EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5758
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Ulam Stability of a Pexiderized Additive-quadratic
Equation

Mehdi Dehghanian1,∗, Yamin Sayyari1, Siriluk Donganont2,∗, Choonkil Park3

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

Abstract. Suppose that E is a normed space. In this work, using Brzdȩk fixed point theorem,
we prove the Hyers-Ulam stability of the Pexiderized additive-quadratic functional equation

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

for all x, y ∈ E.

2020 Mathematics Subject Classifications: 39B72, 39B82, 47H10

Key Words and Phrases: Pexiderized additive-quadratic functional equation, Hyers-Ulam sta-
bility, fixed point

1. Introduction and preliminariess

The concept of stability of functional equations originated from a problem of Ulam
[33]. In continue, Hyers gave a positive answer to the question of Ulam in the context of
Banach spaces in the case of additive mappings, that was the first notable advance and a
step toward more solutions in this field. Also, he answered the question of Ulam for the
case of approximate additive mappings under the assumption that G1 and G2 are Banach
spaces (see [19]).

The method provided by Hyers [19] which produces the additive function will be called
a direct method. This method is the most important and powerful tool to concerning the
stability of system of different functional equations [31]. That is, the exact solution of the
functional equation is explicitly constructed as a limit of a sequence, starting from the

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5758

Email addresses: mdehghanian@sirjantech.ac.ir (M. Dehghanian),
y.sayyari@sirjantech.ac.ir (Y. Sayyari),
siriluk.pa@up.ac.th (S. Donganont), baak@hanyang.ac.kr (C. Park)

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



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 2 of 13

given approximate solution (see [14, 23, 32]). The other method is fixed point method,
that is, the exact solution of the functional equation is explicitly constructed as a fixed
point of some certain map [4, 11–13, 27].

Recently, a number of results concerning the stability have been obtained by differ-
ent ways and been applied to a number of functional equations, functional inequalities
and mappings (see [6, 7, 22, 29, 30]). Also, many mathematicians studied the stabilities
additive-quadratic equation and the Drygas’ equation (see [12, 18, 21]).

A mapping f : E → B is said to be additive if it satisfies

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

for all x, y ∈ E. A mapping f : E → B is called quadratic if f satisfies the functional
equation

f(x+ y) + f(x− y) = 2f(x) + 2f(y)

for all x, y ∈ E.
In [1], Aczél and Dhombres showed that if E is a linear space over a field F of charac-

teristic 0, then q : E → F is a solution of the quadratci functional equation if and only if
there is a unique symmetric biadditive mapping L : E2 → F such that q(x) = L(x, x) for
all x ∈ E.

Various Pexiderized versions of the quadratic functional equation have been studied in
[5, 20]. Various works on stability of the quadratic functional equation can be found in
[9, 16, 28].

In 2011, Brzdȩk et al. [8] gave a simple fixed point theorem. Before stating Brzdȩk
fixed point theorem, let us introduce some hypothesis, which we will use in the sequel.
(A1) E is a nonempty set and B is a Banach space.
(A2) δ1, . . . , δk : E → E and λ1, . . . , λk : E → R+ are given maps.
(A3) H : BE → BE is an operator satisfying the inequality

∥Hg(x)−Hl(x)∥ ≤
k∑

i=1

λi(x)∥g (δi(x))− l (δi(x)) ∥

for all g, h : E → B and x ∈ E.
(A4) Λ : RE

+ → RE
+ is a linear operator defined by

ΛF(x) :=

k∑
i=1

λi(x)F (δi(x))

for F : E → R+ and x ∈ E.

Theorem 1. [8] Suppose that the hypotheses (A1)–(A4) are satisfied. Assume that there
are functions µ : E → R+ and φ : E → B such that, for all x ∈ E,

∥Hφ(x)− φ(x)∥ ≤ µ(x)



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 3 of 13

and

µ∗(x) :=

∞∑
n=0

Λnµ(x) <∞

hold. Then, for all x ∈ E the limit

T (x) := lim
n→∞

Hnφ(x)

exists and the mapping T : E → B is a unique fixed point of H with

∥φ(x)− T (x)∥ ≤ µ∗(x)

for all x ∈ E.

