Startseite Poisson C*-algebra derivations in Poisson C*-algebras
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Poisson C*-algebra derivations in Poisson C*-algebras

  • Yongqiao Wang , Choonkil Park EMAIL logo und Yuan Chang EMAIL logo
Veröffentlicht/Copyright: 10. Dezember 2024
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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 57 Heft 1

Abstract

In this study, we introduce the following additive functional equation:

g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y )

for all λ ∈ C , all unitary elements u , v in a unital Poisson C * -algebra P , and all y ∈ P . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the aforementioned additive functional equation in unital Poisson C * -algebras. Furthermore, we apply to study Poisson C * -algebra homomorphisms and Poisson C * -algebra derivations in unital Poisson C * -algebras.

Keywords: Hyers-Ulam stability; fixed point method; additive functional equation; Poisson C *-algebra derivation in Poisson C *-algebra; Poisson C *-algebra homomorphism in Poisson C *-algebra
MSC 2010: 47B47; 17B63; 17B40; 39B72; 46L05; 47H10; 39B52

1 Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. Rassias [5] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1 . Gajda [6] following the same approach as in Rassias [4] gave an affirmative solution to this question for p > 1 . It was shown by Gajda [6], as well as by Rassias and Šemrl [7] that one cannot prove a Rassias’ type theorem when p = 1 . The counterexamples of Gajda [6], as well as of Rassias and Šemrl [7], have stimulated several mathematicians to invent new definitions of approximately additive or approximately linear mappings. A generalization of the Rassias theorem was obtained by Găvruta [8] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors [9–18].

We recall a fundamental result in the fixed point theory.

Theorem 1.1

[19,20] Let ( X , d ) be a complete generalized metric space and let J : X → X be a strictly contractive mapping with Lipschitz constant α < 1 . Then, for each given element x ∈ X , either

d ( J n x , J n + 1 x ) = + ∞

for all nonnegative integers n or there exists a positive integer n 0 such that

  1. d ( J n x , J n + 1 x ) < + ∞ , ∀ n ≥ n 0 ;

  2. the sequence { J n x } converges to a fixed point y * of J ;

  3. y * is the unique fixed point of J in the set Y = { y ∈ X ∣ d ( J n 0 x , y ) < + ∞ } ;

  4. d ( y , y * ) ≤ 1 1 − α d ( y , J y ) for all y ∈ Y .

In 1996, Isac and Rassias [21] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [22,23]).

We first recall the definition of a Poisson algebra, given by Bai et al. [24].

Definition 1.2

[24] Let P be a vector space equipped with two bilinear operations ∘ , { , } : P × P → P . The triple ( P , ∘ , { , } ) is called a Poisson algebra if ( P , ∘ ) is a associative algebra and ( P , { , } ) is a Lie algebra that satisfy the compatibility condition

(1.1) { x , y ∘ z } = { x , y } ∘ z + y ∘ { x , z }

for all x , y , z ∈ P .

Equation (1.1) is called the Leibniz rule since the adjoint operators of the Lie algebra are derivations of the commutative associative algebra.

Example 1.3

Let L be a Lie algebra with Lie bracket [ ⋅ , ⋅ ] , which is a noncommutative (or commutative) algebra. Let x ∘ y be x y + y x 2 for all x , y ∈ L . Then, ( L , ∘ ) is a commutative algebra. Let { ⋅ , ⋅ } ≔ [ ⋅ , ⋅ ] . Then, we can easily show that ( L , ∘ , { ⋅ , ⋅ } ) is a Poisson algebra.

From now on, we denote x ∘ y by x y for x , y ∈ P and ( P , ∘ , { , } ) by ( P , { , } ) .

Cho et al. [25] studied homomorphisms and derivations in non-Archimedean Lie C * -algebras by using the fixed point method. Motivated by their results, we define a unital Poisson C * -algebra and Poisson C * -algebra derivations and Poisson C * -algebra homomorphisms in unital Poisson C * -algebras as follows.

Definition 1.4

A commutative unital C * -algebra P is called a unital Poisson C * -algebra if ( P , { , } ) is a Poisson algebra.

Definition 1.5

Let P and Q be unital Poisson C * -algebras.

  1. A C -linear mapping g : P → P is a Poisson C * -algebra derivation if g : P → P satisfies

    g ( x y z ) = g ( x ) y z + x g ( y ) z + x y g ( z ) , g ( { x , y z } ) = { g ( x ) , y z } + { x , g ( y ) z } + { x , y g ( z ) } , g ( x * ) = g ( x ) *

    for all x , y , z ∈ P .

  2. A C -linear mapping h : P → Q is a Poisson C * -algebra homomorphism if h : P → Q satisfies

    h ( x y z ) = h ( x ) h ( y ) h ( z ) , h ( { x , y z } ) = { h ( x ) , h ( y ) h ( z ) } , h ( x * ) = h ( x ) *

    for all x , y , z ∈ P .

Throughout this study, assume that P is a unital Poisson C * -algebra with unit e and unitary group U ( P ) ≔ { u ∈ P ∣ u * u = u u * = e } and Q is a unital Poisson C * -algebra with unit e ′ .

Remark 1

  1. If we put x = y = z = e in Definition 1.5 ( i ) , then g ( e ) = 0 . If we put z = e in Definition 1.5 ( i ) , then

    g ( x y ) = g ( x ) y + x g ( y ) , g ( { x , y } ) = { g ( x ) , y } + { x , g ( y ) }

    for all x , y ∈ P . Thus, g : P → P satisfies the definition of Poisson C * -algebra derivation, given in [26, Definition 1.4].

  2. Conversely, assume that g : P → P satisfies the definition of Poisson C * -algebra derivation, given in [26, Definition 1.4]. Then,

    g ( x y z ) = g ( x y ) z + x y g ( z ) = g ( x ) y z + x g ( y ) z + x y g ( z ) , g ( { x , y z } ) = { g ( x ) , y z } + { x , g ( y z ) } = { g ( x ) , y z } + { x , g ( y ) z } + { x , y g ( z ) }

    for all x , y , z ∈ P .

  3. Assume that h : P → Q satisfies the definition of Poisson C * -algebra homomorphism, given in [26, Definition 1.4], i.e., h : P → Q satisfies

    h ( x y ) = h ( x ) h ( y ) , h ( { x , y } ) = { h ( x ) , h ( y ) } , h ( x * ) = h ( x ) *

    for all x , y ∈ P . Then,

    h ( x y z ) = h ( x y ) h ( z ) = h ( x ) h ( y ) h ( z ) , h ( { x , y z } ) = { h ( x ) , h ( y z ) } = { h ( x ) , h ( y ) h ( z ) }

    for all x , y , z ∈ P .

In general, the converse does not hold. Assume that h : P → Q satisfies h ( e ) = e ′ . If we put z = e in Definition 1.5 ( i i ) , then

h ( x y ) = h ( x y e ) = h ( x ) h ( y ) h ( e ) = h ( x ) h ( y ) e ′ = h ( x ) h ( y ) , h ( { x , y } ) = h ( { x , y e } ) = { h ( x ) , h ( y ) h ( e ) } = { h ( x ) , h ( y ) e ′ } = { h ( x ) , h ( y ) }

for all x , y ∈ P . Thus, h : P → Q satisfies the definition of Poisson C * -algebra homomorphism, given in [26, Definition 1.4].

In this study, we solve the additive functional equation (2.1) and prove the Hyers-Ulam stability of the additive functional equation (2.1) in unital Poisson C * -algebras by using the direct method and by the fixed point method. Furthermore, we investigate Poisson C * -algebra derivations and Poisson C * -algebra homomorphisms in unital Poisson C * -algebras associated with the additive functional equation (2.1).

2 Hyers-Ulam stability of the functional equation (2.1): direct method

In this section, we solve and investigate the functional equation (2.1) in unital C * -algebras

(2.1) g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y ) .

Lemma 2.1

[27, Lemma 1] Assume that a mapping g : P → Q satisfies

(2.2) g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y )

for all λ ∈ C , all u , v ∈ U ( P ) , and all y ∈ P . Then, the mapping g : P → Q is C -linear.

Proof

Substituting λ = 1 and u = − v in (2.2), we obtain g ( 2 y ) = 2 g ( y ) for all y ∈ P and so, we obtain g ( 0 ) = 0 .

Substituting λ = 0 in (2.2), we obtain

g ( v + 2 y ) = g ( v ) + 2 g ( y )

and so,

(2.3) g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y ) = λ g ( u ) + g ( v + 2 y )

for all λ ∈ C , all u , v ∈ U ( P ) , and all y ∈ P . Substituting z = v + 2 y in (2.3), we obtain

(2.4) g ( λ u + z ) = λ g ( u ) + g ( z )

for all λ ∈ C , all u ∈ U ( P ) , and all z ∈ P .

Substituting z = 0 in (2.4), we obtain g ( λ u ) = λ g ( u ) for all λ ∈ C and all u ∈ U ( P ) .

Since each x ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j ( λ j ∈ C , u j ∈ U ( P ) ) , it follows from (2.4) that

g ( λ x + z ) = g ( λ ∑ j = 1 m λ j u j + z ) = g ∑ j = 1 m λ λ j u j + z = g λ λ 1 u 1 + ∑ j = 2 m λ λ j u j + z = λ λ 1 g ( u 1 ) + g ∑ j = 2 m λ λ j u j + z ⋮ = λ λ 1 g ( u 1 ) + λ λ 2 g ( u 2 ) + … + g ( λ λ m u m + z ) = λ λ 1 g ( u 1 ) + λ λ 2 g ( u 2 ) + … + λ λ m g ( u m ) + g ( z ) = λ ( λ 1 g ( u 1 ) + λ 2 g ( u 2 ) + … + λ m g ( u m ) ) + g ( z ) = λ ( λ 1 g ( u 1 ) + λ 2 g ( u 2 ) + … + λ m − 1 g ( u m − 1 ) + g ( λ m u m ) ) + g ( z ) = λ ( λ 1 g ( u 1 ) + λ 2 g ( u 2 ) + … + g ( λ m − 1 u m − 1 + λ m u m ) ) + g ( z ) ⋮ = λ ( λ 1 g ( u 1 ) + g ( λ 2 u 2 + … + λ m u m ) ) + g ( z ) = λ ( g ( λ 1 u 1 + λ 2 u 2 + … + λ m − 1 u m − 1 + λ m u m ) ) + g ( z ) = λ g ( x ) + g ( z )

for all λ ∈ C , and all z ∈ P . So the mapping g : P → Q is C -linear.□

Now, we prove the Hyers-Ulam stability of the additive functional equation (2.1) in unital C * -algebras.

Theorem 2.2

[27, Theorem 2] Let φ : P 3 → [ 0 , ∞ ) be a function such that

(2.5) Φ ( u , v , y ) ≔ ∑ j = 1 ∞ 2 j φ u 2 j , v 2 j , y 2 j < ∞

for all u , v ∈ U ( P ) , and all y ∈ P . Let g : P → Q be a mapping satisfying

(2.6) ∥ g ( λ ( 2 s u ) + 2 s v + 2 y ) − λ g ( 2 s u ) − g ( 2 s v ) − 2 g ( y ) ∥ ≤ φ ( 2 s u , 2 s v , y )

for all λ ∈ C , all u , v ∈ U ( P ) , all s ∈ Z with s ≤ 0 , and all y ∈ P . Then, there exists a unique C -linear mapping G : P → Q such that

(2.7) ‖ g ( y ) − G ( y ) ‖ ≤ 1 2 Φ ( u , u , y ) ,

for all u ∈ U ( P ) and all y ∈ P .

Proof

Letting λ = − 1 and replacing u by u 2 and v by u 2 in (2.6), we obtain

‖ g ( 2 y ) − 2 g ( y ) ‖ ≤ φ u 2 , u 2 , y

and so,

g ( y ) − 2 g y 2 ≤ φ u 2 , u 2 , y 2

for all u ∈ U ( P ) and all y ∈ P .

Similarly, we can show that

2 j g y 2 j − 2 j + 1 g y 2 j + 1 ≤ 2 j φ u 2 j + 1 , u 2 j + 1 , y 2 j + 1

for all u ∈ U ( P ) , all y ∈ P , and each positive integer j . Thus,

(2.8) 2 l g y 2 l − 2 m g y 2 m ≤ ∑ j = l m − 1 2 j g y 2 j − 2 j + 1 g y 2 j + 1 ≤ 1 2 ∑ j = l + 1 m 2 j φ u 2 j , u 2 j , y 2 j

for all nonnegative integers m and l with m > l , all u ∈ U ( P ) and all y ∈ P . It follows from (2.8) that the sequence { 2 k g ( y 2 k ) } is Cauchy for all y ∈ P . Since Q is complete, the sequence { 2 k g ( y 2 k ) } converges. So one can define the mapping G : P → Q by

G ( y ) ≔ lim k → ∞ 2 k g y 2 k

for all y ∈ P . Moreover, substituting l = 0 and passing to the limit m → ∞ in (2.8), we obtain (2.7).

It follows from (2.5) and (2.6) that

∥ G ( λ u + v + 2 y ) − λ G ( u ) − G ( v ) − 2 G ( y ) ∥ = lim n → ∞ 2 n g λ u + v + 2 y 2 n − λ g u 2 n − g v 2 n − 2 g y 2 n ≤ lim n → ∞ 2 n φ u 2 n , v 2 n , y 2 n = 0

for all λ ∈ C , all u , v ∈ U ( P ) , and all y ∈ P . So,

G ( λ u + v + 2 y ) = λ G ( u ) + G ( v ) + 2 G ( y )

for all λ ∈ C , all u , v ∈ U ( P ) , and all y ∈ P . By Lemma 2.1, the mapping G : P → Q is C -linear.

The proof of the uniqueness of the mapping G is similar to the proof of [29, Theorem 2.3].□

Theorem 2.3

[27, Theorem 3] Let φ : P 3 → [ 0 , ∞ ) be a function such that

(2.9) Ψ ( u , v , y ) ≔ ∑ j = 0 ∞ 1 2 j φ ( 2 j u , 2 j v , 2 j y ) < ∞

for all u , v ∈ U ( P ) , and all y ∈ P . Let g : P → Q be a mapping satisfying (2.6) for all s ∈ Z with s ≥ 0 . Then, there exists a unique C -linear mapping G : P → Q such that

(2.10) ‖ g ( y ) − G ( y ) ‖ ≤ 1 2 Ψ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P .

Proof

Letting λ = − 1 and replacing v by u in (2.6), we obtain

‖ g ( 2 y ) − 2 g ( y ) ‖ ≤ φ ( u , u , y )

and so,

g ( y ) − 1 2 g ( 2 y ) ≤ 1 2 φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P .

Similarly, we can show that

1 2 j g ( 2 j y ) − 1 2 j + 1 g ( 2 j + 1 y ) ≤ 1 2 j + 1 φ ( 2 j u , 2 j u , 2 j y )

for all u ∈ U ( P ) , all y ∈ P , and each positive integer j . Thus,

(2.11) ‖ 1 2 l g ( 2 l y ) − 1 2 m g ( 2 m y ) ‖ ≤ ∑ j = l m − 1 1 2 j g ( 2 j y ) − 1 2 j + 1 g ( 2 j + 1 y ) ≤ 1 2 ∑ j = l m − 1 1 2 j φ ( 2 j u , 2 j u , 2 j y )

for all nonnegative integers m and l with m > l , all u ∈ U ( P ) , and all y ∈ P . It follows from (2.11) that the sequence { 1 2 k g ( 2 k y ) } is Cauchy for all y ∈ P . Since Q is complete, the sequence { 1 2 k g ( 2 k y ) } converges. So one can define the mapping G : P → Q by

G ( y ) ≔ lim k → ∞ 1 2 k g ( 2 k y )

for all y ∈ P . Moreover, substituting l = 0 and passing to the limit m → ∞ in (2.11), we obtain (2.10).

It follows from (2.6) and (2.9) that

∥ G ( λ u + v + 2 y ) − λ G ( u ) − G ( v ) − 2 G ( y ) ∥ = lim n → ∞ 1 2 n ∥ g ( 2 n ( λ u + v + 2 y ) ) − λ g ( 2 n u ) − g ( 2 n v ) − 2 g ( 2 n y ) ∥ ≤ lim n → ∞ 1 2 n φ ( 2 n u , 2 n v , 2 n y ) = 0

for all λ ∈ C , all u , v ∈ U ( P ) , and all y ∈ P . So,

G ( λ u + v + 2 y ) = λ G ( u ) + G ( v ) + 2 G ( y )

for all λ ∈ C , all u , v ∈ U ( P ) , and all y ∈ P . By Lemma 2.1, the mapping G : P → Q is C -linear.

The proof of the uniqueness of the mapping G is similar to the proof of [29, Theorem 2.3].□

3 Hyers-Ulam stability of Poisson C * -algebra derivations and Poisson C * -algebra homomorphisms in Poisson C * -algebras: Direct method

Using the direct method, we prove the Hyers-Ulam stability of Poisson C * -algebra homomorphisms in unital Poisson C * -algebras associated with the additive functional equation (2.2).

Theorem 3.1

Let φ : P 3 → [ 0 , ∞ ) be a function such that

(3.1) ∑ j = 0 ∞ 8 j φ u 2 j , v 2 j , y 2 j < ∞

for all u , v ∈ U ( P ) and all y ∈ P . Let g : P → Q be a mapping satisfying (2.6) for all s ∈ Z with s ≤ 0 . If the mapping g : P → Q satisfies

(3.2) ‖ g ( 8 s u v w ) − g ( 2 s u ) g ( 2 s v ) g ( 2 s w ) ‖ ≤ φ ( 2 s u , 2 s v , 2 s w ) ,

(3.3) ‖ g ( 2 s u * ) − g ( 2 s u ) * ‖ ≤ φ ( 2 s u , 2 s u , 2 s u ) ,

(3.4) ‖ g ( { 2 s u , 4 s v w } ) − { g ( 2 s u ) , g ( 2 s v ) g ( 2 s w ) } ‖ ≤ φ ( 2 s u , 2 s v , 2 s w ) ,

for all u , v , w ∈ U ( P ) and all s ∈ Z with s ≤ 0 , then there exists a unique Poisson C * -algebra homomorphism H : P → Q such that

(3.5) ‖ g ( y ) − H ( y ) ‖ ≤ 1 2 ∑ j = 1 ∞ 2 j φ u 2 j , u 2 j , y 2 j

for all u ∈ U ( P ) , and all y ∈ P .

Proof

By Theorem 2.2, there exists a unique C -linear mapping H : P → Q satisfying (3.5). The mapping H : P → Q is given by

H ( y ) ≔ lim k → ∞ 2 k g y 2 k

for all y ∈ P .

It follows from (3.1) and (3.2) that

‖ H ( u v w ) − H ( u ) H ( v ) H ( w ) ‖ = lim n → ∞ 8 n g u v w 8 n − g u 2 n g v 2 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n = 0

and so,

H ( u v w ) = H ( u ) H ( v ) H ( w )

for all u , v , w ∈ U ( P ) .

Since H : P → Q is C -linear and each x , y , z ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j , y = ∑ k = 1 l μ k v k , and z = ∑ s = 1 t ν s w s ( λ j , μ k , ν s ∈ C , u j , v k , w s ∈ U ( P ) ) ,

H ( x y z ) = H ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s u j v k w s = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s H ( u j v k w s ) = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s H ( u j ) H ( v k ) H ( w s ) = ∑ j = 1 m λ j H ( u j ) ∑ k = 1 l μ k H ( v k ) ∑ s = 1 t ν s H ( w s ) = H ∑ j = 1 m λ j u j H ∑ k = 1 l μ k v k H ∑ s = 1 t ν s w s = H ( x ) H ( y ) H ( z )

for all x , y , z ∈ P . So, the C -linear mapping H : P → Q satisfies H ( x y z ) = H ( x ) H ( y ) H ( z ) for all x , y , z ∈ P .

It follows from (3.1) and (3.3) that

‖ H ( u * ) − H ( u ) * ‖ = lim n → ∞ 2 n g u * 2 n − g u 2 n * ≤ lim n → ∞ 2 n φ u 2 n , u 2 n , u 2 n ≤ lim n → ∞ 8 n φ u 2 n , u 2 n , u 2 n = 0

and so,

H ( u * ) = H ( u ) *

for all u ∈ U ( P ) .

Since H : P → