Fuzzy stability of multi-additive mappings

Choonkil Park; Abasalt Bodaghi; Mana Donganont

doi:10.1515/dema-2025-0159

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Fuzzy stability of multi-additive mappings

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Published/Copyright: October 9, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The main aim of this study is to establish some stability results concerning the multi-additive mappings by applying the so-called direct (Hyers) method and the alternative fixed approach in the setting of fuzzy normed spaces. In addition, it is proven that if a fuzzy approximate multi-additive mapping is continuous at a point, then it can be approximated by an everywhere continuous multi-additive mapping. Comparing the obtained results by the mentioned ways, we observe that the fixed-point tool gives us more exact approximation of approximately multi-additive mappings in comparison to the direct method.

Keywords: fuzzy normed space; fuzzy Hyers-Ulam-Rassias stability; multi-additive mapping

MSC 2020: 39B82; 39B52; 47H10

1 Introduction and preliminaries

It is known that the topic of functional equations play an important, essential, and fascinating role in nonlinear analysis, employing simple algebraic procedures that lead to intriguing solutions. Ulam stability [1] (proposed for group homomorphisms and answered by Hyers [2]) is a crucial concept in studying functional equations and their solutions. In fact, Hyers gave a first affirmative answer to the question of Ulam for Banach spaces as follows: Let V and W be Banach spaces. Suppose that f : V ⟶ W satisfies

‖ f ( u + v ) − f ( u ) − f ( v ) ‖ ≤ ε ,

for all u , v ∈ V and for some ε ≥ 0 . Then, there exists a unique additive mapping A : V ⟶ W such that

‖ f ( v ) − A ( v ) ‖ ≤ ε ,

for all v ∈ V in which A ( v ) = lim n → ∞ f ( 2 n v ) 2 n . Later, the result of Hyers was significantly generalized by Rassias [3] and Găvruţa [4]. This theory examines whether a function that approximately satisfies a certain functional equation is near to a function that exactly satisfies the equation. After that, the stability problem for functional equations was extended and generalized for miscellaneous mappings and equations, which are available in many articles and books; see for instance [5–9] and more references therein.

A fuzzy norm on a linear space has been introduced by Katsaras [10] and almost simultaneously Wu and Fang [11] defined a concept of fuzzy normed space and presented the generalization of the Kolmogorov normalized theorem for a fuzzy topological vector space. Next Biswas [12] defined and investigated fuzzy inner product spaces in a linear space. Since then, many authors and mathematicians have defined fuzzy norms on a vector space from various points of view [13–16]. Following Cheng and Mordeson [17], who they introduced a definition of fuzzy norm on a vector space in such a manner that the corresponding induced fuzzy metric is of Kramosil and Michalek type [18], Bag and Samanta [13] modified their definition and then investigated some properties of fuzzy normed spaces [19]. The Hyers-Ulam-Rassias stability of the various functional equations in fuzzy normed spaces was studied for instance in [20–24].

Let V be a commutative group, W be a linear space over rational numbers, and n be an integer with n ≥ 2 . A mapping f : V n ⟶ W is called multi-additive if it satisfies the Cauchy’s functional equation A ( x + y ) = A ( x ) + A ( y ) in each variable, that is,

f ( v 1 , … , v i − 1 , v i + v i ′ , v i + 1 , … , v n ) = f ( v 1 , … , v i − 1 , v i , v i + 1 , … , v n ) + f ( v 1 , … , v i − 1 , v i ′ , v i + 1 , … , v n ) ,

for all i ∈ { 1 , … , n } . Note that the mapping f : R n ⟶ R defined by f ( r 1 , … , r n ) = c ∏ j = 1 n r j is a multi-additive mapping, where c is a fixed real number.

It is important for us to know whether the above-mentioned mappings (defined as a system of n functional equations) can be unified and described as an equation. The answer is affirmative. In fact, Ciepliński in [25, Theorem 2] presented the following result.

