Stability of an additive-quadratic functional equation in modular spaces

Abderrahman Baza; Mohamed Rossafi; Choonkil Park; Mana Donganont

doi:10.1515/math-2024-0075

Article Open Access

Stability of an additive-quadratic functional equation in modular spaces

Abderrahman Baza , Mohamed Rossafi , Choonkil Park and Mana Donganont

Published/Copyright: November 29, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

Using the direct method, we prove the Hyers-Ulam-Rassias stability of the following functional equation:

ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) = 0

in ρ -complete convex modular spaces satisfying Fatou property or Δ 2 -condition.

Keywords: Hyers-Ulam-Rassias stability; additive-quadratic functional equation; modular space; Fatou property; Δ2-condition

MSC 2010: 39B82; 39B52

1 Introduction and preliminaries

Nakano [1] introduced the concept of modular linear spaces in 1950. Many authors, including Luxemburg [2], Amemiya [3], Musielak [4], Koshi and Shimogaki [5], Mazur [6], Turpin [7], and Orlicz [8], have now thoroughly proved these hypotheses. There are numerous uses for Orlicz spaces [8] and the idea of interpolation [8] in the context of modular spaces. As suggested by Khamsi [9], several scholars used a fixed-point approach of quasi-contractions to test stability in modular spaces without using the Δ 2 -condition. Recent results on the stability of various functional equations involving the Δ 2 -condition and the Fatou property were reported by Sadeghi [10]. First, we review some vocabulary, notations, and common characteristics of the theory of modulars and modular spaces.

Definition 1.1

Let Y be an arbitrary real vector space. A function ρ : Y → [ 0 , ∞ ) is called a modular if for arbitrary u , v ∈ Y ,

ρ ( u ) = 0 if and only if u = 0 ;
ρ ( α u ) = ρ ( u ) for all scalars α with ∣ α ∣ = 1 ;
ρ ( α u + β v ) ≤ ρ ( u ) + ρ ( v ) for all α and β with α + β = 1 and α , β ≥ 0 .

The pair ( Y , ρ ) is called a ρ -modular space.

If (3) is replaced by
(1.1) ρ ( α u + β v ) ≤ α ρ ( u ) + β ρ ( v )
for all α and β with α + β = 1 and α , β ≥ 0 , then we say that ρ is a convex modular.

A sequence { x n } in Y is called ρ -convergent to x , denoted by x n → x or x = ρ − lim n → ∞ x n if ρ ( x n − x ) → 0 as n → ∞ . A modular ρ is said to have the Fatou property if ρ ( x ) ≤ liminf n → ∞ ρ ( x n ) whenever x = ρ − lim n → ∞ x n .

A modular ρ defines a corresponding modular space, i.e., the set Y ρ given by

Y ρ = { u ∈ Y : ρ ( λ u ) → 0 as λ → 0 } ,

which is a real vector space by Definition 1.1. Moreover, by Definition 1.1 (3), the real vector space Y ρ is not trivial if Y is not trivial.

A modular ρ is said to satisfy the Δ 2 -condition if there exists τ > 0 such that ρ ( 2 u ) ≤ τ ρ ( u ) for all u ∈ Y ρ .

Definition 1.2

Let { u n } and u be in Y ρ .

The sequence { u n } , with u n ∈ Y ρ , is ρ -convergent to u , denoted by u n → u , if ρ ( u n − u ) → 0 as n → ∞ .
The sequence { u n } , with u n ∈ Y ρ , is called ρ -Cauchy if ρ ( u n − u m ) → 0 as n , m → ∞ .
Y ρ is called ρ -complete if every ρ -Cauchy sequence in Y ρ is ρ -convergent.

Proposition 1.3

In modular spaces,

if u n → ρ u and a is a constant vector, then u n + a → ρ u + a ;
if u n → ρ u and v n → ρ v , then α u n + β v n → ρ α u + β v , where α + β ≤ 1 and α , β ≥ 0 .

Remark 1.4

Note that ρ ( t u ) is an increasing function in t for each u ∈ X . Indeed, suppose that 0 < a < b . Then, Definition 1.1 (4) with v = 0 shows that ρ ( a u ) = ρ a b b u ≤ ρ ( b u ) for each u ∈ Y . Moreover, if ρ is a convexe modular on Y and ∣ α ∣ ≤ 1 , then ρ ( α u ) ≤ α ρ ( u ) .

In general, if 0 ≤ λ i ≤ 1 , i = 1 , … , n , then ρ ( λ 1 u 1 + λ 2 u 2 + … + λ n u n ) ≤ λ 1 ρ ( u 1 ) + λ 2 ρ ( u 2 ) + … + λ n ρ ( u n ) .

If { u n } is ρ -convergent to u , then { c u n } is ρ -convergent to c u , where ∣ c ∣ ≤ 1 . However, the ρ -convergent of a sequence { u n } to u does not imply that { α u n } is ρ -convergent to α u for any scalar α with ∣ α ∣ > 1 .

If ρ is a convex modular satisfying Δ 2 -condition with 0 < τ < 2 , then ρ ( u ) ≤ τ ρ ( 1 2 u ) ≤ τ 2 ρ ( u ) for all u . Hence, ρ = 0 . Consequently, we must have τ ≥ 2 if ρ is a convex modular.

