EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6197
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

The Stability of the Generalized Functional Equation

Gang Lyu1, Yingxiu Jiang2,∗, Qi Liu3, Choonkil Park4

1 School of General Education, Guangzhou College of Technology and Business,
Guangzhou 510850, P.R. China
2 Department of Mathematics, Yanbian University, Yanji 133001, P.R. China
3 School of Mathematics and Physics, Anqing Normal University, Anqing 246133,
P.R. China
4 Research Institute for Convergence of Basic Science, Hanyang University, Seoul 04763,
Korea

Abstract. The aim of this paper is to prove the stability (in the sense of Ulam) of the functional
equation:

f(x) = α(x)f(f1(x)) + β(x)f(f2(x)),

where α and β are given real valued functions defined on a nonempty set S such that sup{|α(x)| :
x ∈ S} < 1 and fi(x)(i = 1, 2) are given mappings.

2020 Mathematics Subject Classifications: 39B62, 39B52, 47H10, 46B25

Key Words and Phrases: Functional inequations, Hyers-Ulam stability, Banach space

1. Introduction and preliminaries

The origin of the stability problem for functional equations traces back to a question
posed by Ulam [1]. Hyers [2] made remarkable headway in 1941. In the context of Banach
spaces, he derived highly renowned and captivating results regarding the Cauchy functional
equation. Later, there are also very many generalizations of Cauchy equations (see [3–5]).
For more information concerning various functional equations, see [6–16].

Baker was the first to use the fixed point method to study the Hyers-Ulam stability
of functional equations in [17]. In that work, he actually utilized the following variant of
Banach’s fixed point theorem. For the functional equation

f(t) = (α(t) + β(t))f(ϕ(t)),

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6197

Email addresses: lvgang@gzgs.edu.cn (G. Lyu),
yxjiang@ybu.edu.cn (Y. Jiang), liuq67@aqnu.edu.cn (Q. Liu),
baak@hanyang.ac.kr (C. Park)

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



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 2 of 17

if

∥g(t)− {α(t) + β(t)}g(ϕ(t))∥ ≤ δ,

then there exists a unique mapping f satisfying this functional equation, and

∥f(t)− g(t)∥ ≤ δ

1− λ
.

Sikorska [18] built a direct method (see, e.g., [19, 20]) to improve the approximating
constant for the following mappings

∥f(x)− af(h(x))− bf(−h(x))∥ ≤ ϵ(x),

∥f(x)− af(hn(x))− bf(hn+1(x))∥ ≤ ϵ(x).

It improves the research results of Sikorska [18], breaking through the limitation that
her method could only handle cases where the intermediate function is odd. The paper
proposes a new direct method, which is not only used to solve specific functional equations
but also to construct and study a series of functional equations. This provides new research
tools and ideas for this field. The fixed point theory is a pivotal method for proving
the stability of functional equations[21],widely applied in differential equations[22] and
computer science[23], this theory transforms abstract solvability problems into concrete
analyses of operator properties, remaining indispensable for establishing existence and
uniqueness in functional equation frameworks.

Thoroughly, we explore the dependence relationships and properties of different param-
eters in generalized functional equations. When studying single-variable abstract equa-
tions, we need to analyze the influence of parameters such as α and β on the solutions.
Also, expand the parameters from the real number field to the complex number field to
reveal the characteristics of the equations more comprehensively.

In this paper, we discuss the Hyers-Ulam stability of the following functional equation

∥f(x)− α(x)f(g(x))− β(x)f(h(x))∥ ≤ ϵ(x)

in Banach spaces.
In fact, the above mentioned inequality represents an even more general form. In

Section 2, we introduce certain improvements to the existing approximations. Specifically,
we enhance the existing approximation methods, aiming to obtain more accurate and
comprehensive results. These improvements contribute to a deeper understanding of the
problem and provide more effective tools for subsequent research. In Section 3, we explore
several applications of the stability results. We not only demonstrate the practical value of
the stability theory in different scenarios but also expand its scope of application, showing
its potential in solving real-world problems. For the sake of simplicity, we present our
results for functions with values in Banach spaces. However, with some minor additional
assumptions, these results can be extended and reformulated in more general Banach
space settings. This indicates that our research has the potential to be further generalized,
making it applicable to a wider range of mathematical models and real world applications.