Theorem 1

Let ( V , + ) be a commutative semigroup with the identity element 0 and W be a linear space. A mapping f : V n ⟶ W is multi-additive if and only if it fulfills the equation

f ( v 1 + v 2 ) = ∑ j 1 , … , j n ∈ { 1 , 2 } f ( v j 1 1 , … , v j n n ) ,

where v j = ( v j 1 , … , v j n ) ∈ V n with j ∈ { 1 , 2 } .

In the next result, we give a characterization of multi-additive mappings for the case that V = R d , which was proved in [26, Theorem 13.4.3].

Theorem 2

Let g : R d N ⟶ R be a continuous d-additive function. Then, there exist constants c j 1 … j d ∈ R , j 1 , … , j d = 1 , … , N , such that

g ( r 1 , … , r d ) = ∑ j 1 = 1 N … ∑ j d = 1 N c j 1 … j d r 1 j 1 … r d j d ,

for all r i = ( r i 1 , … , r i N ) and i = 1 , … , d .

More information and details about the structure of multi-additive mappings are available in [26]. Moreover, a lot of stability results for the multi-additive mappings can be found in [27–31]. Recently, Bodaghi [32] introduced and characterized d -dimensional multi-additive mappings and established their Ulam’s stability in the setting of neutrosophic Banach spaces.

In this article, we prove Hyers-Ulam-Rassias stability of the multi-additive mappings by applying the direct and fixed methods in the setting of fuzzy normed spaces. Moreover, we prove that if a fuzzy approximate multi-additive mapping is continuous at a point, then we can approximate it by an everywhere continuous multi-additive mapping. Finally, we find that the fixed-point technique is more exact approximation of approximately multi-additive mappings relative to the direct method.

2 Stability results for multi-additive mappings

In this section, we prove the Hyers-Ulam-Rassias stability of multi-additive mappings in fuzzy normed spaces by the direct and fixed-point methods.

Definition 1

Let V be a real linear space. A function T : V × R ⟶ [ 0 , 1 ] is called a fuzzy norm on V if for all u , v ∈ V and all s , t ∈ R

T ( v , r ) = 0 for r ≤ 0 ;
v = 0 if and only if T ( v , r ) = 1 for all r > 0 ;
T ( r v , t ) = T v , t ∣ r ∣ if r ≠ 0 ;
T ( u + v , s + t ) ≥ min { T ( u , s ) , T ( v , t ) } ;
T ( v , ⋅ ) is a non-decreasing function on R on and lim t → ∞ T ( v , t ) = 1 ;
For v ≠ 0 , T ( v , ⋅ ) is (upper semi)-continuous on R .

The pair ( V , T ) is said to be a fuzzy normed linear space. Note that the fuzzy normed linear space ( V , T ) is exactly a Menger probabilistic normed linear space ( V , T , Δ ) when Δ = min . It is a locally convex first-countable Hausdorff linear topological space; for more details, refer [33]. Moreover, we may regard T ( v , r ) as the truth value of the statement that “the norm of v is less than or equal to the real number t .”

Let ( V , T ) be a fuzzy normed vector space. A sequence { v j } in V is called convergent or converge if there exists a v ∈ V such that lim j → ∞ T ( v j − v , t ) = 1 for all t 0 . In this case, v is called the limit of the sequence { v j } and we denote it by T − lim t → ∞ v j = v . Moreover, a sequence { v j } in V is called Cauchy if for each δ > 0 and each t > 0 , there exists an t 0 ∈ N such that for all t ≥ t 0 and all k > 0 , we have T ( v j + k − v j , t ) > 1 − δ . Obviously, every convergent sequence in a fuzzy normed vector space is Cauchy. Now, if each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping f : V ⟶ W between fuzzy normed vector spaces V and W is continuous at a point v 0 ∈ V if for each sequence { v j } converging to v 0 in V , the sequence { f ( v j ) } converges to f ( v 0 ) . If f : V ⟶ W is continuous at each v ∈ V , then it is said to be continuous on V ; for more details, refer [19].

The following example is presented in [24]. In continuation, we apply this example to arrive at more results.

Example 1

Let ( V , ‖ ⋅ ‖ ) be a normed linear space. For v ∈ V , the function

T ( v , t ) = t ‖ v ‖ + t t > 0 , 0 otherwise

is a fuzzy norm on V and ( V , T ) is a fuzzy normed linear space. Recall that the fuzzy norm above is the standard fuzzy norm induced by norm ‖ ⋅ ‖ .

The next examples convert a normed linear space to a fuzzy normed space, indicated in [21] and [23], respectively.

Example 2

Let ( V , ‖ ⋅ ‖ ) be a normed linear space. For v ∈ V , the function

T ( v , t ) = 1 t ≥ ‖ v ‖ , 0 t < ‖ v ‖

is a fuzzy norm on V and ( V , T ) is a fuzzy normed linear space.

Example 3

Let ( V , ‖ ⋅ ‖ ) be a normed linear space. For v ∈ V , the function

T ( v , t ) = 1 t > ‖ v ‖ , t ‖ v ‖ 0 < t ≤ ‖ v ‖ , 0 otherwise

is a fuzzy norm on V and ( V , T ) is a fuzzy normed linear space.

2.1 Stability results: Direct method

In this subsection, we prove the stability of multi-additive mappings in fuzzy normed spaces by the direct method.

Theorem 3

Let V be a real linear space, ( W , T ) be a fuzzy Banach space, and ( Y , T ′ ) be a fuzzy normed space. Suppose that f : V n ⟶ W is a mapping and ϕ : V n ⟶ Y is a function such that

(1) T f ( v 1 + v 2 ) − ∑ j 1 , … , j n ∈ { 1 , 2 } f ( v j 1 1 , … , v j n n ) , t + s ≥ min { T ′ ( ϕ ( v 1 ) , t ) , T ′ ( ϕ ( v 2 ) , s ) } ,

for all v 1 , v 2 ∈ V n , and t , s > 0 . If the hypothesis

( H ) : ϕ ( 2 v ) = α ϕ ( v ) for a r e a l n u m b e r α with 0 < ∣ α ∣ < 2 n

holds, then there exists a unique multi-additive mapping A : V n ⟶ W fulfills either

(2) T ( f ( v ) − A ( v ) , t ) ≥ M v , 2 n − α 2 n + 1 t

(3) T ( f ( v ) − A ( v ) , t ) ≥ N v , α − 2 n 2 α t ,

for all v ∈ V n and t > 0 , where

(4) M ( v , t ) ≔ T ′ ( ϕ ( v ) , 2 n − 1 t ) and N ( v , t ) ≔ T ′ ϕ ( v ) , t 2 α .

Proof

Substituting v 1 = v 2 and s = t in (1), we have

(5) T ( f ( 2 v 1 ) − 2 n f ( v 1 ) , 2 t ) ≥ min { T ′ ( ϕ ( v 1 ) , t ) , T ′ ( ϕ ( v 1 ) , t ) } = T ′ ( ϕ ( v 1 ) , t ) ,

for all v 1 ∈ V n and t > 0 . For the rest, we set v 1 by v unless otherwise stated explicitly. It follows from relation (5) that

(6) T f ( 2 v ) 2 n − f ( v ) , t ≥ M ( v , t ) ,

for all v ∈ V n and t > 0 , where M ( v , t ) is defined in (4). Switching v by 2 m v in (6) and using this fact that

(7) M ( 2 v , t ) = M v , t α ,

we obtain

T f ( 2 m + 1 v ) 2 ( m + 1 ) n − f ( 2 m v ) 2 m n , α m 2 m n t = T f ( 2 m + 1 v ) 2 n − f ( 2 m v ) , α m t ≥ M ( 2 m v , α m t ) = M ( v , t ) ,

for all v ∈ V n , t > 0 , and m ≥ 0 . Moreover, for each m > l > 0 , we have

(8) T f ( 2 m v ) 2 m n − f ( 2 l v ) 2 l n , ∑ j = l m − 1 α j 2 j n t = T ∑ j = l m − 1 f ( 2 m + 1 v ) 2 ( m + 1 ) n − f ( 2 m v ) 2 m n , ∑ j = l m − 1 α j 2 j n t ≥ min ⋃ j = l m − 1 T f ( 2 m + 1 v ) 2 ( m + 1 ) n − f ( 2 m v ) 2 m n , α j 2 j n t ≥ M ( v , t ) .

Take ε , δ > 0 . It is known that lim t → ∞ M ( v , t ) = 1 , and so there is some t 0 > 0 such that M ( v , t 0 ) > 1 − ε . On the other hand, ∑ j = 0 m − 1 α j 2 j n t 0 < ∞ , and hence there is some n 0 ∈ N such that ∑ j = l m − 1 α j 2 j n t 0 < δ for all m > l ≥ n 0 . It now concludes that

T f ( 2 m v ) 2 m n − f ( 2 l v ) 2 l n , δ ≥ T f ( 2 m v ) 2 m n − f ( 2 l v ) 2 l n , ∑ j = l m − 1 α j 2 j n t 0 ≥ M ( v , t 0 ) ≥ 1 − ε .

The relation above shows that the sequence f ( 2 m v ) 2 m n is Cauchy in ( W , T ) . Due to the completeness of ( W , T ) , the sequence f ( 2 m v ) 2 m n converges to some A ( v ) ∈ W . In other words, there exists a mapping A : V n ⟶ W such that

A ( v ) = lim m → ∞ T − f ( 2 m v ) 2 m n .

Substituting l = 0 in (8), we find

T f ( 2 m v ) 2 m n − f ( v ) , ∑ j = 0 m − 1 α j 2 j n t ≥ M ( v , t ) ,

for all v ∈ V n and all t > 0 . The last inequality implies that

(9) T f ( 2 m v ) 2 m n − f ( v ) , t ≥ M v , t ∑ j = 0 m − 1 α j 2 j n ,

for all v ∈ V n and all t > 0 . We claim that the mapping A is multi-additive. For each v 1 , v 2 ∈ V n and t > 0 , we obtain

T A ( v 1 + v 2 ) − ∑ j 1 , … , j n ∈ { 1 , 2 } A ( v j 1 1 , … , v j n n ) , t ≥ min { T A ( v 1 + v 2 ) − f ( 2 m ( v 1 + v 2 ) ) 2 m n , t 2 n + 1 + 1 , ⋃ j 1 , … , j n ∈ { 1 , 2 } T f ( 2 m v j 1 1 , … , 2 m v j n n ) 2 m n − A ( 2 m v j 1 1 , … , 2 m v j n n ) , t 2 n + 1 + 1 , ⋃ j 1 , … , j n ∈ { 1 , 2 } T f ( 2 m ( v 1 + v 2 ) ) 2 m n − f ( 2 m v j 1 1 , … , 2 m v j n n ) 2 m n , t 2 n + 1 + 1 } .

The first term and the first union on the right-hand side of the above inequality tend to 1 as m → ∞ and the second union, by (1) is greater than or equal to

min T ′ ϕ ( 2 m v 1 ) , 2 m n 2 n + 2 + 2 t , T ′ ϕ ( 2 m v 2 ) , 2 m n 2 n + 2 + 2 t = min T ′ ϕ ( v 1 ) , 2 m n ( 2 n + 2 + 2 ) α m t , T ′ ϕ ( v 2 ) , 2 m n ( 2 n + 2 + 2 ) α m t ,

which goes to 1 when m → ∞ . Hence,

T A ( v 1 + v 2 ) − ∑ j 1 , … , j n ∈ { 1 , 2 } A ( v j 1 1 , … , v j n n ) , t = 1 ,

for all v 1 , v 2 ∈ V n and t > 0 . This means that A is multi-additive by Theorem 1. Here we approximate the difference between f and A in a fuzzy sense. By inequality (9), for each v ∈ V n and t > 0 , we deduce that

