Approximate multi-Cauchy mappings on certain groupoids

Choonkil Park; Abasalt Bodaghi; Mana Donganont

doi:10.1515/math-2025-0142

Article Open Access

Approximate multi-Cauchy mappings on certain groupoids

Choonkil Park , Abasalt Bodaghi and Mana Donganont

Published/Copyright: April 24, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this article, we give a representation of multi-Cauchy mappings on groupoids as an equation and then establish the (Hyers and Găvruţa) stability of such mappings on groupoids. In the case that the range is a subset of Banach space, the stability result will be different.

Keywords: Găvruţa stability; groupoid; Hyers-Ulam stability; multi-Cauchy mapping

MSC 2010: 39B52; 39B72; 39B82

1 Introduction

The first stability problem regarding the Cauchy functional equation from a group G into a metric group H has been proposed by Ulam [1] in 1940. One year later, Hyers [2] was the first author who answered Ulam’s problem when G and H are Banach spaces as follows:

Let G and H be Banach spaces. Suppose that φ : G ⟶ H fulfills

‖ φ ( a + b ) − φ ( a ) − φ ( b ) ‖ ≤ δ ,

for all a , b ∈ G and for some δ ≥ 0 . Then, there exists an additive mapping A : G ⟶ H such that ‖ φ ( x ) − A ( x ) ‖ ≤ δ for all a ∈ G . Next, some generalizations of Hyers’ result for additive and linear mappings have been studied by Aoki [3], Rassias [4], and Găvruţa [5]. More information about the stability of miscellaneous functional equations on various spaces is available, for instance, in books [6–8].

Throughout this article, N , Q , and R are the sets of all positive integers, rationals, and real numbers, respectively, N 0 ≔ N ∪ { 0 } , R + ≔ [ 0 , ∞ ) . Moreover, for the set E , we denote E × E × … × E ︷ n -times by E n .

Over the last two decades, the stability problem for functional equations has been studied by the authors for multiple variables mappings like multi-additive mappings. Here, we indicate their definitions:

Let V be a commutative group, W be a linear space over Q , and n ∈ N with n ≥ 2 . A mapping f : V n ⟶ W is called multi-additive if it satisfies

(1.1) A ( x + y ) = A ( x ) + A ( y ) ,

in each variable. It is shown in [9, Theorem 2] that a mapping f is multi-additive if and only if it satisfies

f ( x 1 + x 2 ) = ∑ i 1 , … , i n ∈ { 1,2 } f ( x i 1 1 , … , x i n n ) ,

where x i = ( x i 1 , … , x i n ) ∈ V n with i ∈ { 1 , 2 } . For more information about the structure of multi-additive mappings and their Ulam’s stability, we refer to [9–11] and [12, Sections 13.4 and 17.2].

Let ( G , • ) and ( H , ∗ ) be groupoids equipped with binary operations. A mapping ϕ : G ⟶ H is a homomorphism between groupoids if

(1.2) ϕ ( a • b ) = ϕ ( a ) ∗ ϕ ( b ) , ( a , b ∈ G ) .

A particular case of (1.2) is the famous Cauchy (additive) functional equation (1.1), where G and H are two semigroups with the operation + .

The stability of the Cauchy functional equation on square-symmetric groupoids was investigated by Páles et al. [13] who generalized the classical theorem of Hyers for the first time. Moreover, Kim [14] established some stability results of the Cauchy functional equation from square-symmetric groupoids into metric square-symmetric groupoids based on the control function proposed by Găvruţa. Some results on the stability of Cauchy and Jensen equations through a fixed point method were investigated in [15]; for more results on the stability of Cauchy equation, we refer to [16–18].

In this article, we introduce the multi-Cauchy mappings as a system of the Cauchy functional equations. Then, we reduce such system to obtain a single equation. Finally, we prove the stability of the multi-Cauchy functional equations on the powers of a square-symmetric groupoid.

2 Representation of multi-Cauchy mappings

Let ( G , • ) and ( H , ∗ ) be groupoids equipped with given binary operations. For each a ∈ G , set 2 a ≔ a • a . Similarly, for each h ∈ H , put 2 h ≔ h ∗ h . The binary operation • is said to be square-symmetric if 2 ( a • b ) ≔ 2 a • 2 b . It is obvious that a commutative semigroup is a square-symmetric groupoid. The converse, however, is not true in general. For example, put G ≔ N , r > 1 a fixed element in G , the binary operation a • b ≔ a + r b ( a , b ∈ G ) . Then, ( G , • ) is a square-symmetric groupoid and • is not associative.