The beginning of this subject was in 1940 at the Mathematics Club of the University of Wisconsin, when Ulam [11] discussed some of the open problems. One of them was the following problem regarding the approximations of group homomorphisms:

Ulam problem: [11] Let G 1 be a group, G 2 be a metric group with metric d ( ⋅ , ⋅ ) , and ε be a positive number. Does there exist a δ > 0 such that if f : G 1 → G 2 satisfies

d ( f ( x y ) , f ( x ) f ( y ) ) ≤ ε

for all x , y ∈ G 1 , then a homomorphism ϕ : G 1 → G 2 exists with

d ( f ( x ) , ϕ ( x ) ) ≤ δ

for all x ∈ G 1 ?

This problem was the fundamental building block of the stability theory of functional equations. Within a year, Hyers [12] provided the first partial answer to Ulam’s question, showing the case of approximately additive mappings when G 1 and G 2 are Banach spaces. This result caught the attention of many mathematicians, beginning with Bourgin [13,14] and then Aoki [15], who presented Ulam’s problem with unbounded Cauchy differences. In 1978, Rassias [16] demonstrated the existence of singular linear mappings close to approximate additive mappings in the case where the relevant inequality is not bounded. We can include these results in the following theorem.

Theorem 1.5

[16] Let ( X , ‖ ⋅ ‖ X ) and ( Y , ‖ ⋅ ‖ Y ) be a normed space and a Banach space, respectively. Let c ≥ 0 and p ≠ 1 be two real numbers. Suppose that f : X → Y is an operator satisfying the inequality

∥ f ( x + y ) − f ( x ) − f ( y ) ∥ Y ≤ c ( ∥ x ∥ X p + ∥ y ∥ X p ) , x , y ∈ X \ { 0 } .

Then, there exists an additive mapping A : X → Y such that

∥ f ( x ) − A ( x ) ∥ Y ≤ c ∣ 1 − 2 p − 1 ∣ ∥ x ∥ X p , x ∈ X \ { 0 } .

In the spirit of Rassias’ approach, Forti [17] and Găvruţa [18] were able to obtain a more general result. They generalized all of the aforementioned stability results by swapping out the Cauchy differences for a positive-valued control function φ . See [19–23] for further information regarding the stability results and applications.

In this article, we study the Hyers-Ulam-Rassias stability of the following functional equation:

(1.2) ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) = 0 for all x , y , z , w ∈ X

in ρ -complete convex modular space satisfying the Fatou property or with Δ 2 -condition.

Throughout the article, assume that X is a real vector space and Y is a ρ -complete convex modular space.

2 Hyers-Ulam-Rassias stability of (1) in modular spaces satisfying the Fatou property

Lemma 2.1

[24] Let X and Y be real vector spaces and ϕ : X 2 → Y be a mapping satisfying ϕ ( 0 , z ) = ϕ ( x , 0 ) = 0 and

ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) = 0 for a l l x , y , z , w ∈ X .

Then, ϕ is additive in the first variable and quadratic in the second variable.

Note that such a mapping ϕ : X 2 → Y is called an additive-quadratic mapping.

Theorem 2.2

Let X be a real vector space and Y be a ρ -complete convex modular space satisfying the Fatou property. Let α : X 2 ⟶ [ 0 , ∞ ) be a function such that

(2.1) ψ ( x , y ) ≔ ∑ j = 1 ∞ 1 4 j α ( 2 j − 1 x , 2 j − 1 y ) ≤ φ ( x , y ) ≔ ∑ j = 1 ∞ 1 2 j α ( 2 j − 1 x , 2 j − 1 y ) < ∞

for all x , y ∈ X . Let ϕ : X 2 → Y ρ be a mapping that satisfies

ϕ ( x , 0 ) = ϕ ( 0 , z ) = 0

and

(2.2) ρ ( ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) ) ≤ α ( x , y ) α ( z , w )

for all x , y , z , w ∈ X . Then, there exists a unique additive-quadratic mapping H : X 2 → Y ρ such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ min { φ ( x , x ) α ( z , 0 ) , α ( x , 0 ) ψ ( z , z ) } for a l l x , z ∈ X .

Proof

Letting y = x and w = 0 in (2.2), we obtain

ρ ( ϕ ( 2 x , z ) − 2 ϕ ( x , z ) ) ≤ α ( x , x ) α ( z , 0 ) .

Then, by convexity of ρ and (1.1), we have

(2.3) ρ 1 2 ϕ ( 2 x , z ) − ϕ ( x , z ) ≤ ρ 1 2 ( ϕ ( 2 x , z ) − 2 ϕ ( x , z ) ) ≤ 1 2 ρ ( ( ϕ ( 2 x , z ) − 2 ϕ ( x , z ) ) ) ≤ 1 2 α ( x , x ) α ( z , 0 )

and by induction, we show that

(2.4) ρ 1 2 k ϕ ( 2 k x , z ) − ϕ ( x , z ) ≤ ∑ j = 1 k 1 2 j α ( 2 j − 1 x , 2 j − 1 x ) α ( z , 0 )

for all x , z ∈ X and every positive integer k . For k = 1 , we obtain (2.3). Suppose that (2.4) holds for a fixed k ∈ N . We have

ρ 1 2 k + 1 ϕ ( 2 k + 1 x , z ) − ϕ ( x , z ) = ρ 1 2 1 2 k ϕ ( 2 k ⋅ 2 x , z ) − ϕ ( 2 x , z ) + 1 2 ϕ ( 2 x , z ) − ϕ ( x , z ) ≤ 1 2 ∑ j = 1 k 1 2 j α ( 2 j x , 2 j x ) α ( z , 0 ) + 1 2 α ( x , x ) α ( z , 0 ) = ∑ j = 1 k + 1 1 2 j α ( 2 j − 1 x , 2 j − 1 x ) α ( z , 0 ) .

