The stability of high ring homomorphisms and derivations on fuzzy Banach algebras

Lin Chen; Xiaolin Luo

doi:10.1515/math-2024-0069

Article Open Access

The stability of high ring homomorphisms and derivations on fuzzy Banach algebras

Lin Chen and Xiaolin Luo

Published/Copyright: October 5, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we focus on exploring the fuzzy version of the Hyers-Ulam-Rassias stability of n -ring homomorphisms and n -ring derivations in the context of fuzzy Banach algebras. Our investigation utilizes the direct method.

Keywords: n-ring homomorphisms; n-ring derivations; Hyers-Ulam-Rassias stability; fuzzy Banach algebras

MSC 2010: 39B72; 39B82; 39A10; 43A22; 46B99

1 Introduction and preliminaries

The questions about the stability of functional equations trace back to Ulam [1], who in the 1940s posed a question about the perturbation of homomorphisms on metric groups. In the context of Banach spaces, Hyers [2] made a significant contribution by providing a concrete solution to Ulam’s problem concerning additive maps. More specifically, let E 1 and E 2 be two Banach spaces and f : E 1 → E 2 be a mapping such that for some ε > 0 ,

‖ f ( x + y ) − f ( x ) − f ( y ) ‖ ⩽ ε

for all x , y ∈ E 1 . Then, there is a unique additive map ϕ : E 1 → E 2 such that

‖ f ( x ) − ϕ ( x ) ‖ ⩽ ε

for all x ∈ E 1 . The method proposed by Hyers, which uses the construction of Cauchy sequences to derive additive mappings, is known as the direct method. Subsequently, Aoki [3] and Rassias [4] developed a generalized version of this approach by relaxing the control conditions. Specifically, assume that a function f : E 1 → E 2 between Banach spaces satisfies the inequality

‖ f ( x + y ) − f ( x ) − f ( y ) ‖ ≤ θ ( ‖ x ‖ p + ‖ y ‖ p )

for some θ ≥ 0 , 0 ≤ p < 1 , and for all x , y ∈ E 1 . Then, there exists a unique additive function ϕ : E 1 → E 2 such that

‖ f ( x ) − ϕ ( x ) ‖ ≤ K ‖ x ‖ p

for all x ∈ E 1 , where K > 0 depends on p as well as on θ . This phenomenon is called Hyers-Ulam-Rassias stability. Gajda [5] showed that the result holds for p > 1 and provided an example demonstrating that the theorem does not hold for p = 1 . This result has been further generalized, leading to a collection of results known as Hyers-Ulam-Rassias stability.

Let A and ℬ be rings, and let f be an additive mapping from A to ℬ . For each positive integer n , f is called an n-ring homomorphism if

f ( a 1 a 2 … a n ) = f ( a 1 ) f ( a 2 ) … f ( a n )

holds for all a 1 , a 2 , … , a n ∈ A ; f is called an n-ring derivation if

f ( a 1 a 2 … a n ) = f ( a 1 ) ∏ i = 2 n a i + a 1 f ( a 2 ) ∏ i = 3 n a i + … + a 1 a 2 … a n − 1 f ( a n )

holds for all a 1 , a 2 , … , a n ∈ A . If n = 2 , then an n -ring homomorphism is the usual ring homomorphism, and an n -ring derivation is the usual ring derivation.

The investigation of approximate ring homomorphisms and approximate ring derivations in Banach algebras was initiated by Badora [6]. Jun and Park [7] were the first to study the stability of derivations in the Banach algebra C n [ 0 , 1 ] . In [8], the author explored the stability of generalized homomorphisms in quasi-Banach algebras. In 2009, Gordji investigated the stability of n -ring homomorphisms and n -ring derivations [9], as well as n -Jordan homomorphisms [10], in Banach algebras. For more comprehensive information on the Hyers-Ulam-Rassias stability of functional equations, we recommend referring to the books [11,12].

The concept of fuzzy norm and fuzzy topological space on a vector space was first introduced by Katsaras [13]. Subsequently, Bag and Samanta [14] presented the notion of a fuzzy norm that corresponds to the fuzzy metric of Kramosil type [15]. They also conducted a study on different properties of fuzzy normed spaces [16]. Let X be a linear space. Recall that the pair ( X , N ) is called a fuzzy normed space if for all u , v ∈ X and s , t ∈ R , the map N : X × R → [ 0 , 1 ] satisfies the following properties:

N ( u , t ) = 0 , for all t ⩽ 0 ;
u = 0 ⇔ N ( u , t ) = 1 for all t > 0 ;
N ( c u , t ) = N ( u , t ∣ c ∣ ) where c ≠ 0 ;
N ( u + v , s + t ) ⩾ min { N ( u , s ) , N ( v , t ) } ;
N ( u , ⋅ ) is a non-decreasing function of R and lim t → ∞ N ( u , t ) = 1 ;
for u ≠ 0 , N ( u , ⋅ ) is continuous on R .