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 3 of 17

2. Main results

In this section, we prove the main theorem of the paper.

Theorem 1. Suppose that X is a linear normed space and Y is a Banach space. Let
f : X → Y be a mapping satisfying

∥f(x)− α(x)f(g(x))− β(x)f(h(x))∥ ≤ ϵ(x) (1)

for all x ∈ X. If α(x) and β(x) are functions from X to R, and g, h : X → X are
mappings and ϵ : X → [0,∞) is a function such that

∞∑
n=0

(Λnϵ)(x) =: ϵ∗(x) <∞

holds and Λ is a linear operator defined by

(Λδ)(x) := |α(x)|δ(g(x)) + |β(x)|δ(h(x))

for a function δ : X → [0,∞) and x ∈ X, then there exists a unique determined mapping
K : X → Y , given by

K(x) = α(x)K(g(x)) + β(x)K(h(x))

such that

∥f(x)−K(x)∥ ≤ ϵ∗(x) (2)

for all x ∈ X.

Proof. Let T : Y X → Y X be an operator satisfying (Tf)(x)) = α(x)f(g(x)) +
β(x)f(h(x)) in (1). Then we get

∥f(x)− (Tf)(x))∥ ≤ ϵ(x). (3)

At the same time, we can see that

∥(Tξ)(x)− (Tζ)(x)∥
≤ |α(x)|∥ξ(g(x))− ζ(g(x))∥+ |β(x)|∥ξ(h(x))− ζ(h(x))∥

(4)

for all ξ, ζ ∈ Y X and x ∈ X.
First, we get by induction that, for all n ∈ N,

∥(Tnf)(x))− (Tn+1f)(x)∥ ≤ (Λnϵ)(x), x ∈ X. (5)

Obviously, from (3), for the case n = 0, (5) holds. Now fix n ∈ N and suppose that the
inequality (5) is valid. Then, using (4) for all x ∈ X, we have

∥(Tn+1f)(x)− (Tn+2f)(x)∥
≤ |α(x)|∥(Tnf)(g(x))− (Tn+1f)(g(x))∥+ |β(x)|∥(Tnf)(h(x))− (Tn+1f)(h(x))∥
≤ |α(x)|(Λnϵ)(g(x)) + |β(x)|(Λnϵ)(h(x)) = (Λn+1ϵ)(x).



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 4 of 17

Thus we complete the proof of (5). For n, k ∈ N , k > 0,

∥(Tnf)(x)− (Tn+kf)(x)∥ ≤
k−1∑
i=0

∥(Tnf)(x)− (Tn+i+1f)(x)∥

≤
n+k−1∑
i=n

(Λiϵ)(x) ≤ ϵ∗(x), x ∈ X.

(6)

From the convergence of the series
∑

(Λnϵ)(x), for every x ∈ X, {(Tnf)(x)}n∈N is a
Cauchy sequence. Since Y is complete, we can define limn→∞(Tnf)(x) := ψ(x). Taking
n = 0 and k → ∞ in (6), we know that (2) holds, and

∥(Tψ)(x)− (Tn+1f)(x)∥ ≤ (Λn+1)ϵ(x), n ∈ N , x ∈ X,

and thus

(Tψ)(x) = lim
n→∞

(Tn+1ψ)(x) = ψ(x), x ∈ X.

In order to prove the uniqueness of ψ, suppose that ψ1, ψ2 ∈ Y X are two fixed points of
T with ∥ψi(x)− f(x)∥ ≤ ϵ∗(x) for all x ∈ X, i = 1, 2. We can easily show that

∥ψ1(x)− ψ2(x)∥ = ∥(Tmψ1)(x)− (Tmψ2)(x)∥ ≤ 2
∞∑

i=m

(Λiϵ)(x), x ∈ X.