Hence, (2.4) holds for every k ∈ N .

Let m , n be positive integers with n > m . Then, we have

(2.5) ρ 1 2 n ϕ ( 2 n x , z ) − 1 2 m ϕ ( 2 m x , z ) = ρ 1 2 m ϕ ( 2 n − m ⋅ 2 m x , z ) 2 n − m − ϕ ( 2 m x , z ) ≤ 1 2 m ∑ j = 1 n − m 1 2 j α ( 2 m + j − 1 x , 2 m + j − 1 x ) α ( z , 0 ) = ∑ k = m + 1 n 1 2 k α ( 2 k − 1 x , 2 k − 1 x ) α ( z , 0 )

for all x , z ∈ X . It follows from (2.5) and (2.1) that ϕ ( 2 n x , z ) 2 n is a Cauchy sequence in Y ρ which is complete, and this guarantees the existence of a mapping A : X 2 → Y ρ such that

(2.6) A ( x , z ) = ρ − lim ϕ ( 2 n x , z ) 2 n , x , z ∈ X .

Now, by the Fatou property, we have

ρ ( A ( x , z ) − ϕ ( x , z ) ) ≤ liminf n → ∞ ρ 1 2 n ϕ ( 2 n x , z ) − ϕ ( x , z ) ≤ ∑ k = 1 ∞ 1 2 k α ( 2 k − 1 x , 2 k − 1 x ) α ( z , 0 ) .

Hence,

(2.7) ρ ( A ( x , z ) − ϕ ( x , z ) ) ≤ φ ( x , x ) α ( z , 0 )

for all x , z ∈ X .

Now, we prove that A is an additive-quadratic mapping. In the first, we have

ρ 1 2 n ϕ ( 2 n ( x + y ) , z + w ) + 1 2 n ϕ ( 2 n ( x − y ) , z − w ) − 2 2 n ϕ ( 2 n x , z ) − 2 2 n ϕ ( 2 n x , w ) ≤ 1 2 n α ( 2 n x , 2 n y ) α ( z , w )

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . Thus, we have

ρ 1 12 ( A ( x + y , z + w ) + A ( x − y , z − w ) − 2 A ( x , z ) − 2 A ( x , w ) ) ≤ 1 6 1 2 ρ A ( x + y , z + w ) − ϕ ( 2 n ( x + y ) , z + w ) 2 n + 1 2 ρ A ( x − y , z − w ) − ϕ ( 2 n ( x − y ) , z − w ) 2 n + ρ A ( x , z ) − ϕ ( 2 n x , z ) 2 n + ρ A ( x , w ) − ϕ ( 2 n x , w ) 2 n + 1 12 ρ ϕ ( 2 n ( x + y ) , z + w ) 2 n + ϕ ( 2 n ( x − y ) , z − w ) 2 n − 2 2 n ϕ ( 2 n x , z ) − 2 2 n ϕ ( 2 n x , w )

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . Then, we obtain

A ( x + y , z + w ) + A ( x − y , z − w ) − 2 A ( x , z ) − 2 A ( x , w ) = 0

and by Lemma 2.1, we conclude that A is an additive-quadratic mapping.

Now, let B : X 2 → Y ρ be another additive-quadratic mapping satisfying (2.7). Then, we have

ρ A ( x , z ) − B ( x , z ) 2 = ρ 1 2 A ( 2 k x , z ) 2 k − ϕ ( 2 k x , z ) 2 k + 1 2 ϕ ( 2 k x , z ) 2 k − B ( 2 k x , z ) 2 k ≤ 1 2 ρ A ( 2 k x , z ) 2 k − ϕ ( 2 k x , z ) 2 k + 1 2 ρ ϕ ( 2 k x , z ) 2 k − B ( 2 k x , z ) 2 k ≤ 1 2 ⋅ 1 2 k { ρ ( A ( 2 k x , z ) − ϕ ( 2 k x , z ) ) + ρ ( ϕ ( 2 k x , z ) − B ( 2 k x , z ) ) } ≤ 1 2 k φ ( 2 k x , 2 k x ) α ( z , 0 ) = ∑ j = 1 ∞ 1 2 k + j α ( 2 k + j − 1 x , 2 k + j − 1 x ) α ( z , 0 ) = ∑ l = k + 1 ∞ 1 2 l α ( 2 l − 1 x , 2 l − 1 x ) α ( z , 0 )

for all x , z ∈ X , which goes to zero as k → ∞ . Thus, A ( x , z ) = B ( x , z ) for all x , z ∈ X .

Conversely, letting y = 0 and w = z in (2.2), we obtain

ρ ( ϕ ( x , 2 z ) − 4 ϕ ( x , z ) ) ≤ α ( z , z ) α ( x , 0 ) .

Thus,

ρ 1 4 ϕ ( x , 2 z ) − ϕ ( x , z ) ≤ 1 4 α ( z , z ) α ( x , 0 ) .

By simple induction, we have

ρ 1 4 k ϕ ( x , 2 k z ) − ϕ ( x , z ) ≤ ∑ j = 1 k 1 4 j α ( 2 j − 1 z , 2 j − 1 z ) α ( x , 0 ) .

Let m , n be positive integers with n > m . Then, we have

(2.8) ρ 1 4 n ϕ ( x , 2 n z ) − 1 4 m ϕ ( x , 2 m z ) = ρ 1 4 m ϕ ( x , 2 n − m ⋅ 2 m z ) 4 n − m − ϕ ( x , 2 m z ) ≤ 1 4 m ∑ j = 1 n − m 1 4 j α ( 2 m + j − 1 z , 2 m + j − 1 z ) α ( x , 0 ) = ∑ k = m + 1 n 1 4 k α ( 2 k − 1 z , 2 k − 1 z ) α ( x , 0 ) .

By (2.8) and (2.1), we conclude that ϕ ( x , 2 z ) 4 n is a Cauchy sequence in Y ρ which is ρ -complete. So, there exists a mapping C : X 2 → Y ρ such that

C ( x , z ) = ρ limit ϕ ( x , 2 n z ) 4 n , x , z ∈ X .

Moreover,

ρ ( C ( x , z ) − ϕ ( x , z ) ) ≤ liminf n → ∞ ρ 1 4 n ϕ ( x , 2 n z ) − ϕ ( x , z ) ≤ ∑ k = 1 ∞ 1 4 k α ( 2 k − 1 z , 2 k − 1 z ) α ( x , 0 ) .

Hence,

(2.9) ρ ( C ( x , z ) − ϕ ( x , z ) ) ≤ ψ ( z , z ) α ( x , 0 ) .

Now, we prove that C is an additive-quadratic mapping.

We have

ρ 1 4 n ϕ ( x + y , 2 n ( z + w ) ) + 1 4 n ϕ ( x − y , 2 n ( z − w ) ) − 2 4 n ϕ ( x , 2 n z ) − 2 4 n ϕ ( x , 2 n w ) ≤ 1 4 n α ( x , y ) α ( 2 n z , 2 n w )

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . Hence, we have

ρ 1 12 ( C ( x + y , z + w ) + C ( x − y , z − w ) − 2 C ( x , z ) − 2 C ( x , w ) ) ≤ 1 6 1 2 ρ C ( x + y , z + w ) − ϕ ( x + y , 2 n ( z + w ) ) 4 n + 1 2 ρ C ( x − y , z − w ) − ϕ ( x − y , 2 n ( z − w ) ) 4 n + ρ C ( x , z ) − ϕ ( x , 2 n z ) 4 n + ρ C ( x , w ) − ϕ ( x , 2 n w ) 4 n + 1 12 ρ ϕ ( x + y , 2 n ( z + w ) ) 4 n + ϕ ( x − y , 2 n ( z − w ) ) 4 n − 2 4 n ϕ ( x , 2 n z ) − 2 4 n ϕ ( x , 2 n w )

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . So, we obtain

C ( x + y , z + w ) + C ( x − y , z − w ) − 2 C ( x , z ) − 2 C ( x , w ) = 0 .

By Lemma 2.1, we deduce that C is an additive-quadratic mapping.

To show the uniqueness of C , let D : X 2 → Y ρ be another additive-quadratic mapping satisfying (2.9). Then, we have

ρ C ( x , z ) − D ( x , z ) 2 = ρ 1 2 C ( x , 2 k z ) 4 k − ϕ ( x , 2 k z ) 4 k + 1 2 ϕ ( x , 2 k z ) 4 k − D ( x , 2 k z ) 4 k ≤ 1 2 ⋅ 1 4 k { ρ ( C ( x , 2 k z ) − ϕ ( x , 2 k z ) ) + ρ ( ϕ ( x , 2 k z ) − D ( x , 2 k z ) ) } ≤ 1 4 k ψ ( 2 k z , 2 k z ) α ( x , 0 ) = ∑ j = 1 ∞ 1 4 k + j α ( 2 k + j − 1 z , 2 k + j − 1 z ) α ( x , 0 ) = ∑ l = k + 1 ∞ 1 4 l α ( 2 l − 1 z , 2 l − 1 z ) α ( x , 0 )

for all x , z ∈ X , which goes to zero as k → ∞ . Thus, C ( x , z ) = D ( x , z ) for all x , z ∈ X .

It follows from (2.7) and (2.9) that

ρ C ( x , z ) − A ( x , z ) 2 = ρ 1 2 C ( 2 n x , z ) 2 n − ϕ ( 2 n x , z ) 2 n + 1 2 ϕ ( 2 n x , z ) 2 n − A ( 2 x , z ) 2 n ≤ 1 2 ⋅ 1 2 n { ρ ( C ( 2 n x , z ) − ϕ ( 2 n x , z ) ) + ρ ( ϕ ( 2 n x , z ) − A ( 2 n x , z ) ) } ≤ 1 2 ⋅ 1 2 n α ( 2 n x , 0 ) ψ ( z , z ) + 1 2 ⋅ 1 2 n φ ( 2 n , 2 n x ) α ( z , 0 )

for all x , z ∈ X , which goes to zero as n → ∞ . This implies that C ( x , z ) = A ( x , z ) = H ( x , z ) for all x , z ∈ X . So, there exists a unique additive-quadratic mapping H : X 2 → Y ρ such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ min { φ ( x , x ) α ( z , 0 ) , α ( x , 0 ) ψ ( z , z ) }

for all x , z ∈ X .□

Corollary 2.3

Let X be a vector space, and Y ρ be a ρ -complete convex modular space. Let 0 < r < 1 and θ be positive real numbers and ϕ : X 2 ⟶ Y ρ be a mapping satisfying