Recently, in [17], the authors introduced C * -algebra-valued fuzzy normed spaces as a generalization of fuzzy normed spaces and studied the fuzzy Hyers-Ulam stability of random integral equations. It is worth mentioning that the above properties of fuzzy normed spaces can also be derived from part one in [17]. Let { u n } be a sequence in fuzzy norm space ( X , N ) . If there exists a u ∈ X such that for all positive number t , we have

(N7) lim n → ∞ N ( u n − u , t ) = 1 ,

then the sequence { u n } is called convergent to u . Write N − lim n → ∞ u n = u . If for any 0 < η < 1 and each t ∈ R , there exist n ′ ∈ N such that

(N8) N ( u n + k − u n , t ) > 1 − η

for all n ⩾ n ′ and k ∈ N , then { u n } is called a Cauchy sequence. Fuzzy normed space ( X , N ) is called a fuzzy norm algebra if for all u , v ∈ X and s 1 , s 2 ∈ R , we have

(N9) N ( u v , s 1 s 2 ) ⩾ N ( u , s 1 ) N ( v , s 2 ) .

A complete (each Cauchy sequence in X converges in X ) fuzzy norm algebra is called a fuzzy Banach algebra.

In 2008, Mirmostafaee and Moslehian [18] introduced three distinct versions of approximately additive functions in fuzzy spaces: the uniform version, the non-uniform version, and the classical form. Similarly to [12, page 2], these approaches offer a reasonable framework for defining the fuzzy version of Hyers-Ulam-Rassias stability of functional equations in fuzzy Banach spaces.

Definition 1.1

Let E 1 be a linear space and ( E 2 , N ) be a fuzzy Banach space. For some p , q ∈ N and for any i ∈ { 1 , … , p } , let g i : E 1 q → E 1 and G : E 2 p × E 1 q → E 2 be functions. Assume that φ , Φ : E 1 q → [ 0 , + ∞ ] are functions satisfying some given conditions. If, for every f : E 1 → E 2 , the following holds:

lim t → + ∞ N ( G ( f ( g 1 ( x 1 , … , x q ) ) , … , f ( g p ( x 1 , … , x q ) ) , x 1 , … , x q ) , t φ ( x 1 , … , x q ) ) = 0 ,

then there exists a function H : E 1 → E 2 such that

G ( H ( g 1 ( x 1 , … , x q ) ) , H ( g p ( x 1 , … , x q ) ) , x 1 , … , x q ) = 0

for all x 1 , … , x q ∈ E 1 , and

lim t → + ∞ N ( f ( x ) − H ( x ) , t Φ ( x 1 , … , x q ) ) = 0

for any x ∈ E . In this case, we say that the functional equation

G ( f ( g 1 ( x 1 , … , x q ) ) , … , f ( g p ( x 1 , … , x q ) ) , x 1 , … , x q ) = 0

has a fuzzy version of Hyers-Ulam-Rassias stability on ( E 1 , E 2 ) .

Many interesting results regarding the fuzzy version of the Hyers-Ulam-Rassias stability of functional equations in fuzzy Banach spaces can be found in [19–24] and references therein. The primary objective of this article is to investigate the fuzzy version of Hyers-Ulam-Rassias stability for n -ring homomorphisms and n -ring derivations on fuzzy Banach algebras with the uniform version.

2 Fuzzy stability of n -ring homomorphisms and n -ring derivations

In this section, we consider the fuzzy version of Hyers-Ulam-Rassias stability of n -ring homomorphisms and n -ring derivations by using the direct method. Before presenting our main results, we will first establish two lemmas that are crucial for the subsequent proofs.

Lemma 2.1

Let X be a Banach space and ( Y , N ) be a fuzzy Banach space. Suppose that f : X → Y is a mapping satisfying

(1) lim t → ∞ N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ p + ‖ y ‖ p ) ) = 1

for some real number θ ≥ 0 , p ∈ [ 0 , 1 ) , uniformly on X × X for all x , y ∈ X and for all real number t > 0 . Then, there exists a unique additive mapping H : X → Y such that

(2) lim t → ∞ N H ( x ) − f ( x ) , 1 1 − 2 p − 1 t θ ‖ x ‖ p = 1 ,

where x ∈ X and t > 0 .