Thus ψ1(x) = ψ2(x).

Remark 1. We can easily prove∥∥∥∥∥f(x)−
n∑

i=0

αi(x)(f(fi(x)))

∥∥∥∥∥ ≤ ϵ(x)

for all x ∈ X. Thus there exists a unique determined mapping K : X → Y , given by

K(x) =
n∑

i=0

αi(x)(K(fi(x)), x ∈ X,

such that

∥f(x)−K(x)∥ ≤ ϵ∗(x)

for all x ∈ X.



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 5 of 17

3. The stability of generalized functional equations in Banach spaces

In the section, we use Theorem 1 to prove the Hyers-Ulam stability of functional
inequalities in Banach spaces. The several mathematicians have investigated the problems
of Hyers-Ulam stability of functional equations (see [19, 20, 24–35]).

Theorem 2. Let ϵ : X3 → [0,∞) be a function with ϵ(0, 0, 0) = 0 such that there exists
an L < 1 with

ϵ(kx, ky, kz) ≤ |k|Lϵ(x, y, z)

for all x, y, z ∈ X and k ∈ R. Suppose that X is a linear normed space and Y is a Banach
space. Let f : X → Y be a mapping satisfying

∥f(Ax+By + Cz) + f(Ax) + γ(x)f(z)

−α(x)f(Ax+ y)− β(x)f(x+ Cz)− γ(x)f(By + z)∥
≤ ϵ(x, y, z),

(7)

for all x, y, z ∈ X, where α, β, γ : X → R are functions and A,B,C ∈ R such that
à :=

∣∣A+B+C
A

∣∣ + ∣∣γ ( x
A

)∣∣ ∣∣ 1
A

∣∣ + ∣∣α (
x
A

)∣∣ ∣∣A+1
A

∣∣ + ∣∣β ( x
A

)∣∣ ∣∣C+1
A

∣∣ + ∣∣γ ( x
A

)∣∣ ∣∣B+1
A

∣∣ < 1. Then
there exists a unique mapping k : X → Y such that

∥f(x)− k(x)∥ ≤ 1

1− ÃL
ϵ(x, x, x)

for all x ∈ X.

Proof. Letting x = y = z in (7), we get

∥f((A+B + C)x) + f(Ax) + γ(x)f(x)

−α(x)f(Ax+ x)− β(x)f(x+ Cx)− γ(x)f(Bx+ x)∥
≤ ϵ(x, x, x).

Thus ∥∥∥∥f ((A+B + C

A

)
x

)
+ f(x) + γ

( x
A

)
f
( x
A

)
−α

( x
A

)
f

(
(A+ 1)x

A

)
− β

( x
A

)
f

(
(C + 1)x

A

)
− γ

( x
A

)
f

(
(B + 1)x

A

)∥∥∥∥
≤ ϵ

( x
A
,
x

A
,
x

A

)
≤

∣∣∣∣ 1A
∣∣∣∣Lϵ(x, x, x).

In Remark 1, let α1(x) = −γ
(
x
A

)
, α2(x) = −1, α3(x) = α

(
x
A

)
, α4 = β

(
x
A

)
, α5(x) =

γ
(
x
A

)
, f1(x) =

x
A , f2(x) =

A+B+C
A , f3(x) =

(A+1)x
A , f4(x) =

(C+1)x
A , f5(x) =

(B+1)x
A .



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 6 of 17

Consider T : Y X → Y X given as

Tf(x) = −f
((

A+B + C

A

)
x

)
− γ

( x
A

)
f
( x
A

)
+α

( x
A

)
f

(
(A+ 1)x

A

)
+ β

( x
A

)
f

(
(C + 1)x

A

)
+ γ

( x
A

)
f

(
(B + 1)x

A

)
for all x ∈ X, f ∈ Y X .