ϕ ( x , 0 ) = ϕ ( 0 , z ) = 0

and

(2.10) ρ ( ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) ) ≤ θ ( ‖ x ‖ r + ‖ y ‖ r ) ( ‖ z ‖ r + ‖ w ‖ r )

for all x , y , z , w ∈ X . Then, there exists a unique additive-quadratic mapping H : X 2 ⟶ Y ρ such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ 2 θ ‖ z ‖ r ‖ x ‖ r 4 − 2 r , x , z ∈ X .

Proof

The proof follows from Theorem 2.2 by taking α ( x , y ) = θ ( ‖ x ‖ r + ‖ y ‖ r ) for all x , y ∈ X and remarking that

min 2 θ 2 − 2 r ‖ x ‖ r ‖ z ‖ r , 2 θ ‖ x ‖ r ‖ z ‖ r 4 − 2 r = 2 θ 4 − 2 r ‖ x ‖ r ‖ z ‖ r

for all x , z ∈ X .□

Now, we obtain a classical Ulam stability.

Corollary 2.4

Let ε be a nonnegative real number, X be a vector space, and Y ρ be a ρ -complete modular space, where ρ is a convex modular. Let ϕ : X 2 → Y ρ be a mapping such that ϕ ( x , 0 ) = ϕ ( 0 , z ) = 0 and

ρ ( ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) ) ≤ ε

for all x , y , z , w ∈ X . Then, there exists a unique additive quadratic mapping H : X 2 → Y ρ such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ ε 3

for all x , z ∈ X .

3 Hyers-Ulam-Rassias stability of (1) in modular spaces satisfying Δ 2 -condition

Theorem 3.1

Let X be a vector space, Y ρ be a ρ -complete convex modular space satisfying the Δ 2 -condition with τ > 0 . Let α : X 2 ⟶ [ 0 , ∞ ) be a function such that

(3.1) φ ( x , y ) = ∑ j = 1 ∞ τ 2 2 j α x 2 j , y 2 j < ∞ and lim n → ∞ τ n α x 2 n , y 2 n = 0 , ψ ( x , y ) = ∑ j = 1 ∞ τ 3 2 j α x 2 j , y 2 j < ∞ and lim n → ∞ τ 2 n α x 2 n , y 2 n = 0 .

Let ϕ : X 2 ⟶ Y ρ be a mapping such that ϕ ( x , 0 ) = ϕ ( 0 , z ) = 0 and

(3.2) ρ ( ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) ) ≤ α ( x , y ) α ( z , w )

for all x , y , z , w ∈ X . Then, there exists a unique additive-quadratic mapping H : X 2 ⟶ Y ρ such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ min 1 2 φ ( x , x ) α ( z , 0 ) , 1 2 τ ψ ( z , z ) α ( x , 0 )

for all x , z ∈ X .

Proof

Letting y = x and w = 0 in (3.2), we obtain

ρ ( ϕ ( 2 x , z ) − 2 ϕ ( x , z ) ) ≤ α ( x , x ) α ( z , 0 ) .

Hence,

ρ ϕ ( x , z ) − 2 ϕ x 2 n , z ≤ α x 2 , x 2 α ( z , 0 ) .

Then, by the Δ 2 -condition and the convexity of ρ , we have

ρ ϕ ( x , z ) − 2 n ϕ x 2 n , z = ρ ∑ j = 1 n 1 2 j 2 2 j − 1 ϕ x 2 j − 1 , z − 2 2 j ϕ x 2 j , z ≤ 1 τ ∑ j = 1 n τ 2 2 j α x 2 j , x 2 j α ( z , 0 )

for all x , z ∈ X . So, for all positive integers m and n with n > m , we have

(3.3) ρ 2 n ϕ x 2 n , z − 2 m ϕ x 2 m , z ≤ τ m ρ 2 n − m ϕ x 2 n , z − ϕ x 2 m , z ≤ τ m − 1 ∑ j = 1 n − m τ 2 2 j α x 2 m + j , x 2 m + j α ( z , 0 ) = 1 τ 2 τ m ∑ l = m + 1 n τ 2 2 l α x 2 l , x 2 l α ( z , 0 ) .

It follows from (3.3) and (3.1) that 2 n ϕ x 2 n , z is a Cauchy sequence in Y ρ , which is complete. Hence, we define a mapping A : X 2 → Y ρ as

A ( x , z ) = ρ − lim n → ∞ 2 n ϕ x 2 n , z ; x , z ∈ X .

Now, we have

ρ ( ϕ ( x , z ) − A ( x , z ) ) ≤ 1 2 ρ 2 ϕ ( x , z ) − 2 n + 1 ϕ x 2 n , z + 1 2 ρ 2 n + 1 ϕ x 2 n , z − 2 A ( x , z ) ≤ τ 2 ρ ϕ ( x , z ) − 2 n ϕ x 2 n , z + τ 2 ρ 2 n ϕ x 2 n , z − A ( x , z ) ≤ 1 2 ∑ j = 1 n τ 2 2 j α x 2 j , x 2 j α ( z , 0 ) + τ 2 ρ 2 n ϕ x 2 n , z − A ( x , z )

for all x , z ∈ X . Passing to the limit n → ∞ , we obtain

(3.4) ρ ( ϕ ( x , z ) − A ( x , z ) ) ≤ 1 2 φ ( x , x ) α ( z , 0 ) .