Proof

For any ε > 0 , by equation (1) and (N6), we can find some t 0 > 0 such that

(3) N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ p + ‖ y ‖ p ) ) ≥ 1 − ε

for all t ≥ t 0 . Let x = y in equation (3) and use (N3), we can obtain

(4) N 1 2 f ( 2 x ) − f ( x ) , t θ ‖ x ‖ p ≥ 1 − ε

for all x ∈ X and all t ≥ t 0 . Replacing x by 2 n − 1 x in equation (4) and using (N3), we have

N 1 2 n f ( 2 n x ) − 1 2 n − 1 f ( 2 n − 1 x ) , 2 ( n − 1 ) ( p − 1 ) t θ ‖ x ‖ p ≥ 1 − ε

for all x ∈ X and all t ≥ t 0 . So, by (N3) and (N4), we have

(5) N 1 2 n f ( 2 n x ) − 1 2 m f ( 2 m x ) , ∑ j = m n − 1 2 j ( p − 1 ) t θ ‖ x ‖ p ≥ min N 1 2 n f ( 2 n x ) − 1 2 n − 1 f ( 2 n − 1 x ) , 2 ( n − 1 ) ( p − 1 ) t θ ‖ x ‖ p , … , N 1 2 m + 1 f ( 2 m + 1 x ) − 1 2 m f ( 2 m x ) , 2 m ( p − 1 ) t θ ‖ x ‖ p ≥ 1 − ε

for all n > m ≥ 0 . For a given real number α > 0 , the convergence of series ∑ j = m ∞ 2 j ( p − 1 ) implies that there exists n 0 ∈ N such that

∑ j = m n − 1 2 j ( p − 1 ) < α

for all n ⩾ n 0 . Thus, by (N5), we can see that

(6) N 1 2 n f ( 2 n x ) − 1 2 m f ( 2 m x ) , α ⩾ 1 − ε

for each n ⩾ n 0 . Hence, { 1 2 n f ( 2 n x ) } is a Cauchy sequence in fuzzy Banach space ( Y , N ) . Since ( Y , N ) is complete, this Cauchy sequence converges to some point H ( x ) ∈ Y . So, we can define H : X → Y by

H ( x ) ≔ N − lim n → ∞ 1 2 n f ( 2 n x ) .

Namely, for all x ∈ X and each t > 0 , we obtain

(7) lim n → ∞ N ( H ( x ) − 1 2 n f ( 2 n x ) , t ) = 1 .

Next, we show that H is an additive mapping. By replacing x with 2 n x , y with 2 n y and t with t 0 in equation (3), we conclude

N 1 2 n f ( 2 n ( x + y ) ) − 1 2 n f ( 2 n x ) − 1 2 n f ( 2 n y ) , 2 n ( p − 1 ) t 0 θ ( ‖ x ‖ p + ‖ y ‖ p ) ⩾ 1 − ε .

Let t > 0 and 0 < ε < 1 be fixed. Since lim n → ∞ 2 n ( p − 1 ) t 0 θ ( ‖ x ‖ p + ‖ y ‖ p ) = 0 , there exists an n 1 ∈ N + such that 2 n ( p − 1 ) t 0 θ ( ‖ x ‖ p + ‖ y ‖ p ) ⩽ t 4 for all n ⩾ n 1 . So, by (N5), we obtain

(8) N 1 2 n f ( 2 n ( x + y ) ) − 1 2 n f ( 2 n x ) − 1 2 n f ( 2 n y ) , t 4 ⩾ 1 − ε

for all x , y ∈ X and n ⩾ n 1 . Thus, for each n ⩾ n 1 , by (N4), we have

N ( H ( x + y ) − H ( x ) − H ( y ) , t ) ⩾ min N H ( x + y ) − 1 2 n f ( 2 n ( x + y ) ) , t 4 , N 1 2 n f ( 2 n x ) − H ( x ) , t 4 , N 1 2 n f ( 2 n y ) − H ( y ) , t 4 , N 1 2 n f ( 2 n ( x + y ) ) − 1 2 n f ( 2 n x ) − 1 2 n f ( 2 n y ) , t 4 .

It follows from equations (7) and (8) that

N ( H ( x + y ) − H ( x ) − H ( y ) , t ) ⩾ 1 − ε .

Hence,

N ( H ( x + y ) − H ( x ) − H ( y ) , t ) = 1

for all t > 0 . By (N2), we obtain

H ( x + y ) − H ( x ) − H ( y ) = 0

for all x ∈ X . This shows that H is additive. Next, our task is to show equation (2) holds. Put m = 0 in equation (5), we conclude

(9) N 1 2 n f ( 2 n x ) − f ( x ) , ∑ j = 0 n − 1 2 j ( p − 1 ) t θ ‖ x ‖ p ⩾ 1 − ε

for all positive integers n . Let u > 0 , by (N4), we have

N H ( x ) − f ( x ) , u + ∑ j = 0 n − 1 2 j ( p − 1 ) t θ ‖ x ‖ p ⩾ min N ( H ( x ) − 1 2 n f ( 2 n x ) , u ) , N 1 2 n f ( 2 n x ) − f ( x ) , ∑ j = 0 n − 1 2 j ( p − 1 ) t θ ‖ x ‖ p .

By equations (7) and (9), we arrive that

N H ( x ) − f ( x ) , u + ∑ j = 0 n − 1 2 j ( p − 1 ) t θ ‖ x ‖ p ⩾ 1 − ε

for a sufficiently large positive integer n . Since the series ∑ j = 0 ∞ 2 j ( p − 1 ) converges to 1 1 − 2 p − 1 , by (N6), the continuity of function N ( H ( x ) − f ( x ) , ⋅ ) , we can see

N H ( x ) − f ( x ) , u + 1 1 − 2 p − 1 t θ ‖ x ‖ p ⩾ 1 − ε .

Let u → 0 , we deduce that

N H ( x ) − f ( x ) , 1 1 − 2 p − 1 t θ ‖ x ‖ p ⩾ 1 − ε .

Hence, equation (2) is true. Finally, we prove the additive mapping H is unique. Assume T : X → Y is another additive mapping satisfying equation (2). Then, T ( x ) ≔ N − lim n → ∞ 1 2 n f ( 2 n x ) . For all x ∈ X and t > 0 , we see

N ( H ( x ) − T ( x ) , t ) ⩾ min N H ( x ) − 1 2 n f ( 2 n x ) , t 2 , N 1 2 n f ( 2 n x ) − T ( x ) , t 2 → 1 ( n → ∞ ) .

Therefore, H ( x ) = T ( x ) for all x ∈ X . The proof is completed.□

By utilizing a similar approach as the proof of Lemma 2.1, we can establish the following result for the case when p > 1 .

Lemma 2.2

Let X be a Banach space and ( Y , N ) be a fuzzy Banach space. Let θ be nonnegative real numbers and p > 1 . Suppose that f : X → Y is a mapping such that

lim t → ∞ N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ p + ‖ y ‖ p ) ) = 1

uniformly on X × X for all x , y ∈ X and all real number t > 0 . Then, there exists a unique additive mapping H : X → Y such that

lim t → ∞ N ( H ( x ) − f ( x ) , 1 2 p − 1 − 1 t θ ‖ x ‖ p ) = 1 ,

where x ∈ X and t > 0 .

The first main result in this section is given below, which is analogous to Theorem 2.1 in [9].

Theorem 2.3

Let X be a Banach algebra and ( Y , N ) be a fuzzy Banach algebra. Then, the n-ring homomorphism from X to Y has a fuzzy version of Hyers-Ulam-Rassias stability. Specifically, let θ be non-negative real number, p > 0 , p ≠ 1 . Suppose that f : X → Y is a mapping such that

lim t → ∞ N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ p + ‖ y ‖ p ) ) = 1

uniformly on X × X for all x , y ∈ X , t > 0 , and

(10) lim t → ∞ N f ∏ i = 1 n x i − ∏ i = 1 n f ( x i ) , t θ ∏ i = 1 n ‖ x i ‖ p = 1

uniformly on X × … × X ︸ n for all x 1 , x 2 , … , x n ∈ X and t > 0 , then there exists a unique n-ring homomorphism H : X → Y such that

(11) lim t → ∞ N H ( x ) − f ( x ) , 1 ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p = 1

for all x ∈ X and t > 0 .

Proof

By Lemmas 2.1 and 2.2, we can show that there exists a unique additive mapping H : X → Y such that

lim t → ∞ N H ( x ) − f ( x ) , 1 ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p = 1

for all x ∈ X and t > 0 . Hence, according to the property of fuzzy norm, for any ε > 0 , we can find some t 0 > 0 such that

(12) N H ( x ) − f ( x ) , 1 ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p ≥ 1 − ε

for all t ≥ t 0 . Let s = ∣ 1 − p ∣ 1 − p . Obviously, it has that s = 1 as 0 ≤ p < 1 and s = − 1 as p > 1 . Putting x = k s x ( k ∈ N and k ≠ 0 ) in equation (12), we have

N H ( k s x ) − f ( k s x ) , k s p ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p ≥ 1 − ε .

Whence, we can deduce that

N 1 k s H ( k s x ) − 1 k s f ( k s x ) , k s ( p − 1 ) ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p ≥ 1 − ε

for all x ∈ X and all t > 0 . As H is an additive mapping, we have

N H ( x ) − 1 k s f ( k s x ) , k s ( p − 1 ) ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p ≥ 1 − ε .

Therefore, we can define H by

(13) H ( x ) ≔ N − lim k → ∞ 1 k s f ( k s x ) .