We can define the operator Λ : RX
+ → RX

+ , given by

Λf(x) = f

((
A+B + C

A

)
x

)
+
∣∣∣γ ( x

A

)∣∣∣ f ( x
A

)
+
∣∣∣α( x

A

)∣∣∣ f ((A+ 1)x

A

)
+
∣∣∣β ( x

A

)∣∣∣ f ((C + 1)x

A

)
+

∣∣∣γ ( x
A

)∣∣∣ f ((B + 1)x

A

)
for all x ∈ X. In particular,

Λϵ(x, x, x) = ϵ

((
A+B + C

A

)
x,

(
A+B + C

A

)
x,

(
A+B + C

A

)
x

)
+

∣∣∣γ ( x
A

)∣∣∣ ϵ( x
A
,
x

A
,
x

A

)
+

∣∣∣α( x
A

)∣∣∣ ϵ((A+ 1)x

A
,
(A+ 1)x

A
,
(A+ 1)x

A

)
+

∣∣∣β ( x
A

)∣∣∣ ϵ((C + 1)x

A
,
(C + 1)x

A
,
(C + 1)x

A

)
+

∣∣∣γ ( x
A

)∣∣∣ ϵ((B + 1)x

A
,
(B + 1)x

A
,
(B + 1)x

A

)
≤

∣∣∣∣A+B + C

A

∣∣∣∣Lϵ (x, x, x)
+

∣∣∣γ ( x
A

)∣∣∣ ∣∣∣∣ 1A
∣∣∣∣Lϵ (x, x, x)

+
∣∣∣α( x

A

)∣∣∣ ∣∣∣∣A+ 1

A

∣∣∣∣Lϵ (x, x, x)
+

∣∣∣β ( x
A

)∣∣∣ ∣∣∣∣C + 1

A

∣∣∣∣Lϵ (x, x, x)
+

∣∣∣γ ( x
A

)∣∣∣ ∣∣∣∣B + 1

A

∣∣∣∣Lϵ (x, x, x)
≤

(∣∣∣∣A+B + C

A

∣∣∣∣+ ∣∣∣γ ( x
A

)∣∣∣ ∣∣∣∣ 1A
∣∣∣∣+ ∣∣∣α( x

A

)∣∣∣ ∣∣∣∣A+ 1

A

∣∣∣∣
+
∣∣∣β ( x

A

)∣∣∣ ∣∣∣∣C + 1

A

∣∣∣∣+ ∣∣∣γ ( x
A

)∣∣∣ ∣∣∣∣B + 1

A

∣∣∣∣)Lϵ (x, x, x)
= ÃLϵ (x, x, x) .



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 7 of 17

Since Λ is linear, we can get

Λnϵ(x, x, x) = (ÃL)nϵ(x, x, x), x ∈ X, n ∈ N0.

Since à < 1, the series
∑∞

n=0 Λ
nϵ(x, x, x) is convergent for every x ∈ X and

ϵ∗(x, x, x) =
∞∑
n=0

Λnϵ(x) =
∞∑
n=0

(
ÃL

)n
ϵ(x, x, x)

=
1

1− ÃL
ϵ(x, x, x), x ∈ X.

By Theorem 1, there exists a mapping l : X → Y such that

l(x) = lim
n→∞

Tnf(x),

l(x) = −l
((

A+B + C

A

)
x

)
− γ

( x
A

)
l
( x
A

)
+α

( x
A

)
l

(
(A+ 1)x

A

)
+ β

( x
A

)
l

(
(C + 1)x

A

)
+ γ

( x
A

)
l

(
(B + 1)x

A

)
,

∥f(x)− k(x)∥ ≤ 1

1− ÃL
ϵ(x, x, x).

This completes the proof.

