1−Ld(f,J f) and d(g,G) ≤ 1
1−Ld(g,J g) which imply

∥f(x)− F (x)∥ ≤ 2 + L

4(1− L)
∆(x, x), ∥g(x)−G(x)∥ ≤ 2 + L

2(1− L)
∆(x, x)

for all x ∈ B.
Using (4) and (5), we conclude that

∥F (x+ y) + F (x− y)−G(x)−G(y)∥

= lim
n→+∞

4n
∥∥∥∥f (

x+ y

2n

)
+ f

(
x− y

2n

)
− g

( x

2n

)
− g

( y

2n

)∥∥∥∥
≤ lim

n→+∞
4n∆

( x

2n
,
y

2n

)
≤ lim

n→+∞
Ln∆(x, y) = 0

and

∥G
(
x+ y

2

)
+G

(
x− y

2

)
− F (x)− F (y)∥

= lim
n→+∞

4n
∥∥∥∥g(x+ y

2 · 2n

)
+ g

(
x− y

2 · 2n

)
− f

( x

2n

)
− f

( y

2n

)∥∥∥∥
≤ lim

n→+∞
4n∆

( x

2n
,
y

2n

)
≤ lim

n→+∞
Ln∆(x, y) = 0

for all x, y ∈ B, since L < 1. Hence{
F (x+ y) + F (x− y) = G(x) +G(y),

G
(x+y

2

)
+G

(x−y
2

)
= F (x) + F (y)

for all x, y ∈ B. Therefore by Lemma 1, the mappings F,G : B → B are quadratic.

Corollary 1. Let p and q be nonnegative real numbers with p + q > 4 and f, g : B → B
be mappings satisfying g(0) = 0 and

∥f(x+ y) + f(x− y)− g(x)− g(y)∥ ≤ ∥x∥p∥y∥q,

∥g
(x+y

2

)
+ g

(x−y
2

)
− f(x)− f(y)∥ ≤ ∥x∥p∥y∥q

for all x, y ∈ B. Then there exist unique quadratic mappings F,G : B → B such that

∥F (x)− f(x)∥ ≤ 2p+q + 8

2(2p+q − 16)
∥x∥p+q,

∥G(x)− g(x)∥ ≤ 2p+q + 8

2p+q − 16
∥x∥p+q

for all x ∈ B.



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 7 of 10

Proof. The proof follows from Theorem 2 by taking L = 24

2p+q and ∆(x, y) = ∥x∥p∥y∥q
for all x, y ∈ B.

Corollary 2. Let p and θ be nonnegative real numbers with p > 4 and f, g : B → B be
mappings satisfying g(0) = 0 and

∥f(x+ y) + f(x− y)− g(x)− g(y)∥ ≤ θ(∥x∥p + ∥y∥p),

∥g
(x+y

2

)
+ g

(x−y
2

)
− f(x)− f(y)∥ ≤ θ(∥x∥p + ∥y∥p)

for all x, y ∈ B. Then there exist unique quadratic mappings F,G : B → B such that

∥F (x)− f(x)∥ ≤ 2p + 2

2p − 4
θ∥x∥p,

∥G(x)− g(x)∥ ≤ 2(2p + 2)

2p − 4
θ∥x∥p

for all x ∈ B.

Proof. The proof follows from Theorem 2 by taking L = 4
2p and ∆(x, y) = θ(∥x∥p +

∥y∥p) for all x, y ∈ B.

3. Stability of quadratic F -hom-ders in Banach algebras

In this section, by using the fixed point technique, we prove the Hyers-Ulam stability
of quadratic F -hom-ders in complex Banach algebras.

Theorem 3. Suppose that ∆ : B2 → [0,∞) is a function such that there exists an L < 1
with

∆(x, y) ≤ L

16
∆(2x, 2y) (9)

for all x, y ∈ B. Let f, g : B → B be mappings satisfying g(0) = 0 and{
∥f(x+ y) + f(x− y)− g(x)− g(y)∥ ≤ ∆(x, y),

∥g
(x+y

2

)
+ g

(x−y
2

)
− f(x)− f(y)∥ ≤ ∆(x, y)

(10)

and

∥g(x)g(y)− f(x)g(y)− g(x)f(y)∥ ≤ ∆(x, y) (11)

for all x, y ∈ B. Then there exist unique quadratic mappings F,G : B → B satisfying (6)
and (7) and G is a quadratic F -hom-der.



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 8 of 10

Proof. Since

∆(x, y) ≤ L

16
∆(2x, 2y) ≤ L

4
∆(2x, 2y)

for all x, y ∈ B, by Theorem 2, there exist unique mappings F,G : B → B satisfying (6)
and (7) which are given by

lim
n→+∞

4nf
( x

2n

)
= F (x), lim

n→+∞
4ng

( x

2n

)
= G(x)

for all x ∈ B.
It follows from (11) that

∥G(x)G(y)− F (x)G(y)−G(x)F (y)∥

= lim
n→+∞

16n
∥∥∥g ( x

2n

)
g
( y

2n

)
− f

( x

2n

)
g
( y

2n

)
− g

( x

2n

)
f
( y

2n

)∥∥∥
≤ lim

n→+∞
16n∆

( x

2n
,
y

2n

)
≤ lim

n→+∞
Ln∆(x, y) = 0

for all x, y ∈ B. So

G(x)G(y) = F (x)G(y) +G(x)F (y)

for all x, y ∈ B. Thus the quadratic mapping G is a quadratic F -hom-der.

Corollary 3. Let p and q be nonnegative real numbers with p + q > 4 and f, g : B → B
be mappings satisfying g(0) = 0 and{

∥f(x+ y) + f(x− y)− g(x)− g(y)∥ ≤ ∥x∥p∥y∥q,
∥g

(x+y
2

)
+ g

(x−y
2

)
− f(x)− f(y)∥ ≤ ∥x∥p∥y∥q

and

∥g(xy)− f(x)g(y)− g(x)f(y)∥ ≤ ∥x∥p∥y∥q

for all x, y ∈ B. Then there exist unique quadratic mappings F,G : B → B such that G is
a quadratic F -hom-der and

∥F (x)− f(x)∥ ≤ 2p+q + 8

2(2p+q − 16)
∥x∥p+q,

∥G(x)− g(x)∥ ≤ 2p+q + 8

2p+q − 16
∥x∥p+q

for all x ∈ B.



C. Park, S. Donganont, S. W. Min / Eur. J. Pure Appl. Math, 18 (1) (2025), 5719 9 of 10

Proof. The proof follows from Theorem 3 by taking ∆(x, y) = ∥x∥p∥y∥q for all x, y ∈ B.
Choosing L = 24−p−q, we obtain the desired result.

Corollary 4. Let p and θ be nonnegative real numbers with p > 4 and f, g : B → B be
mappings satisfying g(0) = 0 and{

∥f(x+ y) + f(x− y)− g(x)− g(y)∥ ≤ θ(∥x∥p + ∥y∥q),
∥g

(x+y
2

)
+ g

(x−y
2

)
− f(x)− f(y)∥ ≤ θ(∥x∥p + ∥y∥q)

and

∥g(xy)− f(x)g(y)− g(x)f(y)∥ ≤ θ(∥x∥p + ∥y∥q)

for all x, y ∈ B. Then there exist unique quadratic mappings F,G : B → B such that G is
a quadratic F -hom-der and

∥F (x)− f(x)∥ ≤ 2p + 2

2p − 4
θ∥x∥p,

∥G(x)− g(x)∥ ≤ 2(2p + 2)

2p − 4
θ∥x∥p

for all x ∈ B.

Proof. The proof follows from Theorem 3 by taking ∆(x, y) = θ(∥x∥p + ∥y∥q) for all
x, y ∈ B. Choosing L = 24−p, we obtain the desired result.

4. Conclusion and future work

We solved the system of quadratic functional equations (2) and we defined quadratic
f -hom-ders in Banach algebras and investigated the Hyers-Ulam stability of quadratic
f -hom-ders in Banach algebras. We will define cubic f -hom-ders and quartic f -hom-ders
in Banach algebras, fuzzy Banach algebras and non-Archimedean Banach algebras and
investigate the Hyers-Ulam stability of them.

Acknowledgements

This research was supported by University of Phayao and Thailand Science Research and
Innovation Fund (Fundamental Fund 2025, Grant No. 5020/2567).

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