T ( A ( v ) − f ( v ) , t ) ≥ min T A ( v ) − f ( 2 m v ) 2 m n , t 2 , T f ( 2 m v ) 2 m n − f ( v ) , t 2 ≥ M v , t 2 ∑ j = 0 ∞ α j 2 j n = M v , 2 n − α 2 n + 1 t ,

and so inequality (2) is obtained. For the uniqueness of A , assume that A is another additive mapping from V n into W , which fulfills inequality (2). For each v ∈ V n and t > 0 , we obtain

(10) T ( A ( v ) − A ( v ) , t ) ≥ min T A ( v ) − f ( v ) , t 2 , T A ( v ) − f ( v ) , t 2 ≥ M v , 2 n − α 2 n + 1 t .

Since A and A are multi-additive, we have A ( 2 v ) = 2 n A ( v ) and A ( 2 v ) = 2 n A ( v ) . Thus, A ( v ) = 1 2 m n A ( 2 m v ) and A ( v ) = 1 2 m n A ( 2 m v ) , and so by (10), we reach to

T ( A ( v ) − A ( v ) , t ) = T ( A ( 2 m v ) − A ( 2 m v ) , 2 m n t ) ≥ M v , 2 n − α 2 n + 1 2 n α m t ,

for all v ∈ V n , t > 0 , and m ∈ N . By assumption and property (T5), we find lim m → ∞ 2 n α m = + ∞ . Therefore, the right-hand side of the above inequality tends to 1 as m goes to infinity. This shows that that A ( v ) = A ( v ) for all v ∈ V n .

To obtain the other inequality, we note that the assumption (H) implies that ϕ v 2 = 1 α ϕ ( v ) in which ∣ α ∣ > 2 n . It follows from (5) that

(11) T f ( v ) − 2 n f v 2 , t ≥ N ( v , t ) ,

for all v ∈ V n and all t > 0 , where N ( v , t ) is defined in (4). Switching v by 2 m v in (11) and applying the property N v 2 , t = N ( v , α t ) , we obtain

T 2 m n f v 2 m − 2 ( m + 1 ) n f v 2 m + 1 , 2 m n α m t ≥ N ( v , t ) ,

for all v ∈ V n , t > 0 , and m ≥ 0 . In addition, for each m > l > 0 , we obtain

(12) T 2 l n f v 2 l − 2 m n f v 2 m , ∑ j = l m − 1 2 j n α j t ≥ N ( v , t ) .

Let ε , δ > 0 . Since lim t → ∞ N ( v , t ) = 1 , and so there is some t 0 > 0 such that N ( v , t 0 ) > 1 − ε . On the other hand ∑ j = 0 m − 1 2 j n α j t 0 < ∞ , and so for some n 0 ∈ N , we obtain ∑ j = l m − 1 2 j n α j t 0 < δ for all m > l ≥ n 0 . These discussions necessitate that

T 2 l n f v 2 l − 2 m n f v 2 m , δ ≥ N ( v , t 0 ) ≥ 1 − ε .

Therefore, the sequence 2 m n f v 2 m is Cauchy in ( W , T ) . By assumptions, this sequence converges to a mapping A : V n ⟶ W such that

A ( v ) = lim m → ∞ T − 2 m n f v 2 m .