Theorem 2. [2] Let f, h : E → B be mappings satisfying

∥f(x+ y) + f(x− y) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥ ≤ ϵ

for some ϵ > 0 and for all x, y ∈ E. Then there exist an additive mapping A : E → B and
a unique quadratic mapping Q : E → B such that

∥h(x)− h(0)−A(x)∥ ≤ 13ϵ,

∥f(x)− f(0)−Q(x)∥ ≤ 24ϵ

for all x ∈ E.

Motivated by the above results on the Hyers-Ulam stability of additive functional
equations, quadratic functional equations, cubic functional equations and quartic func-
tional equations, in the current work, we try to examine the Hyers-Ulam stability of the
following Pexiderized additive-quadratic functional equation

ϕ(u+ 3v)− 5ϕ(u+ 2v)− ϕ(u− 2v) + 10ϕ(u+ v) + 5ϕ(u− v)− 10ϕ(u)− 120ϕ(v) = 0, (1)

in Banach spaces by means of Brzdȩk’s fixed point approach.
A concept employed by Brzdȩk, a lot of articles on hyperstability have been written

on this topic and we refer to [24, 25].
Throughout the paper N0 denotes the set of all non-negative integers.

2. Some auxiliary results

In this section, we establish lemmas for the proof of Hyers-Ulam stability of the func-
tional equation (1).

The next theorem is an example of a very classical result in Hyers-Ulam stability.



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 4 of 13

Theorem 3. [17] Let h : E → B be a mapping satisfying

∥h(x+ y)− h(x)− h(y)∥ ≤ η(∥x∥p + ∥y∥p)

for some η ≥ 0, p > 0, p ̸= 1 and for all x, y ∈ E. Then there is a unique additive mapping
A : E → B such that

∥h(x)−A(x)∥ ≤ 2η

|2p − 2|
∥x∥p (2)

for all x ∈ E.

Researchers obtained new results on Ulam stability of some functional equations using
the Banach limit (see [3, 15]).

In the continue, we need the following lemma, whose proof is similar to the proof of
Theorem 3, and so we will omit it.

Lemma 1. Let h : E → B be a mapping satisfying

∥h(x+ y)− h(x)− h(y)∥ ≤ η(∥x∥p + ∥y∥p) + θ∥x− y∥p

for some η, θ ≥ 0, p > 0, p ̸= 1 and for all x, y ∈ E. Then there is a unique additive
mapping A : E → B satisfying (2).

Lemma 2. Let f : E → B be a mapping satisfying

∥f(x+ y) + f(x− y)− 2f(x)− 2f(y)∥ ≤ η(∥x∥p + ∥y∥p) + θ∥x− y∥p (3)

for some η, θ ≥ 0, p > 0, p ̸= 2 and for all x, y ∈ E. Then there exists a unique quadratic
mapping Q : E → B such that

∥f(x)−Q(x)∥ ≤ 2η

|2p − 4|
∥x∥p

for all x ∈ E.

Proof. Putting x = y = 0 in (3), we obtain f(0) = 0.
Setting y = x in (3) and dividing by 4, we obtain

∥f(x)− 1

4
f(2x)∥ ≤ η

2
∥x∥p (4)

for all x ∈ E.
Let H : BE → BE and µ : E → R+ be defined by

Hg(x) = 1

4
g(2x), g ∈ BE

and

µ(x) =
η

2
∥x∥p



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 5 of 13

for all x ∈ E. Then

∥Hf(x)− f(x)∥ ≤ µ(x)

for all x ∈ E. Hence

∥Hg(x)−Hl(x)∥ ≤ 1

4
∥g(2x)− l(2x)∥

For all g, h ∈ BE and x ∈ E, H : BE → BE satisfies the condition (A3) with λ1(x) =
1
4

and δ1(x) = 2x. By (A4), the operator Λ : RE
+ → RE

+ is defined by:

ΛF(x) =
1

4
F(2x), F ∈ RE

+

for all x ∈ E. Hence

Λµ(x) =
1

4
µ(2x) = 2p−2µ(x), µ ∈ RE

+

for all x ∈ E. Since Λ is linear,

Λnµ(x) = 2n(p−2)µ(x), n ∈ N0

for all x ∈ E.
If p < 2, then the series

∑∞
n=0 Λ

nη(x) is convergent for all x ∈ E and

µ∗(x) =
∞∑
n=0

Λnµ(x) =
∞∑
n=0

2n(p−2)µ(x) =
2η

4− 2p
∥x∥p

for all x ∈ E. By Theorem 1, there exists a mapping Q : E → B such

Q(x) = lim
n→∞

Hnf(x), Q(x) =
1

4
Q(2x)

and

∥f(x)−Q(x)∥ ≤ 2η

4− 2p
∥x∥p

for all x ∈ E.
Next, by induction on n, one can see that

∥Hnf(x+ y) +Hnf(x− y)− 2Hnf(x)− 2Hnf(y)∥
≤ 2n(p−2) [η (∥x∥p + ∥y∥p) + θ∥x− y∥p]

for all x, y ∈ E and n ∈ N0. By letting n→ ∞ we conclude that Q is a quadratic mapping.
Now, we consider the case p > 2. Replacing x and y by x

2 in (3), we have∥∥∥f(x)− 4f
(x
2

)∥∥∥ ≤ 2η

2p
∥x∥p



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 6 of 13

for all x ∈ E. Consider

Hg(x) = 4g
(x
2

)
, g ∈ BE ,

ΛF(x) = 4F
(x
2

)
, F ∈ RE

+

and µ(x) = 2η
2p ∥x∥

p for all x ∈ E. Also,

Λµ(x) = 22−pµ(x)

for all x ∈ E. Since p > 2, the serie
∑∞

n=0 Λ
nµ(x) is convergent for all x ∈ E and

µ∗(x) =

∞∑
n=0

Λnµ(x) =
2η

2p − 4
∥x∥p

for all x ∈ E. So, by Theorem 1 there is Q : E → B such that

Q(x) = lim
n→∞

Hnf(x), Q(x) = 4Q
(x
2

)
and

∥f(x)−Q(x)∥ ≤ 2η

2p − 4
∥x∥p

for all x ∈ E. It follows from (3) and by induction n ∈ N0 that

∥Hnf(x+ y) +Hnf(x− y)− 2Hnf(x)− 2Hnf(y)∥
≤ 2n(2−p) [η (∥x∥p + ∥y∥p) + θ∥x− y∥p]

for all x, y ∈ E. Therefore, Q satisfies the quadratic functional equation.
To prove the uniqueness of Q for the case p < 2, assume that Q1, Q2 : E → B satisfy

the quadratic functional equation on E and

∥f(x)−Q1(x)∥ ≤ η1∥x∥p, ∥f(x)−Q2(x)∥ ≤ η2∥x∥p

for some η1, η2 ≥ 0 and for all x ∈ E. Then

∥Q1(x)−Q2(x)∥ ≤ (η1 + η2)∥x∥p

for all x ∈ E. Hence

Q1(x) =
1

4
Q1(2x), Q2(x) =

1

4
Q2(2x)

for all x ∈ E. Thus

∥Q1(x)−Q2(x)∥ ≤ 1

4
∥Q1(2x)−Q2(2x)∥ ≤ 2p

4
(η1 + η2)∥x∥p



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 7 of 13

for all x ∈ E. By induction on n ∈ N0 we see that

∥Q1(x)−Q2(x)∥ ≤
(
2p

4

)n

(η1 + η2)∥x∥p

which tends to 0 as n→ ∞ for all x ∈ E. This implies

Q1(x) = Q2(x)

for all x ∈ E. The proofs of the cases p > 2 runs as before.

3. Main results

In this section, we investigate the Hyers-Ulam stability of the Pexiderized additive-
quadratic functional equation (1) in Banach spaces.

Theorem 4. Let f, h : E → B be mapings satisfying

∥f(x+ y) + f(x− y) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥ ≤ ϵ (∥x∥p + ∥y∥p) (5)

for some ϵ ≥ 0, p > 0, p ̸= 1, 2 and for all x, y ∈ E. Then there exist an additive mapping
A : E → B and a unique quadratic mapping Q : E → B such that

∥h(x)− h(0)−A(x)∥ ≤ 8 + 24−p

|2p − 2|
ϵ∥x∥p,

∥f(x)− f(0)−Q(x)∥ ≤ 10 + 24−p

|2p − 4|
ϵ∥x∥p

for all x ∈ E.

Proof. Interchanging x with y in (5), we obtain

∥f(x+ y) + f(y − x) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥ ≤ ϵ (∥x∥p + ∥y∥p) (6)

for all x, y ∈ E. From (5) and (6) it follows that

∥f(x− y)− f(y − x)∥ ≤ 2ϵ (∥x∥p + ∥y∥p)

for all x, y ∈ E. Putting y = 0 in the above inequality, we get

∥f(x)− f(−x)∥ ≤ 2ϵ∥x∥p (7)

for all x ∈ E.
Substituting x and then −x in the place of y in (5), we have

∥f(2x) + f(0) + h(2x)− 4f(x)− 2h(x)∥ ≤ 2ϵ∥x∥p, (8)

∥f(0) + f(2x) + h(0)− 2f(x)− 2f(−x)− h(x)− h(−x)∥ ≤ 2ϵ∥x∥p (9)



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 8 of 13

for all x ∈ E. From (7), (8) and (9), we obtain

∥h(2x)− h(x) + h(−x)− h(0)∥ (10)

≤ ∥f(2x) + f(0) + h(2x)− 4f(x)− 2h(x)∥
+∥f(0) + f(2x) + h(0)− 2f(x)− 2f(−x)− h(x)− h(−x)∥+ 2∥f(x)− f(−x)∥

≤ 2ϵ∥x∥p + 2ϵ∥x∥p + 4ϵ∥x∥p = 8ϵ∥x∥p

for all x ∈ E.
Replacing x by −x in the last inequality, we obtain

∥h(−2x) + h(x)− h(−x)− h(0)∥ ≤ 8ϵ∥x∥p (11)

for all x ∈ E.
It follows from (10) and (11) that

∥h(2x) + h(−2x)− 2h(0)∥ ≤ 16ϵ∥x∥p,

which implies

∥h(x) + h(−x)− 2h(0)∥ ≤ 16

2p
ϵ∥x∥p (12)

for all x ∈ E. Replacing y by −y in (5), we have

∥f(x+ y)+ f(x− y)+h(x− y)− 2f(x)− 2f(−y)−h(x)−h(−y)∥ ≤ ϵ (∥x∥p + ∥y∥p) (13)

for all x, y ∈ E. By (5), (7), (12) and (13), we have

∥h(x+ y)− h(x− y)− 2h(y) + 2h(0)∥ (14)

≤ ∥f(x+ y) + f(x− y) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥
+∥f(x+ y) + f(x− y) + h(x− y)− 2f(x)− 2f(−y)− h(x)− h(−y)∥
+2∥f(y)− f(−y)∥+ ∥h(y) + h(−y)− 2h(0)∥

≤ ϵ (∥x∥p + ∥y∥p) + ϵ (∥x∥p + ∥y∥p) + 4ϵ∥y∥p + 16

2p
ϵ∥y∥p

= ϵ
(
2∥x∥p +

(
6 + 24−p

)
∥y∥p

)
for all x, y ∈ E. Interchanging x with y in (14), we get

∥h(x+ y)− h(y − x)− 2h(x) + 2h(0)∥ ≤ ϵ
(
2∥y∥p +

(
6 + 24−p

)
∥x∥p

)
(15)

for all x, y ∈ E. By (12), (14) and (15), we have

∥2h(x+ y)− 2h(x)− 2h(y) + 2h(0)∥
≤ ∥h(x+ y)− h(x− y)− 2h(y) + 2h(0)∥+ ∥h(x+ y)− h(y − x)− 2h(x) + 2h(0)∥
+∥h(x− y) + h(y − x)− 2h(0)∥

≤
(
8 + 24−p

)
ϵ (∥x∥p + ∥y∥p) + 24−pϵ∥x− y∥p,



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 9 of 13

which implies

∥h(x+ y)− h(x)− h(y) + h(0)∥ ≤
(
4 + 23−p

)
ϵ (∥x∥p + ∥y∥p) + 23−pϵ∥x− y∥p (16)

for all x, y ∈ E. Define ĥ : E → B by ĥ(x) := h(x) − h(0) for all x ∈ E. Then we can
rewrite the inequality (16) in the form

∥ĥ(x+ y)− ĥ(x)− ĥ(y)∥ ≤
(
4 + 23−p

)
ϵ (∥x∥p + ∥y∥p) + 23−pϵ∥x− y∥p

for all x, y ∈ E. Applying Lemma 1 to ĥ, we get a unique additive mapping A1 : E → B
such that

∥ĥ(x)−A1(x)∥ ≤ 8 + 24−p

|2p − 2|
ϵ∥x∥p,

which implies

∥h(x)− h(0)−A1(x)∥ ≤ 8 + 24−p

|2p − 2|
ϵ∥x∥p

for all x ∈ E. Letting x = y = 0 in (5), we have

2f(0) + h(0) = 0. (17)

Next, using (5), (16) and (17), we get

∥f(x+ y) + f(x− y)− 2f(x)− 2f(y) + 2f(0)∥
≤ ∥f(x+ y) + f(x− y) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥
+∥h(x+ y)− h(x)− h(y) + h(0)∥

≤ ϵ (∥x∥p + ∥y∥p) +
(
4 + 23−p

)
ϵ (∥x∥p + ∥y∥p) + 23−pϵ∥x− y∥p

=
(
5 + 23−p

)
ϵ (∥x∥p + ∥y∥p) + 23−pϵ∥x− y∥p

for all x, y ∈ E.
Similarly, we define f̂ : E → B by f̂(x) := f(x)− f(0) for all x ∈ E. Then we obtain

∥f̂(x+ y) + f̂(x− y)− 2f̂(x)− 2f̂(y)∥ ≤
(
5 + 23−p

)
ϵ (∥x∥p + ∥y∥p) + 23−pϵ∥x− y∥p

for all x, y ∈ E. By Lemma 2, there exists a unique quadratic mapping Q : E → B such
that

∥f̂(x)−Q(x)∥ ≤ 10 + 24−p

|2p − 4|
ϵ∥x∥p,

which implies

∥f(x)− f(0)−Q(x)∥ ≤ 10 + 24−p

|2p − 4|
ϵ∥x∥p

for all x ∈ E, which ends our proof.

The following example shows that for p = 1 the Pexider additive-quadratic functional
equation (1) is not stable (see [26]).



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 10 of 13

Example 1. Define ψ : R → R by

ψ(x) =


−b x ≤ −1

bx −1 < x < 1

b x ≥ 1

where a, b > 0 and assume that f, h : R → R are defined by

f(x) = ax2 and h(x) =
∞∑
n=0

φ (2nx)

2n

for all x ∈ R. We show that

∥f(x+ y) + f(x− y) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥ ≤ 8b (|x|+ |y|)

for all x, y ∈ R, but there are no constant k ≥ 0 and no mapping A : R → R satisfying (1)
and

|h(x)− h(0)−A(x)| ≤ k|x|

for all x ∈ R.

In the following we show that for p = 2 the equation (1) is not stable (see [10]).

Example 2. Define φ : R → R by

φ(x) =

{
ax2 −1 < x < 1

a |x| ≥ 1

where a, b > 0 and assume that f, h : R → R are defined by

f(x) =
∞∑
n=0

φ (2nx)

4n
and h(x) = bx

for all x ∈ R. We show that

∥f(x+ y) + f(x− y) + h(x+ y)− 2f(x)− 2f(y)− h(x)− h(y)∥ ≤ 32a
(
|x|2 + |y|2

)
for all x, y ∈ R, but there are no constant k ≥ 0 and no mapping Q : R → R satisfying (1)
and

|f(x)− f(0)−Q(x)| ≤ k|x|2

for all x ∈ R.



M. Dehghanian et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5758 11 of 13

4. Conclusion and Future Works

In this work, using Brzdȩk fixed point theorem, we proved the Hyers-Ulam stability
of the Pexiderized additive-quadratic functional equation (1) in Banach spaces. We can
apply the method to study the Hyers-Ulam stability problems of the Pexiderized additive-
quadratic functional equation (1) in fuzzy Banach spaces, matrix Banach spaces, Hilbert
C∗-modules and fuzzy Hilbert C∗-modules, in future work.

Acknowledgements

The authors are thankful to the editors and the anonymous reviewers for many valuable
suggestions to improve this paper.

Funding

S. Donganont was supported by the University of Phayao and Thailand Science Re-
search and Innovation Fund (Fundamental Fund 2025, Grant No. 5020/2567).
Conflict of interest
The authors declare that they have no competing interests.

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