Theorem 3. Suppose that X is a linear normed space, Y is a Banach space and θ ≥ 0.
Let f : X → Y be a mapping satisfying

∥f(ax+ by) + f(x− y)−Af(x)−Bf(y)−Df(−y)∥ ≤ θ(∥x∥+ ∥y∥) (8)

for all x, y ∈ X. If a, b, A, B and D satisfy |A+B|+|D|
|a+b| < 1 and |A+B+D− 2| ≠ 0, then

there exists a unique mapping k : X → Y such that

∥f(x)− k(x)∥ ≤ 2θ∥x∥
|a+ b| − (|A+B|+ |D|)

for all x ∈ X.

Proof. Letting x = y = 0 in (8), we get f(0) = 0. Letting y = x in (8), we get∥∥∥∥f(x)− (A+B)f

(
1

a+ b
x

)
−Df

(
− 1

a+ b
x

)∥∥∥∥ ≤ 2θ∥x∥
|a+ b|

(9)

for all x ∈ X.
Consider T : Y X → Y X and ϵ : X → R+,

Jf(x) = (A+B)f

(
1

a+ b
x

)
+Df

(
− 1

a+ b
x

)



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 8 of 17

for all x ∈ X, f ∈ Y X and

ϵ(x) =
2θ∥x∥
|a+ b|

, x ∈ X.

The inequality (9) becomes

∥Jf(x)− f(x)∥ ≤ ϵ(x),∀x ∈ X.

For every g, h ∈ Y X and x ∈ X,

∥Tg(x)− Th(x)∥ =

∥∥∥∥(A+B)g

(
1

a+ b
x

)
+Dg

(
− 1

a+ b
x

)
−(A+B)h

(
1

a+ b
x

)
−Dh

(
− 1

a+ b
x

)∥∥∥∥
≤ |A+B|

∥∥∥∥g( 1

a+ b
x

)
− h

(
1

a+ b
x

)∥∥∥∥
+ |D|

∥∥∥∥g(− 1

a+ b
x

)
− h

(
− 1

a+ b
x

)∥∥∥∥ .
Thus we can define f1(x) =

1
a+bx, f2(x) = − 1

a=bx, L1(x) = |A + B|, L2(x) = |D| and the

operator Λ : RX
+ → RX

+ , given by

Λη(x) = |A+B|η
(

1

a+ b
x

)
+ |D|η

(
− 1

a+ b
x

)
for all x ∈ X. In particular,

Λϵ(x) = |A+B|ϵ
(

1

a+ b
x

)
+ |D|ϵ

(
− 1

a+ b
x

)
= |A+B|θ 2∥x∥

|a+ b|2
+ |D|θ 2∥x∥

|a+ b|2

=

{
|A+B|+ |D|

|a+ b|

}
ϵ(x).

Since Λ is linear, we can get

Λnε(x) =

{
|A+B|+ |D|

|a+ b|

}n

ϵ(x), x ∈ X, n ∈ N0.



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 9 of 17

Since |A+B|+|D|
|a+b| < 1, the series

∑∞
n=0 Λ

nϵ(x) is convergent for every x ∈ X and

ε∗(x) =
∞∑
n=0

Λnϵ(x) =
∞∑
n=0

{
|A+B|+ |D|

|a+ b|

}n

ϵ(x)

=
1

1− |A+B|+|D|
|a+b|

ϵ(x)

=
1

1− |A+B|+|D|
|a+b|

2θ∥x∥
|a+ b|

=
2θ∥x∥

|a+ b| − (|A+B|+ |D|)
, x ∈ X.

By Theorem 1, there exists a mapping k : X → Y such that

k(x) = lim
n→∞

Jnf(x),

k(x) = (A+B)k

(
1

a+ b
x

)
+Dk

(
− 1

a+ b
x

)
,

∥f(x)− k(x)∥ ≤ 2θ∥x∥
|a+ b| − (|A+B|+ |D|)

.

This completes the proof.

4. The stability of the generalized function inequations

This section primarily investigates the stability of functional equations, improves ex-
isting results through a novel direct method, and explores the dependency relationships
among different parameters in generalized functional equations along with their related
properties. It also solves functional inequalities and constructs as well as studies inequali-
ties related to functional equations. Park [36] introduced additive ρ-functional inequalities
and employed the direct method to establish the Hyers-Ulam stability of these inequal-
ities within Banach spaces. Subsequently, in 2016, Choi et al. [37] investigated addi-
tive ρ-functional inequalities in normed spaces. Leveraging the fixed-point method, they
demonstrated the Hyers-Ulam stability of two distinct additive ρ-functional inequalities

∥f(x+ y)− f(x)− f(y)∥ ≤
∥∥∥∥ρ(2f (x+ y

2

)
− f(x)− f(y)

)∥∥∥∥
and ∥∥∥∥2f (x+ y

2

)
− f(x)− f(y)

∥∥∥∥ ≤ ∥ρ(f(x+ y)− f(x)− f(y))∥

(where ρ < 1) in normed spaces. In 2023, Nawaz et al. [38] analyzed the Hyers-Ulam
stability of cubic and quartic ρ-functional inequalities in fuzzy matrix spaces. They used



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 10 of 17

the fixed-point method to study the following functional inequalities:
(f(2x+ y) + f(2x− y)− 2f(x+ y)− 2f(x− y)− 12f(x)

−ρ(4f(x+ y
2 ) + 4(f(x− y

2 )− f(x+ y)− f(x− y))− 6f(x), r)| ≥ r
r+φ(x,y) ,

f(2x+ y) + f(2x− y)− 4f(x+ y)− 4f(x− y)− 24f(x) + 6f(y)

−ρ(8f(x+ y
2 ) + 8(f(x− y

2 )− 2f(x+ y)− 2f(x− y))− 12f(x) + 3f(y), r) ≥ r
r+φ(x,y)

where ρ ̸= 2 is a real number.
Functional equations play a fundamental role in mathematics and its applications,

particularly in areas such as information theory, economics, and decision sciences. One
important equation is the generalized Drygas equation:

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

whose solution is called a Drygas mapping. The general solution of the above functional
equation was given by Ebanks, Kannappan and Sahoo [39] as

f(x) = Q(x) +A(x),

where A is an additive mapping and Q is a quadratic mapping offering insights into the
underlying economic behavior[40].The following theorem explores the solutions and sta-
bility properties of the generalized Drygas equation, extending its applications to broader
mathematical and practical contexts.

Theorem 4. Let ϕ : X2 → [0,∞) be a function with ϕ(0, 0) = 0 such that there exists an
L < 1 with

ϕ(ax, ay) ≤ |a|Lϕ(x, y)

for all x, y ∈ X. Suppose that X is a linear normed space, Y is a Banach space, and
ρ : X → (0, 1] is a function. Let f : X → Y be an odd mapping satisfying

∥f(ax+ by) + f(ax− by)−Af(x)−Bf(y)−Df(−y)∥
≤ ∥ρ(x)(f(ax+ by)− f(ax)− f(by))∥+ ϕ(x, y)

(10)

for all x, y ∈ X. If L, A and a satisfy 2|a|L
|A| < 1, then there exists a unique additive

mapping k : X → Y such that

∥f(x)− k(x)∥ ≤ |A|
|A| − 2L|a|

ϕ(x, 0)

for all x ∈ X.

Proof. Letting y = 0 in (10), we get

∥f(x)− 2

A
f(ax)∥ ≤ 1

|A|
ϕ(x, 0), x ∈ X.



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 11 of 17

Consider the mapping J : Y X → Y X such that

Jξ(x) =
2

A
ξ(ax), x ∈ X, ξ ∈ Y X .

Then we get

∥Jf(x)− f(x)∥ ≤ 1

|A|
ϕ(x, 0), x ∈ X.

For every ξ, µ ∈ Y X ,

∥Jg(x)− Jh(x)∥ =
2

|A|
∥g(ax)− h(ax)∥.