Now, we prove that A is an additive-quadratic mapping.

In the first, we have

ρ 2 n ϕ x + y 2 n , z + w + 2 n ϕ x − y 2 n , z − w − 2 ⋅ 2 n ϕ x 2 n , z − 2 ⋅ 2 n ϕ x 2 n , w ≤ τ n α x 2 n , y 2 n α ( z , w )

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . So, we have

ρ ( A ( x + y , z + w ) + A ( x − y , z − w ) − 2 A ( x , z ) − 2 A ( x , w ) ) ≤ 1 8 ρ 8 A ( x + y , z + w ) − 2 n ϕ x + y 2 n , z + w + 1 8 ρ 8 A ( x − y , z − w ) − 2 n ϕ x − y 2 n , z − w + 1 8 ρ 16 A ( x , z ) − 2 n ϕ x 2 n , z + 1 8 ρ 16 A ( x , w ) − 2 n ϕ x 2 n , w + 1 8 ρ 8 2 n ϕ x + y 2 n , z + w + 2 n ϕ x − y 2 n , z − w − 2 ⋅ 2 n ϕ x 2 n , z − 2 ⋅ 2 n ϕ x 2 n , w ≤ τ 3 8 ρ A ( x + y , z + w ) − 2 n ϕ x + y 2 n , z + w + τ 3 8 ρ A ( x − y , z − w ) − 2 n ϕ x − y 2 n , z − w + τ 4 8 ρ A ( x , z ) − 2 n ϕ x 2 n , z + τ 4 8 ρ A ( x , w ) − 2 n ϕ x 2 n , w + τ 3 8 ρ 2 n ϕ x + y 2 n , z + w + 2 n ϕ x − y 2 n , z − w − 2 ⋅ 2 n ϕ x 2 n , z − 2 ⋅ 2 n ϕ x 2 n , w

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . Hence, we conclude that

A ( x + y , z + w ) + A ( x − y , z − w ) − 2 A ( x , z ) − 2 A ( x , w ) = 0

and by Lemma 2.1, we deduce that A is an additive-quadratic mapping.

Now, let B : X 2 → Y ρ be another additive-quadratic mapping satisfying (3.4). Then, we have

ρ ( A ( x , z ) − B ( x , z ) ) ≤ 1 2 ρ 2 n + 1 A x 2 n , z − 2 n + 1 ϕ x 2 n , z + 1 2 ρ 2 n + 1 ϕ x 2 n , z − 2 n + 1 B x 2 n , z ≤ τ n + 1 2 ρ A x 2 n , z − ϕ x 2 n , z + τ n + 1 2 ρ ϕ x 2 n , z − B x 2 n , z ≤ τ n + 1 2 φ x 2 n , x 2 n α ( z , 0 ) = 2 τ n − 1 ∑ l = n + 1 ∞ τ 2 2 l α x 2 l , x 2 l α ( z , 0 )

for all x , z ∈ X , which goes to zero as n → ∞ . Thus, we have

A ( x , z ) = B ( x , z ) for all x , z ∈ X .

Conversely, letting y = 0 and w = z in (3.2), we obtain

ρ ( ϕ ( x , 2 z ) − 4 ϕ ( x , z ) ) ≤ α ( z , z ) α ( x , 0 ) .

Hence,

ρ ϕ ( x , z ) − 4 ϕ x , z 2 ≤ α z 2 , z 2 α ( x , 0 ) .

By the Δ 2 -condition and the convexity of ρ , we have (remarking that ∑ j = 1 k − 1 1 2 k ≤ 1 )

ρ ϕ ( x , z ) − 4 n ϕ x , z 2 n = ρ ∑ j = 1 n 1 2 j 2 3 j − 2 ϕ x , z 2 j − 2 3 j ϕ x , z 2 j ≤ 1 τ 2 ∑ j = 1 n τ 3 2 j α z 2 j , z 2 j α ( x , 0 )

for all x , z ∈ X . Now, we have

ρ 4 m ϕ x , z 2 m − 4 n + m ϕ x , z 2 n + m ≤ τ 2 m ρ ϕ x , z 2 m − 4 n ϕ x , z 2 n + m ≤ τ 2 m − 2 ∑ j = 1 n τ 3 2 j α z 2 m + j , z 2 m + j α ( x , 0 ) ≤ 2 m τ m + 2 ∑ l = m + 1 n + m τ 3 2 l α z 2 l , z 2 l α ( x , 0 )

for all x , z ∈ X , which goes to zero as m → ∞ , since 2 τ ≤ 1 . Hence, the sequence 4 n ϕ x , z 2 n is a ρ -Cauchy sequence in Y ρ , which is ρ -complete. So, we have a mapping C : X 2 → Y ρ such that C ( x , z ) = ρ − lim n → ∞ 4 n ϕ x , z 2 n for x , z ∈ X . By the Δ 2 -condition, we have