Recall that a groupoid ( G , • ) is divisible ( • is divisible) if for each a ∈ G , there exists a unique element a ′ ∈ G such that 2 a ′ = a . For convenience, we will write a 2 ≔ a ′ or 1 2 a ≔ a ′ . To simplify the notation, for each a in a groupoid G ≔ ( G , • ) and each n ∈ N , we write 2 0 a ≔ a and 2 n + 1 a ≔ 2 ( 2 n a ) . If, in addition, G is divisible, then we also write a 2 0 ≔ a and a 2 n + 1 ≔ 1 2 a 2 n for all n ∈ N . To reach our aims in this article, we consider the following mapping:

σ • : G ⟶ G ; a ↦ 2 a and λ ∗ : H ⟶ H ; h ↦ 2 h .

Note that the square-symmetric of • implies that the mapping σ • is an endomorphism. According to the aforementioned fact, the binary operation • such that σ • is an automorphism (bijective) is divisible.

For each ( t 1 , … , t n ) , ( s 1 , … , s n ) ∈ G n , we define the pointwise binary operation on G n as follows:

( t 1 , … , t n ) • ( s 1 , … , s n ) = ( t 1 • s 1 , … , t n • s n ) .

Consider the mapping Γ • : G n ⟶ G n defined by Γ • ( s 1 , … , s n ) ≔ ( σ • ( s 1 ) , … , σ • ( s n ) ) . Moreover, for each m ∈ N , we have

Γ • m ( s 1 , … , s n ) ≔ ( σ • m ( s 1 ) , … , σ • m ( s n ) ) = ( 2 m s 1 , … , 2 m s n ) .

From now on, we denote ( 2 m s 1 , … , 2 m s n ) by 2 m ( s 1 , … , s n ) when no confusion can arise. It is easily seen that if σ • is an automorphism, then so is Γ • .

A mapping f : G n ⟶ H is called multi-Cauchy (multi-additive) if, for each j ∈ { 1 , … , n } and all a i ∈ G , the mapping a ↦ f ( a 1 , … , a j − 1 , a , a j + 1 , … , a n ) satisfies equation (1.2), i.e., for each 1 ≤ j ≤ n ,

f ( a 1 , … , a j − 1 , a j • a ′ j , a j + 1 , … , a n ) = f ( a 1 , … , a j − 1 , a j , a j + 1 , … , a n ) ∗ f ( a 1 , … , a j − 1 , a ′ j , a j + 1 , … , a n ) .

Let n ∈ N with n ≥ 2 and a i n = ( a i 1 , a i 2 , … , a i n ) ∈ G n , where i ∈ { 1 , 2 } . We shall denote a i n by a i when there is no risk of ambiguity. For a 1 , a 2 ∈ G n , we write f ( a 1 ) ∗ f ( a 2 ) ≔ ∗ k ∈ { 1,2 } f ( a k ) .

Assume that a square-symmetric groupoids G and H have the identities e G and e H , respectively. We consider the following condition on H :

(K) For m ∈ N and h ∈ H , if m h = e H , then h = e H .

There are plenty of known examples of groups with the condition (K), e.g., the group ( Z , + ) . A mapping f : G n ⟶ H has the identity condition, if f ( a ) = e H for any a ∈ G n with at least one variable that is equal to e G . It is easily checked that every multi-Cauchy mapping f : G n ⟶ H has identity condition provided that H has condition (K).

Set n ≔ { 1 , … , n } . For a subset m = { j 1 , … , j i } of n with 1 ≤ j 1 < … < j i ≤ n and g = ( g 1 , … , g n ) ∈ V n ,

m g ≔ ( e G , … , e G , g j 1 , e G , … , e G , g j i , e G , … , e G ) ∈ G n

denotes the vector that coincides with g in exactly those components, which are indexed by the elements of m and whose other components are set equal e G .

Lemma 2.1

Let ( G , • ) be a groupoid and and ( H , ∗ ) be a commutative group. Suppose that a mapping f : G n ⟶ H satisfies

(2.1) f ( a 1 • a 2 ) = ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n n ) .

If H has the condition (K), then f has the identity condition.

Proof

We argue by induction on m that f ( m g ) = e H for 0 ≤ m ≤ n − 1 . Let m = 0 . Substituting a 1 = a 2 = ( e G , … , e G ) into (2.1), we have

f ( e G , … , e G ) = 2 n f ( e G , … , e G ) .

Our assumption implies that f ( e G , … , e G ) = e H . Assume that f ( a m − 1 ) = e H . We will prove that f ( a m ) = e H . Without loss of generality, we assume that the first m variables are not equal to e G . Replacing ( a 1 , a 2 ) by ( a m , e G ) in equation (2.1), we obtain f ( m a ) = 2 n − m f ( a m ) and so f ( a m ) = e H . This completes the proof.□

Proposition 2.2

Let ( G , • ) be a groupoid and and ( H , ∗ ) be a commutative group. If a mapping f : G n ⟶ H is multi-Cauchy, then it fulfills (2.1). The converse is true if H has the condition (K).

Proof

Let f be multi-Cauchy mapping. We prove by induction on n that it satisfies equation (2.1). For n = 1 , f fulfills (1.2). Assume that (2.1) is valid for some positive integer n − 1 . We will show that it is true for n . We have

f ( a 1 n • a 2 n ) = f ( a 1 n − 1 • a 2 n − 1 , a 1 n ) ∗ f ( a 1 n − 1 • a 2 n − 1 , a 2 n ) = ( ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n − 1 , n − 1 , a 1 n ) ) ∗ ( ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n − 1 , n − 1 , a 2 n ) ) = ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n n ) ,

which shows that (2.1) holds for n . For the converse, by substituting a 2 = ( e H , … , e H , a 2 j , e H , … , e H ) into (2.1) and using Lemma 2.1, we obtain

f ( a 11 , … , a 1 , j − 1 , a 1 j • a 2 j , … , a 1 n ) = f ( a 1 ) ∗ f ( a 11 , … , a 1 , j − 1 , a 2 j , … , a 1 n ) ,

and so the proof is now complete.□

3 Stability results for (2.1)

In this section, we prove some stabilities regarding equation (2.1).

Let ( X , d ) be a metric space. The Lipschitz modulus of a mapping ϕ : X ⟶ X is denoted by Lip ( ϕ ) and defined by

Lip ( ϕ ) ≔ sup d ( ϕ ( x ) , ϕ ( y ) ) d ( x , y ) ∣ x , y ∈ X , x ≠ y .

It is simply to check that Lip ( ϕ ∘ ψ ) ≤ Lip ( ϕ ) Lip ( ψ ) .

One of the main goals of this article is the upcoming theorem, which is the Găvruţa stability of equation (2.1).

Theorem 3.1

Let ( G , • ) be a square-symmetric groupoid and ( H , ∗ , d ) be a complete metric divisible semigroup. Suppose that a mapping f : G n ⟶ H fulfills

(3.1) d ( f ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n n ) ) ≤ φ ( a 1 , a 2 ) ,

for all a 1 , a 2 ∈ G n in which φ : G n × G n ⟶ R + is a function that satisfies

(3.2) ∑ m = 0 ∞ Lip ( λ ∗ − m ) φ ( 2 m a 1 , 2 m a 2 ) < ∞ ,

for all a 1 , a 2 ∈ G n . Then, for each a ∈ G n ,

(3.3) F ( a ) ≔ lim m → ∞ λ ∗ − m + 1 ∘ f ( 2 m a )

converges and the mapping F : G n ⟶ H is a solution of (2.1) such that

(3.4) d ( f ( a ) , F ( a ) ) ≤ Lip ( λ ∗ − 1 ) Φ ( a ) ,

for all a ∈ G n , where Φ ( a ) ≔ ∑ m = 0 ∞ Lip ( λ ∗ − m ) φ ( 2 m a , 2 m a ) . In addition, if { Lip ( λ ∗ − m ) Lip ( λ ∗ m ) } is bounded, then F is a unique solution of (2.1) for which the mapping a ↦ d ( f ( a ) , F ( a ) ) is bounded with bound Lip ( λ ∗ − 1 ) Φ ( a ) .

Proof

Consider a family of mappings F m : G n ⟶ H defined via F 0 ≔ f and F m ≔ λ ∗ − m ∘ f ∘ Γ • m , for all m ∈ N . We shall show that for each fixed and arbitrary a ∈ G n , the sequence { F m } is convergent. Replacing a 1 and a 2 by 2 m a 1 in (3.1), we obtain

d ( f ( 2 m a 1 • 2 m a 1 ) , f ( 2 m a 1 ) ∗ f ( 2 m a 1 ) ) ≤ φ ( 2 m a 1 , 2 m a 1 ) ,

for all a 1 ∈ G n . For the rest of proof, we set a 1 by a unless otherwise stated explicitly. The last inequality can be rewritten as form

(3.5) d ( f ( 2 m + 1 a ) , λ ∗ ∘ f ( 2 m a ) ) ≤ φ ( 2 m a , 2 m a ) .

It follows from (3.5) that

d ( F m + 1 ( a ) , F m ( a ) ) = d ( λ ∗ − ( m + 1 ) ∘ f ( 2 m + 1 a ) , λ ∗ − m + 1 ∘ λ ∗ ∘ f ( 2 m a ) ) ≤ Lip ( λ ∗ − ( m + 1 ) ) d ( f ( 2 m a ) , λ ∗ ∘ f ( 2 m a ) ) ≤ Lip ( λ ∗ − ( m + 1 ) ) φ ( 2 m a , 2 m a ) ≤ Lip ( λ ∗ − 1 ) Lip ( λ ∗ − m ) φ ( 2 m a , 2 m a ) ,

for all a ∈ G n . It concludes from the aforementioned relation that for each l , k ∈ N with k > l ,

d ( F k ( a ) , F l ( a ) ) ≤ ∑ j = l k − 1 d ( F j + 1 ( a ) , F j ( a ) ) ≤ ∑ j = l k − 1 Lip ( λ ∗ − ( j + 1 ) ) φ ( 2 j a , 2 j a ) .

The convergence of series (3.2) implies that the last term in the aforementioned inequalities goes to zero as l → ∞ , which shows that { F m } is a Cauchy sequence, and hence, it converges to mapping F : G n ⟶ H as defined in (3.3), which is also well defined. Furthermore, for each m ∈ N , we have

d ( F m + 1 ( a ) , f ( a ) ) ≤ ∑ j = 0 m d ( F j + 1 ( a ) , F j ( a ) ) ≤ ∑ j = 0 m Lip ( λ ∗ − ( j + 1 ) ) φ ( 2 j a , 2 j a ) ≤ Lip λ ∗ − 1 ∑ j = 0 m Lip ( λ ∗ − j ) φ ( 2 j a , 2 j a ) .

Taking m → ∞ , we reach to (3.4). Replacing ( a 1 , a 2 ) by ( 2 m a 1 , 2 m a 2 ) in (3.1) and using the endomorphism of Γ • , we find

d ( f ∘ ( Γ • m ( a 1 • a 2 ) ) , ∗ j 1 , … , j n ∈ { 1,2 } f ( 2 m a j 1 1 , … , 2 m a j n n ) ) ≤ φ ( 2 m a 1 , 2 m a 2 ) ,

for all a 1 , a 2 ∈ G n . On the other hand, λ ∗ is an automorphism of ( H , ∗ ) and so λ ∗ − m is an automorphism as well, for all m ∈ N . This discussion necessitates that

d ( F m ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } F m ( a j 1 1 , … , a j n n ) ) = d ( λ ∗ − m ( f ∘ Γ • m ( a 1 • a 2 ) ) , ∗ j 1 , … , j n ∈ { 1,2 } λ ∗ − m ∘ f ∘ Γ • m ( a j 1 1 , … , a j n n ) ) = d ( λ ∗ − m ( f ∘ Γ • m ( a 1 • a 2 ) ) , λ ∗ − m ( ∗ j 1 , … , j n ∈ { 1,2 } f ∘ Γ • m ( a j 1 1 , … , a j n n ) ) ) ≤ Lip ( λ ∗ − m ) d ( f ∘ Γ • m ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } f ∘ Γ • m ( a j 1 1 , … , a j n n ) ) ≤ Lip ( λ ∗ − m ) φ ( 2 m a 1 , 2 m a 2 ) .

Letting the limit as m → ∞ in the aforementioned relation and applying the definition of F m , we see that F satisfies (2.1). For the uniqueness of F , assume that ℱ is an arbitrary solution of (2.1) such that the mapping a ↦ d ( f ( a ) , ℱ ( a ) ) , is bounded with bound Lip λ ∗ − 1 Φ ( a ) . Substituting a 1 = a 2 ≔ a into (2.1), we obtain

F ∘ Γ • = λ ∗ ∘ F , ℱ ∘ Γ • = λ ∗ ∘ ℱ .

One can prove by induction that

λ ∗ − m ∘ F ∘ Γ • m = F , λ ∗ − m ∘ ℱ ∘ Γ • m = ℱ .

Now, for each a ∈ G n we have

d ( F ( a ) , ℱ ( a ) ) = d ( λ ∗ − m ∘ F ( 2 m a ) , λ ∗ − m ∘ ℱ ( 2 m a ) ) ≤ Lip ( λ ∗ − m ) d ( F ( 2 m a ) , ℱ ( 2 m a ) ) ≤ Lip ( λ ∗ − m ) [ d ( F ( 2 m a ) , f ( 2 m a ) ) + d ( f ( 2 m a ) , ℱ ( 2 m a ) ) ] ≤ 2 Lip ( λ ∗ − m ) ∑ j = 0 ∞ Lip ( λ ∗ − ( j + 1 ) ) φ ( Γ • j ( Γ • m ) ( a ) , Γ • j ( Γ • m ) ( a ) ) ≤ 2 Lip ( λ ∗ − m ) Lip ( λ ∗ m ) ∑ j = 0 ∞ Lip ( λ ∗ − ( m + j + 1 ) ) φ ( 2 m + j a , 2 m + j a ) = 2 Lip ( λ ∗ − m ) Lip ( λ ∗ m ) ∑ j = m ∞ Lip ( λ ∗ − ( j + 1 ) ) φ ( 2 j a , 2 j a ) .

Due to the boundedness of { Lip ( λ ∗ − m ) Lip ( λ ∗ m ) } , it follows that F = ℱ , when m → ∞ . This completes the proof.□

The next corollary is a direct consequence of Theorem 3.1 concerning the stability of equation (2.1).

Corollary 3.2

Let ( G , • ) be a square-symmetric groupoid and ( H , ∗ , d ) be a complete metric divisible semigroup. Let f : G n ⟶ H satisfy (3.1) and φ : G n × G n ⟶ R + be a function such that for some real number L with 0 ≤ L < 1 , the inequality

(3.6) d ( λ ∗ − m ( h 1 ) ) , λ ∗ − m ( h 2 ) φ ( 2 m a 1 , 2 m a 2 ) ≤ L d ( λ ∗ − m + 1 ( h 1 ) ) , λ ∗ − m + 1 ( h 2 ) φ ( 2 m − 1 a 1 , 2 m − 1 a 2 ) ,

holds for all a 1 , a 2 ∈ G n , h 1 , h 2 ∈ H , and n ∈ N . Then, for each a ∈ G n , limit (3.3) exists and the mapping F : G n ⟶ H is a unique solution of (2.1) such that

d ( f ( a ) , F ( a ) ) ≤ 1 1 − L φ ( a , a ) ,

for all a ∈ G n .

Proof

Substituting a 1 = a 2 = a and m = 1 into (3.6), we have

d ( λ ∗ − 1 ( h 1 ) ) , λ ∗ − 1 ( h 2 ) φ ( 2 a , 2 a ) ≤ L d ( h 1 , h 2 ) φ ( a , a ) ,

or equivalently,

Lip ( λ ∗ − 1 ) φ ( 2 a , 2 a ) ≤ L φ ( a , a ) ,

for all a ∈ G n . By induction and using (3.6), one can obtain

Lip ( λ ∗ − m ) φ ( 2 m a , 2 m a ) ≤ L m φ ( a , a ) ,

for all a ∈ G n . Therefore, the bound of (3.4) defined in Theorem 3.1 does not exceed 1 1 − L Lip ( λ ∗ − 1 ) φ ( a , a ) .□

In the following result, we show Hyers’ stability of equation (2.1).

Corollary 3.3

Let δ > 0 . Let ( G , • ) be a square-symmetric groupoid and ( H , ∗ , d ) be a complete metric divisible semigroup. Suppose that a mapping f : G n ⟶ H fulfills

d ( f ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n n ) ) ≤ δ ,

for all a 1 , a 2 ∈ G n . Then, for each a ∈ G n ,

F ( a ) ≔ lim m → ∞ λ ∗ − m + 1 ∘ f ( 2 m a )

converges, and the mapping F : G n ⟶ H is a solution of (2.1) such that

d ( f ( a ) , F ( a ) ) ≤ δ Lip ( λ ∗ − 1 ) ∑ m = 0 ∞ Lip ( λ ∗ − m ) ,

for all a ∈ G n . Moreover, F is a unique solution of (2.1) such that the mapping a ↦ d ( f ( a ) , F ( a ) ) , is bounded.

Proof

Let φ ( a 1 , a 2 ) = δ . Then, the result will be found by Theorem 3.1.□

In analogy with Theorem 3.1, we have the next alternative result for the stability of (2.1). In this theorem, we add the condition divisibility for the square-symmetric groupoid ( G , • ) .

Theorem 3.4

Let ( G , • ) be a square-symmetric divisible groupoid and ( H , ∗ , d ) be a complete metric divisible semigroup. Suppose that a mapping f : G n ⟶ H fulfills

(3.7) d ( f ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n n ) ) ≤ φ ( a 1 , a 2 ) ,

for all a 1 , a 2 ∈ G n in which φ : G n × G n ⟶ R + is a function satisfying

(3.8) ∑ m = 0 ∞ Lip ( λ ∗ m ) φ ( 2 − m a 1 , 2 − m a 2 ) < ∞ ,

for all a 1 , a 2 ∈ G n . Then, for each a ∈ G n ,

(3.9) F ( a ) ≔ lim m → ∞ λ ∗ m + 1 ∘ f ( 2 − m a ) ,

converges and the mapping F : G n ⟶ H is a solution of (2.1) such that

(3.10) d ( f ( a ) , F ( a ) ) ≤ Lip λ ∗ − 1 Ψ ( a ) ,

for all a ∈ G n , where Ψ ( a ) ≔ ∑ m = 0 ∞ Lip ( λ ∗ m ) φ ( 2 − m a , 2 − m a ) . In addition, if { Lip ( λ ∗ − m ) Lip ( λ ∗ m ) } is bounded, then F is a unique solution of (2.1) for which the mapping a ↦ d ( f ( a ) , F ( a ) ) , is bounded with bound Lip λ ∗ − 1 Ψ ( a ) .

Proof

Define the mappings F m : G n ⟶ H defined via F 0 ≔ f and F m ≔ λ ∗ m ∘ f ∘ Γ • − m , for all m ∈ N . Replacing a 1 and a 2 by 2 − m a 1 in (3.7), we have

d ( f ( 2 − m a 1 • 2 − m a 1 ) , f ( 2 − m a 1 ) ∗ f ( 2 − m a 1 ) ) ≤ φ ( 2 − m a 1 , 2 − m a 1 ) ,

for all a 1 ∈ G n . For the rest of proof, we set a 1 by a unless otherwise stated explicitly. We reform the last relation as follows:

(3.11) d ( f ( 2 − m + 1 a ) , λ ∗ ∘ f ( 2 − m a ) ) ≤ φ ( 2 − m a , 2 − m a ) .

By (3.11), we have

d ( F m − 1 ( a ) , F m ( a ) ) = d ( λ ∗ m − 1 ∘ f ( 2 − m + 1 a ) , λ ∗ m − 1 ∘ λ ∗ ∘ f ( 2 − m a ) ) ≤ Lip ( λ ∗ m − 1 ) d ( f ( 2 − m + 1 a ) , λ ∗ ∘ f ( 2 − m a ) ) ≤ Lip ( λ ∗ m − 1 ) φ ( 2 − m a , 2 − m a ) ,

for all a ∈ G n . A direct result from the aforementioned relation shows that for each l , k ∈ N ,

d ( F k ( a ) , F k + l ( a ) ) ≤ ∑ j = l l − 1 d ( F k + j ( a ) , F k + j + 1 ( a ) ) ≤ ∑ j = 0 l − 1 Lip ( λ ∗ k + j ) φ ( 2 − k − j − 1 a , 2 − k − j − 1 a ) ≤ ∑ j = k k + l − 1 Lip ( λ ∗ − 1 ) Lip ( λ ∗ j + 1 ) φ ( 2 − j − 1 a , 2 − j − 1 a ) .

It follows from the convergence of series (3.8) that the last term in the aforementioned inequalities intends to zero as k → ∞ , which shows that { F m } is a Cauchy sequence in H . Therefore, a mapping F : G n ⟶ H as defined in (3.9) is well defined. Moreover, for each m ∈ N , we have

d ( F m + 1 ( a ) , f ( a ) ) ≤ ∑ j = 0 m d ( F j + 1 ( a ) , F j ( a ) ) ≤ ∑ j = 0 m Lip ( λ ∗ j ) φ ( 2 − j − 1 a , 2 − j − 1 a ) ≤ ∑ j = 1 ∞ Lip ( λ ∗ j − 1 ) φ ( 2 − j a , 2 − j a ) ≤ Lip λ ∗ − 1 ∑ j = 1 ∞ Lip ( λ ∗ j ) φ ( 2 − j a , 2 − j a ) .