Substituting l = 0 in (12), we find

T f ( v ) − 2 m n f v 2 m , ∑ j = 0 m − 1 2 j n α j t ≥ N ( v , t ) ,

for all v ∈ V n and all t > 0 . It now follows from the last inequality that

(13) T f ( v ) − 2 m n f v 2 m , t ≥ N v , t ∑ j = 0 m − 1 2 j n α j ,

or all v ∈ V n and all t > 0 . Similar to the above, one can obtain inequality (3) by (13). The proof of being multi-additive A and its uniqueness can be a standard fashion taken from the previous part. This completes the proof.□

Example 4

Let V be a real normed space, ( W , T ) be a fuzzy Banach space, and ( Y , T ′ ) be a fuzzy normed space, where T and T ′ are fuzzy norm as considered in Example 1. Suppose that f : V n ⟶ W is a mapping that satisfying

T f ( v 1 + v 2 ) − ∑ j 1 , … , j n ∈ { 1 , 2 } f ( v j 1 1 , … , v j n n ) , t + s ≥ min T ′ ∑ j = 1 n ‖ v 1 j ‖ r y 0 , t , T ′ ∑ j = 1 n ‖ v 2 j ‖ r y 0 , s ,

for all v 1 , v 2 ∈ V n and all t , s > 0 , where y 0 is a fixed vector in Y . If 0 < r ≠ n , then by Theorem 3, there exists a unique multi-additive mapping A : V n ⟶ W such that

(14) ‖ f ( v ) − A ( v ) ‖ ≤ 4 ‖ y 0 ‖ ∑ j = 1 n ‖ v 1 j ‖ r ∣ 2 n − 2 r ∣ ,

for all v ≔ v 1 ∈ V n .

It is easily seen that if in the definition of multi-additive mapping, we consider a large number for n , then the right side of inequality (14) goes to 1 and fuzzy difference closes to zero. In the following example, we prove the Hyers’ stability of multi-additive mappings, for which the argument is taken from the proof of Theorem 2.3 from [21].

Example 5

Given δ > 0 . Let V be a real normed space, ( W , T ) be a fuzzy Banach space, and ( Y , T ′ ) be a fuzzy normed space. Suppose that f : V n ⟶ W is a mapping that satisfies

T f ( v 1 + v 2 ) − ∑ j 1 , … , j n ∈ { 1 , 2 } f ( v j 1 1 , … , v j n n ) , t + s ≥ min { T ′ ( δ y 0 , t ) , T ′ ( δ y 0 , s ) } ,

for all v 1 , v 2 ∈ V n and all t , s > 0 , where y 0 is a fixed vector in Y . By Theorem 3, there exists a unique multi-additive mapping A : V n ⟶ W such that

T ( f ( v ) − A ( v ) , t ) ≥ T ′ y 0 , 2 n − 1 4 δ t ,

for all v ∈ V n and all t > 0 . Here we consider the following norm on V n

‖ ( u 1 , … , u n ) ‖ 1 = ∑ j = 1 n ‖ u j ‖ , ( u 1 , … , u n ∈ V ) .

We shall show that if f is continuous at a point, then A is continuous on V n . Consider an arbitrary and fixed-point v 0 = ( v 01 … , v 0 n ) in V n . Moreover, for each j ∈ { 1 , … , n } , define the mapping f j * : V ⟶ W through f j * ( v ) = f ( v 01 , … , v 0 , j − 1 , v , v 0 , j + 1 , … , v 0 n ) . It is easily verified that f is continuous at v 0 if and only if f j * is continuous at v 0 j for all j ∈ { 1 , … , n } . In fact, f is continuous on V n if and only if f j * is continuous on V for all j ∈ { 1 , … , n } (the same for A and the corresponding A j * ). Since A is multi-additive, it is enough to show that A is continuous at 0 = ( 0 , … , 0 ) . Now, suppose the assertion is false, that A is not continuous at 0 (as v 0 in the consideration above). Hence, there is a k ∈ { 1 , … , n } such that A k * is not continuous at 0. Let λ = 2 n − 1 12 δ . Then, there exists a sequence { v m k } m ⊆ V such that lim m → ∞ v m k = 0 but T − lim m → ∞ A k * ( v m k ) ≠ 0 . Passing to a subsequence if necessary, we may assume that lim m → ∞ v m k = 0 and there are t 0 and ε > 0 such that T ( A k * ( v m k ) , t 0 ) < 1 − ε for all m . It is known that lim m → ∞ T ′ ( y 0 , λ t ) = 1 , there is t 1 such that T ′ ( y 0 , λ t 1 ) ≥ 1 − ε . Hence, there is a positive integer p such that t 1 p < t 0 . Therefore,