ρ ( ϕ ( x , z ) − C ( x , z ) ) ≤ 1 2 ρ 2 ϕ ( x , z ) − 2 ⋅ 4 n ϕ x , z 2 n + 1 2 ρ 2 ⋅ 4 n ϕ x , z 2 n − 2 C ( x , z ) ≤ τ 2 ρ ϕ ( x , z ) − 4 n ϕ x , z 2 n + τ 2 ρ 4 n ϕ x , z 2 n − C ( x , z ) ≤ 1 2 τ ∑ j = 1 n τ 3 2 j α z 2 j , z 2 j α ( x , 0 ) + τ 2 ρ 4 n ϕ x , z 2 n − C ( x , z )

for all x , z ∈ X . Passing to the limit n → ∞ , we obtain

(3.5) ρ ( ϕ ( x , z ) − C ( x , z ) ) ≤ 1 2 τ ψ ( z , z ) α ( x , 0 ) .

To prove that C is an additive-quadratic mapping, we start with

ρ 4 n ϕ x + y , z + w 2 n + 4 n ϕ x − y , z − w 2 n − 2 ⋅ 4 n ϕ x , z 2 n − 2 ⋅ 4 n ϕ x , w 2 n ≤ τ 2 n α ( x , y ) α z 2 n , w 2 n

for all x , y , z , w ∈ X , which goes to zero as n → ∞ . Thus, we define a mapping A : X 2 → Y ρ as

A ( x , z ) = ρ − lim n → ∞ 2 n ϕ x 2 n , z ,

i.e., lim n → ∞ ρ A ( x , z ) − 2 n ϕ x 2 n , z = 0 for all x , z ∈ X . Then, we have

for all x , z ∈ X . Passing to the limit n → ∞ , we obtain

ρ ( ϕ ( x , z ) − A ( x , z ) ) ≤ 1 2 φ ( x , x ) α ( z , 0 ) ,

since φ ( x , x ) = ∑ j = 1 n τ 2 2 j α x 2 j , x 2 j for all x , y , z , w ∈ X , which goes to zero as n → ∞ . So, we have

C ( x + y , z + w ) + C ( x − y , z − w ) − 2 C ( x , z ) − 2 C ( x , w ) = 0

and by Lemma 2.1, we deduce that C is an additive-quadratic mapping. To show that C is unique, let D be another additive-quadratic mapping satisfying (3.5). Then, we have

ρ ( C ( x , z ) − D ( x , z ) ) ≤ 1 2 ρ 2 ⋅ 4 n C x , z 2 n − 2 ⋅ 4 n ϕ x , z 2 n + 1 2 ρ 2 ⋅ 4 n ϕ x , z 2 n − 2 ⋅ 4 n D x , z 2 n ≤ τ 2 n + 1 2 ρ C x , z 2 n − ϕ x , z 2 n + τ 2 n + 1 2 ρ ϕ x , z 2 − D x , z 2 n ≤ τ 2 n 2 ψ z 2 n , z 2 n α ( x , 0 ) = 2 n − 1 τ n ∑ l = n + 1 ∞ τ 3 2 l α z 2 l , z 2 l α ( x , 0 )

for all x , z ∈ X , which goes to zero as n → ∞ . This implies that C ( x , z ) = D ( x , z ) for all x , z ∈ X . It follows from (3.5) that

ρ 2 n ϕ x 2 n , z − 2 n C x 2 n , z 2 ≤ τ n 2 ρ ϕ x 2 n , z − C x 2 n , z ≤ τ n 4 τ α x 2 n , 0 ψ ( z , z )

for all x , z ∈ X , which goes to zero as n → ∞ . Since C : X 2 → Y ρ is additive in the first variable, we obtain

ρ 1 2 A ( x , z ) − 1 2 C ( x , z ) ≤ 0 ,

and thus, we conclude that

A ( x , z ) = C ( x , z ) = H ( x , z ) for all x , z ∈ X .

Hence, there exists a unique additive-quadratic mapping

H : X 2 → Y ρ

such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ min 1 2 τ ψ ( z , z ) α ( x , 0 ) , 1 2 φ ( x , x ) α ( z , 0 )

for all x , z ∈ X . This completes the proof.□

Corollary 3.2

Let X be a vector space, Y ρ be a ρ -complete convex modular space satisfying Δ 2 -condition. Let r > log 2 τ 2 2 and θ be positive real numbers, and ϕ : X 2 → Y ρ be a mapping satisfying ϕ ( x , 0 ) = ϕ ( 0 , z ) = 0 and

ρ ( ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) ) ≤ θ ( ‖ x ‖ r + ‖ y ‖ r ) ( ‖ z ‖ r + ‖ w ‖ r )

for all x , y , z , w ∈ X . Then, there exists a unique additive-quadratic mapping H : X 2 → Y ρ such that

ρ ( ϕ ( x , z ) − H ( x , z ) ) ≤ θ τ 2 ‖ x ‖ r ‖ z ‖ r 2 r + 1 − τ 2

for all x , z ∈ X .

Proof

The proof follows from Theorem 3.1 by taking

α ( x , y ) = θ ( ‖ x ‖ r + ‖ y ‖ r )

for all x , y ∈ X and remarking that

min θ τ 2 ‖ x ‖ r ‖ z ‖ r 2 r + 1 − τ 2 , θ τ 2 ‖ x ‖ r ‖ z ‖ r 2 r + 1 − τ 3 = θ τ 2 ‖ x ‖ r ‖ z ‖ r 2 r + 1 − τ 2

for all x , z ∈ X .□

4 Conclusion

Using the direct method, in Theorem 2.2, we have proved the Hyers-Ulam-Rassias stability of the functional equation (1) in ρ -complete convex modular spaces satisfying the Fatou property. Moreover, using the direct method, in Theorem 3.1, we have proved the Hyers-Ulam-Rassias stability of the functional equation (1) in ρ -complete convex modular spaces satisfying the Δ 2 -condition.

Acknowledgements

We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.

Funding information: The authors declare that there is no funding available for this article.
Author 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.
Conflict of interest: The authors declare that they have no competing interests.

References

[1] H. Nakano, Modulated Semi-ordered Linear Spaces, Maruzen Company, Ltd., Tokyo, 1950. Search in Google Scholar

[2] W. A. J. Luxemburg, Banach Function Spaces, Ph.D. Thesis, Technische Hogeschool Delft, Delft, the Netherlands, 1955. Search in Google Scholar

[3] I. Amemiya, On the representation of complemented modular lattices, J. Math. Soc. Japan 9 (1957), 263–279. 10.2969/jmsj/00920263Search in Google Scholar

[4] J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin, Heidelberg, 1983. 10.1007/BFb0072210Search in Google Scholar

[5] S. Koshi and T. Shimogaki, On F-norms of quasi-modular spaces, J. Fac. Sci. Hokkaido Univ. Ser. I Math. 15 (1961), no. 3–4, 202–218. 10.14492/hokmj/1530756196Search in Google Scholar

[6] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186. 10.1007/BF02684339Search in Google Scholar

[7] P. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math. 1 (1978), 331–353. Search in Google Scholar

[8] W. Orlicz, Collected Papers, Part I, II, PWN-Polish Scientific Publishers, Warsaw, Poland, 1988. Search in Google Scholar

[9] M. A. Khamsi, Quasicontraction mappings in modular spaces without Δ2-condition, Fixed Point Theory Appl. 2008 (2008), 916187, DOI: https://doi.org/10.1155/2008/916187. 10.1155/2008/916187Search in Google Scholar

[10] G. Sadeghi, A fixed-point approach to stability of functional equations in modular spaces, Bull. Malays. Math. Sci. Soc. 37 (2014), 333–344. Search in Google Scholar

[11] S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960. Search in Google Scholar

[12] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. 10.1073/pnas.27.4.222Search in Google Scholar PubMed PubMed Central

[13] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385–397. 10.1215/S0012-7094-49-01639-7Search in Google Scholar

[14] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237. 10.1090/S0002-9904-1951-09511-7Search in Google Scholar

[15] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. 10.2969/jmsj/00210064Search in Google Scholar

[16] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300, DOI: https://doi.org/10.1090/S0002-9939-1978-0507327-1. 10.1090/S0002-9939-1978-0507327-1Search in Google Scholar

[17] G. L. Forti, An existence and stability theorem for a class of functional equations, Stochastica 4 (1980), 23–30. 10.1080/17442508008833155Search in Google Scholar

[18] P. Gǎvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431–436, DOI: https://doi.org/10.1006/jmaa.1994.1211. 10.1006/jmaa.1994.1211Search in Google Scholar

[19] A. Charifi, R. Lukasik, and D. Zeglami, A special class of functional equations, Math. Slovaca 68 (2018), no. 2, 397–404, DOI: https://doi.org/10.1515/ms-2017-0110. 10.1515/ms-2017-0110Search in Google Scholar

[20] O. H. Ezzat, Functional equations related to higher derivations in semiprime rings, Open Math. 19 (2021), no. 1, 1359–1365, DOI: https://doi.org/10.1515/math-2021-0123. 10.1515/math-2021-0123Search in Google Scholar

[21] S. Paokanta, M. Dehghanian, C. Park, and Y. Sayyari, A system of additive functional equations in complex Banach algebras, Demonstr. Math. 56 (2023), no. 1, 20220165, DOI: https://doi.org/10.1515/dema-2022-0165. 10.1515/dema-2022-0165Search in Google Scholar

[22] C. Park, K. Tamilvanan, G. Balasubramanian, B. Noori, and A. Najati, On a functional equation that has the quadratic-multiplicative property, Open Math. 18 (2020), no. 1, 837–845, DOI: https://doi.org/10.1515/math-2020-0032. 10.1515/math-2020-0032Search in Google Scholar

[23] A. Turab, N. Rosli, W. Ali, and J. J. Nieto, The existence and uniqueness of solutions to a functional equation arising in psychological learning theory, Demonstr. Math. 56 (2023), no. 1, 20220231, DOI: https://doi.org/10.1515/dema-2022-0231. 10.1515/dema-2022-0231Search in Google Scholar

[24] I. Hwang and C. Park, Ulam stability of an additive-quadratic functional equation in Banach spaces, J. Math. Inequal. 14 (2020), no. 2, 421–436, DOI: https://dx.doi.org/10.7153/jmi-2020-14-27. 10.7153/jmi-2020-14-27Search in Google Scholar

Received: 2023-11-11

Revised: 2024-08-09

Accepted: 2024-09-22

Published Online: 2024-11-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0075

Keywords for this article

Hyers-Ulam-Rassias stability; additive-quadratic functional equation; modular space; Fatou property; Δ2-condition

Creative Commons

BY 4.0