Letting m → ∞ , we obtain (3.10). Replacing ( a 1 , a 2 ) by ( 2 − m a 1 , 2 − m a 2 ) in (3.7) and using the endomorphism of Γ • m , we obtain

d ( F m ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } F m ( a j 1 1 , … , a j n n ) ) = d ( λ ∗ m ( f ∘ Γ • − m ( a 1 • a 2 ) ) , ∗ j 1 , … , j n ∈ { 1,2 } λ ∗ m ∘ f ∘ Γ • − m ( a j 1 1 , … , a j n n ) ) = d ( λ ∗ m ( f ∘ Γ • − m ( a 1 • a 2 ) ) , λ ∗ m ( ∗ j 1 , … , j n ∈ { 1,2 } f ∘ Γ • − m ( a j 1 1 , … , a j n n ) ) ) ≤ Lip ( λ ∗ m ) d ( f ∘ Γ • − m ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } f ∘ Γ • − m ( a j 1 1 , … , a j n n ) ) ≤ Lip ( λ ∗ m ) φ ( 2 − m a 1 , 2 − m a 2 ) .

Taking m → ∞ in the aforementioned relation and using the definition of F m , we find that F fulfills (2.1). Here, we show the uniqueness of F . Assume that ℱ is an arbitrary solution of (2.1) such that the mapping a ↦ d ( f ( a ) , ℱ ( a ) ) , is bounded with bound Lip λ ∗ − 1 Ψ ( a ) . Similar to the proof of Theorem 3.1, one can prove by induction that

λ ∗ m ∘ F ∘ Γ • − m = F , λ ∗ m ∘ ℱ ∘ Γ • − m = ℱ .

Note that { Lip ( λ ∗ − m ) Lip ( λ ∗ m ) } is bounded and it is assumed that ℱ is another solution of (2.1) such that a ↦ d ( f ( a ) ) , d ( ℱ ( a ) ) is bounded with bound Lip λ ∗ − 1 Ψ ( a ) . Thus, for each a ∈ G n , we have

d ( F ( a ) , ℱ ( a ) ) = d ( λ ∗ m ∘ F ( 2 − m a ) , λ ∗ m ∘ ℱ ( 2 − m a ) ) ≤ Lip ( λ ∗ m ) d ( F ( 2 − m a ) , ℱ ( 2 − m a ) ) ≤ Lip ( λ ∗ m ) [ d ( F ( 2 − m a ) , f ( 2 − m a ) ) + d ( f ( 2 − m a ) , ℱ ( 2 − m a ) ) ] ≤ 2 Lip ( λ ∗ − 1 ) Lip ( λ ∗ m ) ∑ j = 0 ∞ Lip ( λ ∗ j ) φ ( Γ • − j ( Γ • − m ) ( a ) , Γ • − j ( Γ • − m ) ( a ) ) ≤ 2 Lip ( λ ∗ − 1 ) Lip ( λ ∗ m ) Lip ( λ ∗ − m ) ∑ j = 0 ∞ Lip ( λ ∗ m + j ) φ ( 2 − m − j a , 2 − m − j a ) = 2 Lip ( λ ∗ − 1 ) Lip ( λ ∗ m ) Lip ( λ ∗ − m ) ∑ j = m + 1 ∞ Lip ( λ ∗ j ) φ ( 2 − j a , 2 − j a ) .

Since the sequence { Lip λ ∗ − m Lip ( λ ∗ m ) } is bounded, this proves that F = ℱ , and therefore, the proof is finished.□

Corollary 3.5

Let ( G , • ) be a square-symmetric groupoid and ( H , ∗ , d ) be a complete metric divisible semigroup. Let f : G n ⟶ H satisfy (3.7) and φ : G n × G n ⟶ R + be a function such that for some real number L with 0 ≤ L < 1 , the inequality

(3.12) d ( λ ∗ m ( h 1 ) ) , λ ∗ m ( h 2 ) φ ( 2 − m a 1 , 2 − m a 2 ) ≤ L d ( λ ∗ m − 1 ( h 1 ) ) , λ ∗ m − 1 ( h 2 ) φ ( 2 − m + 1 a 1 , 2 − m + 1 a 2 )

holds for all a 1 , a 2 ∈ G n , h 1 , h 2 ∈ H , and n ∈ N Then, for each a ∈ G n , limit (3.9) exists and the mapping F : G n ⟶ H is a unique solution of (2.1) such that

d ( f ( a ) , F ( a ) ) ≤ L 1 − L φ ( a , a ) ,

for all a ∈ G n .

Proof

Substituting a 1 = a 2 = a and m = 1 into (3.12), we obtain

d ( λ ∗ ( h 1 ) ) , λ ∗ ( h 2 ) φ ( 2 − 1 a , 2 − 1 a ) ≤ L d ( h 1 , h 2 ) φ ( a , a ) ,

for all a ∈ G n . It follows from the aforementioned equality that

Lip ( λ ∗ ) φ ( 2 − 1 a , 2 − 1 a ) ≤ L φ ( a , a ) ,

for all a ∈ G n . By induction, we obtain, due to (3.12)

Lip ( λ ∗ m ) φ ( 2 − m a , 2 − m a ) ≤ L m φ ( a , a ) ,

for all a ∈ G n . Hence, the bound of (3.4) defined in Theorem 3.4 does not exceed L 1 − L Lip ( λ ∗ − 1 ) φ ( a , a ) .□

In analogue to Corollary 3.3, equation (2.1) has Hyers’ stability as follows.

Corollary 3.6

Let δ > 0 . Let ( G , • ) be a square-symmetric groupoid and ( H , ∗ , d ) be a complete metric divisible semigroup. Suppose that a mapping f : G n ⟶ H fulfills

d ( f ( a 1 • a 2 ) , ∗ j 1 , … , j n ∈ { 1,2 } f ( a j 1 1 , … , a j n n ) ) ≤ δ ,

for all a 1 , a 2 ∈ G n . Then, for each a ∈ G n ,

F ( a ) ≔ lim m → ∞ λ ∗ m + 1 ∘ f ( 2 − m a ) ,

converges, and the mapping F : G n ⟶ H is a solution of (2.1) such that

d ( f ( a ) , F ( a ) ) ≤ δ Lip ( λ ∗ − 1 ) ∑ m = 0 ∞ Lip ( λ ∗ m ) ,

for all a ∈ G n . Moreover, F is a unique solution of (2.1) for which the mapping a ↦ d ( f ( a ) , F ( a ) ) , is bounded.

Proof

Letting φ ( a 1 , a 2 ) = δ , we obtain the desired result from Theorem 3.4.□

Let X be a Banach space over K , which is either R or C . We define the binary operation ∗ on X by x ∗ y = r x + s y , where r , s ∈ K , which are fixed. For the element ( a l 1 1 , … , a l n n ) ∈ G n , we put t i = Card { l j : l j = 1 } . Clearly, 0 ≤ s i ≤ n . With these explanations, for a multi-Cauchy mapping f : G n ⟶ X , equation (2.1) converts to

(3.13) f ( a 1 • a 2 ) = ∑ l 1 , … , l n ∈ { 1,2 } 0 ≤ t i ≤ n r t i s n − t i f ( a l 1 1 , … , a l n n ) ,

for all a 1 , a 2 ∈ G n .

Corollary 3.7

Let ( G , • ) be a square-symmetric groupoid and X be a Banach space over K . Suppose that a mapping f : G n ⟶ X fulfills

f ( a 1 • a 2 ) − ∑ l 1 , … , l n ∈ { 1,2 } 0 ≤ t i ≤ n r t i s n − t i f ( a l 1 1 , … , a l n n ) ≤ φ ( a 1 , a 2 ) ,

for all a 1 , a 2 ∈ G n in which φ : G n × G n ⟶ R + is a function satisfying one of the following conditions:

∣ r + s ∣ ≠ 0 and ∑ m = 0 ∞ ∣ r + s ∣ − m φ ( 2 m a 1 , 2 m a 2 ) < ∞ ;
( G , • ) is divisible and ∑ m = 1 ∞ ∣ r + s ∣ m φ ( 2 − m a 1 , 2 − m a 2 ) < ∞ ,

for all a 1 , a 2 ∈ G n . Then, there exists a uniquely determined solution of (3.13) such that

∥ f ( a ) − F ( a ) ∥ ≤ T ,

for all a ∈ G n , where

T ≔ ∑ m = 0 ∞ ∣ r + s ∣ − m φ ( 2 m a 1 , 2 m a 2 ) , i f ( i ) h o l d s , ∑ m = 1 ∞ ∣ r + s ∣ m φ ( 2 m a 1 , 2 − m a 2 ) , i f ( i i ) h o l d s .

Proof

It is easily checked that Lip ( λ ∗ m ) = ∣ r + s ∣ m and Lip ( λ ∗ − m ) = ∣ r + s ∣ − m . Therefore, the desired results can be obtained by Theorems 3.1 and 3.4.□

Acknowledgement

The authors sincerely thank the anonymous reviewers for their careful reading, and constructive comments to improve the quality of the first draft of this article substantially.

Funding information: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Not applicable. No datasets were generated or analyzed during the current study.

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Received: 2024-04-10

Accepted: 2025-03-11

Published Online: 2025-04-24

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

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https://doi.org/10.1515/math-2025-0142

Keywords for this article

Găvruţa stability; groupoid; Hyers-Ulam stability; multi-Cauchy mapping

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