Now, we only need to show that H is n -ring homomorphism. For x 1 , x 2 , … , x n ∈ X , replacing x with ∏ i = 1 n x i in equation (13), we obtain

(14) H ∏ i = 1 n x i = N − lim k → ∞ 1 k s f k s ∏ i = 1 n x i ,

where k ∈ N + . For any ε > 0 , by equation (10) and (N6), we have

(15) N f ∏ i = 1 n x i − ∏ i = 1 n f ( x i ) , t θ ∏ i = 1 n ‖ x i ‖ p ⩾ 1 − ε

for all t ⩾ t 0 ′ for some real number t 0 ′ > 0 . By replacing x 1 with k s x 1 in equation (15) and using (N3), we conclude

(16) N 1 k s f ( k s x 1 ) ∏ i = 2 n x i − 1 k s f ( k s x 1 ) ∏ i = 2 n f ( x i ) , k s ( p − 1 ) t θ ∏ i = 1 n ‖ x i ‖ p ⩾ 1 − ε

for all x 1 , x 2 , … , x n ∈ X and all t ⩾ t 0 ′ . Since fuzzy norm algebra is with continuous product (it can refer to the article [24]), by equation (13), we have

(17) H ( x 1 ) ∏ i = 2 n f ( x i ) = N − lim k → ∞ 1 k s f ( k s x 1 ) ∏ i = 2 n f ( x i ) .

Hence, let v > 0 , by equations (16) and (17),

N 1 k s f ( k s x 1 ) ∏ i = 2 n x i − H ( x 1 ) ∏ i = 2 n f ( x i ) , k s ( p − 1 ) t θ ∏ i = 1 n ‖ x i ‖ p + v ⩾ min N 1 k s f ( ( k s x 1 ) ∏ i = 2 n x i ) − 1 k s f ( k s x 1 ) ∏ i = 2 n f ( x i ) , k s ( p − 1 ) t θ ∏ i = 1 n ‖ x i ‖ p , N 1 k s f ( k s x 1 ) ∏ i = 2 n f ( x i ) − H ( x 1 ) ∏ i = 2 n f ( x i ) , v ⩾ 1 − ε .

Thus,

(18) H ( x 1 ) ∏ i = 2 n f ( x i ) = N − lim k → ∞ 1 k s f ( k s x 1 ) ∏ i = 2 n x i .

Combining equations (14) and (18), we obtain

(19) H ∏ i = 1 n x i = H ( x 1 ) ∏ i = 2 n f ( x i )

for all x 1 , x 2 , … , x n ∈ X . Since H is an additive mapping, when 0 ≤ p < 1 , we can obtain

(20) H k ∏ i = 1 n x i = k H ∏ i = 1 n x i .

If p > 1 , we put ∏ i = 1 n x i = 1 k ∏ i = 1 n x i in equation (20), then we have

H ∏ i = 1 n x i = k H 1 k ∏ i = 1 n x i .

Hence, when p > 1 , we conclude

(21) 1 k H ∏ i = 1 n x i = H 1 k ∏ i = 1 n x i .

Thereby, combining equations (20) and (21), we see that

(22) H k s ∏ i = 1 n x i = k s H ∏ i = 1 n x i .

By equations (19) and (22), we deduce that

H ( x 1 ) f ( k s x 2 ) ∏ i = 3 n f ( x i ) = H x 1 k s x 2 ∏ i = 3 n x i = H k s ∏ i = 1 n x i = k s H ∏ i = 1 n x i

for all x 1 , x 2 , … , x n ∈ X , k ∈ N . So,

H ∏ i = 1 n x i = H ( x 1 ) 1 k s f ( k s x 2 ) ∏ i = 3 n f ( x i ) .

Since fuzzy norm algebra is with continuous product (it can refer to the article [24]), by equation (13), we have

H ∏ i = 1 n x i = N − lim k → ∞ H ( x 1 ) 1 k s f ( k s x 2 ) ∏ i = 3 n f ( x i ) = H ( x 1 ) H ( x 2 ) ∏ i = 3 n f ( x i ) .

Now, proceed in the same way for x 3 , x 4 , … , x n , we can prove that

H ( x 1 x 2 … x n ) = H ( x 1 ) H ( x 2 ) … H ( x n )

for all x 1 , x 2 , … , x n ∈ X . Finally, the uniqueness property of H follows from the proof of additivity in Lemma 2.1 and 2.2. The proof is completed.□

The next theorem gives the stability of n -ring derivations on fuzzy Banach algebras, which is analogous to Theorem 2.3 in [9].

Theorem 2.4

Let ( X , N ) be a fuzzy Banach algebra. Then, the n-ring derivation from X to X has a fuzzy version of Hyers-Ulam-Rassias stability. Specifically, let θ be nonnegative real numbers, p > 0 , p ≠ 1 . Suppose that f : X → X is a function such that

lim t → ∞ N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ p + ‖ y ‖ p ) ) = 1

uniformly on X × X for all x , y ∈ X , t > 0 , and

(23) lim t → ∞ N f ∏ i = 1 n x i − f ( x 1 ) ∏ i = 2 n x i − x 1 f ( x 2 ) ∏ i = 3 n x i − … − ∏ i = 1 n − 2 x i f ( x n − 1 ) x n − ∏ i = 1 n − 1 x i f ( x n ) , t θ ∏ i = 1 n ‖ x i ‖ p = 1

uniformly on X × … × X ︸ n for all x 1 , x 2 , … , x n ∈ X , t > 0 , then there exists a unique n-ring derivation D : X → X such that

(24) lim t → ∞ N D ( x ) − f ( x ) , 1 ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p = 1 ,

where x ∈ X and t > 0 .

Proof

By Lemmas 2.1 and 2.2, we can show that there exists a unique additive mapping D : X → X such that

lim t → ∞ N D ( x ) − f ( x ) , 1 ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p = 1 ,

where x ∈ X and t > 0 . So, for a given ε > 0 , there exists a t 0 > 0 satisfying

(25) N D ( x ) − f ( x ) , 1 ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p ⩾ 1 − ε

for all t ⩾ t 0 . By equation (23), there exists a t 0 ′ > 0 such that

(26) N f ∏ i = 1 n x i − f ( x 1 ) ∏ i = 2 n x i − … − ∏ i = 1 n − 1 x i f ( x n ) , t θ ∏ i = 1 n ‖ x i ‖ p ⩾ 1 − ε

for all t ⩾ t 0 ′ and x 1 , x 2 , … , x n ∈ X . Now, we only need to prove that the mapping D is n -ring derivation. Let ω ∈ N , ω ≠ 0 and s = ∣ 1 − p ∣ 1 − p . Replacing x with ω s x in equation (25) and using (N3), we have

N 1 ω s D ( ω s x ) − 1 ω s f ( ω s x ) , ω s ( p − 1 ) ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p ⩾ 1 − ε

for all x ∈ X . Since s = − 1 as p > 1 and s = 1 when 0 ≤ p < 1 , lim ω → ∞ ω s ( p − 1 ) ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p = 0 . Therefore, for a given β > 0 , there exists an ω 0 > 0 such that

ω s ( p − 1 ) ∣ 1 − 2 p − 1 ∣ t θ ‖ x ‖ p < β

for all ω ⩾ ω 0 . Hence, by additivity of D and (N5),

(27) N ( D ( x ) − 1 ω s f ( ω s x ) , ω s β ) ≥ N 1 ω s D ( ω s x ) − 1 ω s f ( ω s x ) , β ⩾ 1 − ε .

Thus,

(28) D ( x ) = N − lim ω → ∞ 1 ω s f ( ω s x ) .

Replacing x with ω s x in equation (27), we have

(29) N ( D ( x ) − 1 ω 2 s f ( ω 2 s x ) , ω s β ) = N 1 ω s D ( ω s x ) − 1 ω 2 s f ( ω 2 s x ) , β ⩾ 1 − ε .

By additivity of D , we see

D ( x ) = N − lim ω → ∞ ω − 2 s f ( ω 2 s x ) .

Replacing x with ω s x in equation (29), we can obtain

D ( x ) = N − lim ω → ∞ ω − 3 s f ( ω 3 s x ) .

Similarly, by repeating the same process, we can show that

(30) D ( x ) = N − lim ω → ∞ ω − n s f ( ω n s x ) .

Repalcing x i with ω s x i in equation (26) and using (N3),

N ω − n s f ω n s ∏ i = 1 n x i − ω − s f ( ω s x 1 ) ∏ i = 2 n x i − ω − s x 1 f ( ω s x 2 ) ∏ i = 3 n x i − ω − s ∏ i = 1 n − 2 x i f ( ω s x n − 1 ) x n − … − ω − s ∏ i = 1 n − 1 x i f ( ω s x n ) , ω n s ( p − 1 ) t θ ∏ i = 1 n ‖ x i ‖ p ⩾ 1 − ε .

It follows from lim ω → ∞ ω n s ( p − 1 ) t θ ∏ i = 1 n ‖ x i ‖ p = 0 that for some ω 1 > 0 , we have

ω n s ( p − 1 ) t θ ∏ i = 1 n ‖ x i ‖ p < t n + 2

for all ω ⩾ ω 1 . Hence, for all x 1 , x 2 , … , x n ∈ X , ω ⩾ ω 1 and t > t 0 ′ , we have

(31) N ω − n s f ( ω n s ∏ i = 1 n x i ) − ω − s f ( ω s x 1 ) ∏ i = 2 n x i − … − ω − s ∏ i = 1 n − 1 x i f ( ω s x n ) , t n + 2 ⩾ 1 − ε .

Using (N4), we see

N D ∏ i = 1 n x i − D ( x 1 ) ∏ i = 2 n x i − x 1 D ( x 2 ) ∏ i = 3 n x i − … − ∏ i = 1 n − 1 x i D ( x n ) , t ⩾ min N D ∏ i = 1 n x i − ω − n s f ω n s ∏ i = 1 n x i , t n + 2 , N ω − s f ( ω s x 1 ) ∏ i = 2 n x i − D ( x 1 ) ∏ i = 2 n x i , t n + 2 , … , N ω − s ∏ i = 1 n − 1 x i f ( ω s x n ) − ∏ i = 1 n − 1 x i D ( x n ) , t n + 2 , N ω − n s f ω n s ∏ i = 1 n x i − ω − s f ( ω s x 1 ) ∏ i = 2 n x i − … − ω − s ∏ i = 1 n − 1 x i f ( ω s x n ) , t n + 2 .

Combining equations (28), (30), and (31), we obtain

N D ∏ i = 1 n x i − D ( x 1 ) ∏ i = 2 n x i − x 1 D ( x 2 ) ∏ i = 3 n x i − … − ∏ i = 1 n − 1 x i D ( x n ) , t ⩾ 1 − ε

for a sufficiently large positive integer ω > max { ω 0 , ω 1 } and some t > t 0 ′ . Therefore, we conclude

D ∏ i = 1 n x i − D ( x 1 ) ∏ i = 2 n x i − x 1 D ( x 2 ) ∏ i = 3 n x i − … − ∏ i = 1 n − 1 x i D ( x n ) = 0

for all x 1 , x 2 , … , x n ∈ X . This completes the proof.□

The following example demonstrates that Theorem 2.4 does not necessarily hold for p = 1 . It is essentially a modified version of Example 3.5 in [25].

Example 2.5

Let X be a Banach algebra, x 0 ∈ X , α , β are real numbers such that ∣ α ∣ ≥ 1 − ∏ i = 1 n ‖ x i ‖ and ∣ β ∣ ≤ ‖ x i ‖ ( i = 1 , 2 , … , n ) , where x i ∈ X . Let f = α x + β x 0 ‖ x ‖ for all x ∈ X . Moreover, for each fuzzy norm N on X , for all x , y ∈ X and for all t > 0 , we have

N ( f ( x + y ) − f ( x ) − f ( y ) , t ( ‖ x ‖ + ‖ y ‖ ) ) = N ( β x 0 ‖ x + y ‖ − β x 0 ‖ x ‖ − β x 0 ‖ y ‖ , t ( ‖ x ‖ + ‖ y ‖ ) ) = N β x 0 , t ( ‖ x ‖ + ‖ y ‖ ) ‖ x + y ‖ − ‖ x ‖ − ‖ y ‖ ≥ N ( β x 0 , t ) .

Hence, by item (N5), we can obtain

(32) lim t → ∞ N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ + ‖ y ‖ ) ) = 1 ,

uniformly on X × X . Also,

N f ∏ i = 1 n x i − ∏ i = 1 n f ( x i ) , t ∏ i = 1 n ‖ x i ‖ = N ( 1 − α n ) α ∏ i = 1 n x i + β x 0 ‖ ∏ i = 1 n x i ‖ − β n x 0 n ∏ i = 1 n ‖ x i ‖ − ∑ i = 1 n α x i ( β x 0 ) n − 1 ∏ j = 1 , j ≠ i n ‖ x j ‖ − ∑ i 1 ≠ i 2 n α x i 1 x i 2 ( β x 0 ) n − 2 ∏ j = 1 , j ≠ i 1 , i 2 n ‖ x j ‖ − … − ∑ i 1 ≠ i 2 ≠ … ≠ i n − 1 n α x i 1 x i 2 … x i n − 1 β x 0 ‖ x i n ‖ , t ∏ i = 1 n ‖ x i ‖ ⩾ min N ( 1 − α n ) α ∏ i = 1 n x i , t ∏ i = 1 n ‖ x i ‖ 2 n + 1 , N β x 0 ‖ ∏ i = 1 n x i ‖ , t ∏ i = 1 n ‖ x i ‖ 2 n + 1 , N β n x 0 n ∏ i = 1 n ‖ x i ‖ , t ∏ i = 1 n ‖ x i ‖ 2 n + 1 , N ∑ i = 1 n α x i ( β x 0 ) n − 1 ∏ j = 1 , j ≠ i n ‖ x j ‖ , t ∏ i = 1 n ‖ x i ‖ 2 n + 1 , N ∑ i 1 ≠ i 2 n α x i 1 x i 2 ( β x 0 ) n − 2 ∏ j = 1 , j ≠ i 1 , i 2 n ‖ x j ‖ , t ∏ i = 1 n ‖ x i ‖ 2 n + 1 , … , N ∑ i 1 ≠ i 2 ≠ … ≠ i n − 1 n α x i 1 x i 2 … x i n − 1 β x 0 ‖ x i n ‖ , t ∏ i = 1 n ‖ x i ‖ 2 n + 1 ,

where x 1 , x 2 , … , x n ∈ X and t ∈ R . Apparently, 2 n + 1 ∈ N + , so let a positive integer m = 2 n + 1 . Taking into account the following inequalities:

N ( 1 − α n ) α ∏ i = 1 n x i , t ∏ i = 1 n ‖ x i ‖ m = N α ∏ i = 1 n x i , t ∏ i = 1 n ‖ x i ‖ m ∣ 1 − α n ∣ ⩾ N α ∏ i = 1 n x i , t m , N β x 0 ‖ ∏ i = 1 n x i ‖ , t ∏ i = 1 n ‖ x i ‖ m = N β x 0 , t ∏ i = 1 n ‖ x i ‖ m ‖ ∏ i = 1 n x i ‖ ⩾ N β x 0 , t m , N β n x 0 n ∏ i = 1 n ‖ x i ‖ , t ∏ i = 1 n ‖ x i ‖ m = N β n x 0 n , t m , N α x i ( β x 0 ) n − 1 ∏ j = 1 , j ≠ i n ‖ x j ‖ , t ∏ i = 1 n ‖ x i ‖ m = N α x i β n − 2 x 0 n − 1 , t ‖ x i ‖ m ∣ β ∣ ⩾ N α x i β n − 2 x 0 n − 1 , t m , N α x i 1 x i 2 ( β x 0 ) n − 2 ∏ j = 1 , j ≠ i 1 , i 2 n ‖ x j ‖ , t ∏ i = 1 n ‖ x i ‖ m = N α x i 1 x i 2 β n − 4 x 0 n − 2 , t ‖ x i 1 ‖ ‖ x i 2 ‖ m ∣ β ∣ 2 ⩾ N α x i 1 x i 2 β n − 4 x 0 n − 2 , t m , ⋮ N α x i 1 x i 2 … x i n − 1 β x 0 ‖ x i n ‖ , t ∏ i = 1 n ‖ x i ‖ m = N α x i 1 x i 2 … x i n − 1 x 0 β 2 − n , t ∏ i = 1 n ‖ x i ‖ m ∣ β ∣ n − 1 ‖ x i n ‖ ⩾ N α x i 1 x i 2 … x i n − 1 x 0 β 2 − n , t m ,

it can be easily seen that lim t → ∞ N ( f ( ∏ i = 1 n x i ) − ∏ i = 1 n f ( x i ) , t θ ∏ i = 1 n ‖ x i ‖ ) = 1 uniformly on X × X , whence the conditions of Theorem 2.3 are fulfilled. Next, we suppose that there exists a unique n -ring homomorphism satisfying the conditions of Theorem 2.3. By equation (32), for a given ε > 0 , we can find some t 0 > 0 such that

(33) N ( f ( x + y ) − f ( x ) − f ( y ) , t θ ( ‖ x ‖ + ‖ y ‖ ) ) ≥ 1 − ε

for all x , y ∈ X and all t ≥ t 0 . By using the simple induction on n , we shall show that

(34) N ( f ( 2 n x ) − 2 n f ( x ) , t n 2 n θ ( ‖ x ‖ ) ) ≥ 1 − ε .

Evidently, let x = y in equation (33), we can obtain equation (34) when n = 1 . Now, we suppose that equation (34) holds for some positive integer n . Then,

N ( f ( 2 n + 1 x ) − 2 n + 1 f ( x ) , t ( n + 1 ) 2 n + 1 ( ‖ x ‖ ) ) ≥ min { N ( f ( 2 n + 1 x ) − 2 f ( 2 n x ) , t 2 n + 1 ‖ x ‖ ) , N ( 2 f ( 2 n x ) − 2 n + 1 f ( x ) , t n 2 n + 1 ‖ x ‖ ) } ≥ 1 − ε .

This completes the induction argument. We observe that

lim n → ∞ N ( H ( x ) − f ( x ) , t n ( ‖ x ‖ ) ) ≥ 1 − ε ,

whence

(35) lim n → ∞ N ( H ( x ) − f ( x ) , t n ( ‖ x ‖ ) ) = 1 .

One may regard N ( x , t ) as the truth value of the statement “the norm of x is less than or equal to the real number t .” So equation (35) is a contradiction with the non-fuzzy sense. This means that there is no such H .

3 Conclusion

In this work, we investigate the fuzzy version of Hyers-Ulam-Rassias stability for n -ring homomorphisms and n -ring derivations in fuzzy Banach algebras using a direct method. The concept of the best Ulam constant was first introduced in [26], and determining the best Ulam constant for functional equations is an intriguing problem. However, there are few results in the literature concerning the best fuzzy version of the Ulam constant. Our future research will focus on defining this fuzzy version and determining the best fuzzy Ulam constant for n -ring homomorphisms and n -ring derivations.

Acknowledgements

The authors would like to thank the referee for a very thorough reading of this article and many helpful comments that improved this article.

Funding information: This work was supported by the National Natural Science Foundation of China No. 12061018.
Author contributions: All authors discussed the results and contributed to the final manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-11-02

Revised: 2024-09-01

Accepted: 2024-09-10

Published Online: 2024-10-05

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

Articles in the same Issue

https://doi.org/10.1515/math-2024-0069

Keywords for this article

n-ring homomorphisms; n-ring derivations; Hyers-Ulam-Rassias stability; fuzzy Banach algebras

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