Thus J satisfies the inequality (4) with f1(x) = 2ax and L1(x) = 2
|A| . Next, we define

Λ : RX
+ → RX

+ by

Λϕ(x, y) =
2

|A|
ϕ(ax, ay) <

2|a|L
|A|

ϕ(x, y), x ∈ X,ϕ ∈ RX
+ .

Since Λ is linear, we get

Λnϕ(x, y) =

(
2|a|L
|A|

)n

ϕ(x, y)

and

ε∗(x, y) =
∞∑
n=0

(
2|a|L
|A|

)n

ϕ(x, y) =
|A|

|A| − 2L|a|
ϕ(x, y).

By Theorem 1, there exists a mapping k : X → Y such that

k(x) = lim
n→∞

Jnf(x),

k(x) =
2

A
k(ax),

∥f(x)− k(x)∥ ≤ |A|
|A| − 2L|a|

ϕ(x, 0).

Next, we prove that k satisfies the generalized Drygas equation. Replacing x by x
a and

y by y
b in (10), we have∥∥∥f(x+ y) + f(x− y)−Af

(x
a

)
−Bf

(y
b

)
−Df

(
−y
b

)∥∥∥
≤ ρ(x) ∥f (x+ y)− f (x)− f (y)∥+ ϕ

(x
a
,
y

b

)



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 12 of 17

for all x, y ∈ X. Then, from the above inequality, we get

∥f(x+ y) + f(x− y)− 2f(x)∥

≤ ρ(x) ∥f (x+ y)− f (x)− f (y)∥+ ϕ
(x
a
,
y

b

)
+ ϕ

(x
a
, 0
)
+ ϕ

(
0,
y

b

)
.

Hence

∥Jf(x+ y) + Jf(x− y)− 2Jf(x)− Jf(y)− Jf(−y)∥

=
2

|A|
∥f (a(x+ y)) + f (a(x− y))− 2f (ax)− f (ay)− f (−ay)∥

≤ 2

|A|
ρ(x)∥f(x+

a

b
y)− f(x)− f(

a

b
y)∥+ 2

|A|
ϕ(x, y) +

2

|A|
ϕ(x, 0) +

2

|A|
ϕ(0, y)

and so

∥k(x+ y) + k(x− y)− k(x)− k(y)− k(−y)∥
= lim

n→∞
∥Jnf(x+ y) + Jnf(x− y)− 2Jnf(x)− Jnf(y)− Jnf(−y)∥

≤ lim
n→∞

∥Jnf(x+
a

b
y)− Jnf(x)− Jnf(

a

b
y)∥

+ lim
n→∞

(Jnϕ(x, y) + Jnϕ(x, 0) + Jnϕ(0, y))

for all n ∈ N0 and x, y ∈ X. Letting n→ ∞, we obtain

∥k(x+ y) + k(x− y)− 2k(x)∥ ≤ ρ(x)∥k(x+ y)− k(x)− k(y)∥ (11)

for all x, y ∈ X. Letting y = x in (11), we get

∥k(2x)− 2k(x)∥ ≤ ρ(x)∥k(2x)− 2k(x)∥

for all x ∈ X. Moreover, k(2x) = 2k(x) for all x ∈ X.
Letting l = x+ y,m = x− y in (11), we get∥∥∥∥k(l) + k(m)− 2k

(
l +m

2

)∥∥∥∥ ≤ ρ(x)

∥∥∥∥k(l)− k

(
l +m

2

)
− k

(
l −m

2

)∥∥∥∥ (12)

for all l,m ∈ X. By (11) and (12),

∥k(l) + k(m)− k(l +m)∥ ≤ 1

2
(ρ(x))2∥k(l) + k(m)− k(l +m)∥

for all l,m ∈ X. Thus k(l +m) = k(l) + k(m) for all l,m ∈ X. So f is additive.

Theorem 5. Let ψ : X2 → [0,∞) be a function such that there exists an L < 1 with

ψ (ax, ay) ≤ a2Lψ(x, y)



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 13 of 17

for all x, y ∈ X. Let f : X → Y be an even mapping with f(0) = 0 satisfying

∥f(ax+ by) + f(ax− by)−Af(x)−Bf(y)−Df(−y)∥
≤ ∥ρ(x)(f(ax+ by) + f(ax− by)− 2f(ax)− 2f(by))∥+ ψ(x, y)

(13)

for all x, y ∈ X. Assume that L,A and a satisfy 2a2L
|A| < 1, and ρ : X → (0, 1] is a function.

Then there exists a unique quadratic mapping H : X → Y such that

∥f(x)−H(x)∥ ≤ |A|
|A| − 2La2

ψ(x, 0)

for all x ∈ X.

Proof. Letting y = 0 in (13), we get

∥f(x)− 2

A
f(ax)∥ ≤ 1

|A|
ψ(x, 0), x ∈ X.

Similar to Theorem 4, we can prove that

H(x) = lim
n→∞

Jnf(x),

H(x) =
2

A
H(ax),

∥f(x)−H(x)∥ ≤ |A|
|A| − 2La2

ψ(x, 0).

Moreover,

∥f(x+ y) + f(x− y)− 2f(x)− f(y)− f(−y)∥
≤ ∥ρ(x)(f(x+ y) + f(x− y)− 2f(x)− 2f(y))∥
+ ψ(x/a, y/b) + ψ(x/a, 0) + ψ(0, y/b).

Next, we prove H(x) is quadratic.
By the definition of H(x), we have

∥H(x+ y) +H(x− y)− 2H(x)− 2H(y)∥
≤ ∥ρ(x)(H(x+ y) +H(x− y)− 2H(x)− 2H(y))∥

for all x, y ∈ X. So H(x+y)+H(x−y) = 2H(x)+2H(y) for all x, y ∈ X. This completes
the proof.

5. Conclusion

This paper focuses on the generalized Drygas functional equation and deeply explores
its Hyers - Ulam stability in the context from a real vector space to a Banach space. A



G. Lyu et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6197 14 of 17

series of key results have been obtained by using the fixed point method. Regarding the
stability of the functional equation, under different conditions (such as the specific restric-
tions on a, b, A,B,D), the close relationship between the mapping satisfying the relevant
inequality and the unique generalized Drygas mapping is clearly given, and the error esti-
mate is accurately provided. This not only deepens the theoretical understanding of the
properties of this equation but also provides an important methodological reference and
theoretical basis for subsequent studies on the stability of such equations. In the study
of the functional inequality, it is successfully proven that there exist unique additive and
quadratic mappings under specific conditions, and the corresponding error estimates are
obtained. These conclusions are of great significance in theoretical research fields such as
mathematical analysis and function approximation. They provide new perspectives and
methods for function classification and property characterization. In practical applica-
tions, such as in the design of optimization algorithms and signal processing, they provide
powerful mathematical tools, which can effectively improve algorithm performance and
optimize signal processing effects.

The research results of this paper expand the research boundaries of the stability theory
of functional equations and open up new directions for the future development of this field.
Follow up research can further explore the properties of the generalized Drygas functional
equation and inequality in more complex spaces or under different types of operators. At
the same time, it is necessary to strengthen the applied research in interdisciplinary fields,
fully explore its potential value, and promote the coordinated development of related fields.

Declarations

Conflict of interest
The authors declare that they have no competing interests.

Fundings
This work was supported by National Natural Science Foundation of China (No. 11761074),
Project of Jilin Science and Technology Development for Leading Talent of Science and
Technology Innovation in Middle and Young and Team Project (No. 20200301053RQ), the
Natural Science Foundation of Jilin Province (No. YDZJ202101ZYTS136) and the Sci-
entific Research Project of Guangzhou College of Technology and Business in 2024 (No.
KYZD202404).

Authors’ contributions
The authors equally conceived of the study, participated in its design and coordination,
drafted the manuscript, participated in the sequence alignment, and read and approved
the final manuscript.

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