(15) T ( A k * ( p v m k ) , t 1 ) = T A k * ( v m k ) , t 1 p ≤ T ( A k * ( v m k ) , t 0 ) < 1 − ε .

On the other hand,

T ( A k * ( p v m k ) , t 1 ) ≥ min { T A k * ( p v m k ) − f k * ( p v m k ) , t 1 3 , T f k * ( p v m k ) − f k * ( 0 ) , t 1 3 , T f k * ( 0 ) , t 1 3 } .

The first and the third terms of the right-hand side are greater than or equal to T ′ ( y 0 , λ t 1 ) and the second tends to 1 as m → ∞ . Therefore, for sufficiently large m ,

T ( A k * ( p v m k ) , t 1 ) ≥ T ′ ( y 0 , λ t 1 ) ≥ 1 − ε .

This leads us to a contradiction with (15).

2.2 Stability results: The fixed-point method

In this subsection, we present the Hyers-Ulam-Rassias stability of multi-additive mappings by a fixed-point method.

Let X be a set. A function d : X × X ⟶ [ 0 , ∞ ] is called a generalized metric on X if and only if d satisfies the following statements:

d ( x , y ) = 0 if and only if x = y for all x , y ∈ X ;
d ( x , y ) = d ( y , x ) for all x , y ∈ X ;
d ( x , z ) ≤ d ( x , y ) + d ( y , z ) for all x , y , z ∈ X .

The pair ( X , d ) is called a generalized metric space. Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. For a metric space ( X , d ) , the Lipschitz modulus of a mapping ϕ : X ⟶ X is defined through

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

In the next theorem, we bring a fundamental result from [34] in fixed-point theory, which is useful to our purpose in this subsection; we remind that an extension of this result was stated in [35].

Theorem 4

(The fixed-point alternative) Let ( Ω , d ) be a complete generalized metric space and J : Ω → Ω be a mapping with Lipschitz constant 0 < L < 1 . Then, for each element y ∈ Ω , either d ( J n y , J n + 1 y ) = ∞ for all n ≥ 0 , or there exists a natural number n 0 such that

d ( J n y , J n + 1 y ) < ∞ for all n ≥ n 0 ;
the sequence { J n y } is convergent to a fixed point y * of J ;
y * is the unique fixed-point of J in the set
Λ = { y ∈ Ω : d ( J n 0 y , y ) < ∞ } ;
d ( y , y * ) ≤ 1 1 − L d ( y , J y ) for all y ∈ Λ .

The following result is analogous to Theorem 3 in which we find a different approximation for approximately multi-additive mappings in fuzzy Banach spaces.

Theorem 5

Let V be a real linear space, ( W , T ) be a fuzzy Banach space, and ( Y , T ′ ) be a fuzzy normed space. Suppose that f : V n ⟶ W is a mapping and ϕ : V n ⟶ Y is a function satisfying (1). If the hypothesis (H) in Theorem 3 is true, then there exists a unique multi-additive mapping A : V n ⟶ W satisfies either

T ( f ( v ) − A ( v ) , t ) ≥ M v , 2 n − α 2 n t ,

T ( f ( v ) − A ( v ) , t ) ≥ N v , α − 2 n α t ,

for all v ∈ V n and t > 0 , where M ( v , t ) and N ( v , t ) are defined in (4).

Proof

Let us define a complete generalized metric space ( Ω , d ) , where Ω ≔ W V n (the set of all mappings from V n to W ) and

d ( g , h ) ≔ inf { C ∈ [ 0 , ∞ ) : T ( g ( v ) − h ( v ) , C t ) ≥ M ( v , t ) , ∀ v ∈ V n , t > 0 } ,

where, as usual, inf ∅ = + ∞ , for which g , h ∈ Ω , where M ( v , t ) is defined in (4). It is known that ( Ω , d ) is a complete generalized metric space; refer [20, Theorem 2.1] and [36, Theorem 2.6]. Define the mapping J : Ω ⟶ Ω via

J f ( v ) ≔ 1 2 n f ( 2 v ) ,

for all v ∈ V n . We claim that J is a strictly contractive operator with the Lipschitz constant α 2 n . Take g , h ∈ Ω , v ∈ V n , and C ∈ [ 0 , ∞ ] with d ( g , h ) ≤ C . Then, T ( g ( v ) − h ( v ) , C t ) ≥ M ( v , t ) , and so by property (7), we have

T J g ( v ) − J h ( v ) , α 2 n C t = T 1 2 n g ( 2 v ) − 1 2 n h ( 2 v ) , α 2 n C t = T ( g ( 2 v ) − h ( 2 v ) , α C t ) ≥ M ( 2 v , α t ) = M ( v , t )

for all v ∈ V n and t > 0 . Therefore, d ( J g , J h ) ≤ α 2 n C . This shows that d ( J g , J h ) ≤ α 2 n d ( g , h ) , as claimed. Similar to the proof of Theorem 3, we obtain

T f ( 2 v ) 2 n − f ( v ) , t ≥ M ( v , t ) ,

for all v ∈ V n and t > 0 , where M ( v , t ) is defined in (4). Therefore, d ( J f , f ) ≤ 1 . We can now apply Theorem 4 for the space ( Ω , d ) , the operator J , n 0 = 0 and y = f to deduce that the sequence ( J m f ) m ∈ N is convergent in ( Ω , d ) and its limit, A is a fixed-point of J , that is, A ( v ) = 1 2 n A ( 2 v ) . Moreover, we have d ( J m f , A ) → 0 , which implies that

A ( v ) = T − lim l → ∞ f ( 2 m v ) 2 m n ,

for all v ∈ V n . Furthermore, d ( f , A ) ≤ 1 1 − L d ( f , J f ) , which necessitates that

d ( f , A ) ≤ 1 1 − α 2 n = 2 n 2 n − α .

If { δ l } is a decreasing sequence converging to 2 n 2 n − α , then T ( f ( v ) − A ( v ) , δ l t ) ≥ M ( v , t ) , for all v ∈ V n , t > 0 , and l ∈ N . It concludes that

T ( f ( v ) − A ( v ) , t ) ≥ M v , t δ l ,

for all v ∈ V n , t > 0 , and l ∈ N . Since M is left continuous, we deduce that

T ( f ( v ) − A ( v ) , t ) ≥ M v , 2 n 2 n − α ,

for all v ∈ V n , t > 0 , and l ∈ N . The multi-additivity of A can be proven like in Theorem 3. In addition, the uniqueness of A follows from the fact that A is the unique fixed-point of J with the property that there exists K ∈ ( 0 , ∞ ) such that

T ( f ( v ) − A ( v ) , K t ) ≥ M ( v , t ) ,

for all v ∈ V n and t > 0 , and now the proof is complete.□

In view of results in this section, we see that the fixed-point method gives us more exact approximation n comparison to direct method, used in Theorem 3.

Acknowledgment

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

Funding information: The authors state no funding involved.
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 declare that they have no conflict of interest regarding the publication of the research article.
Data availability statement: No data were used to support this study.

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Received: 2024-04-21

Revised: 2024-11-28

Accepted: 2025-06-30

Published Online: 2025-10-09

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

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0159

Keywords for this article

fuzzy normed space; fuzzy Hyers-Ulam-Rassias stability; multi-additive mapping

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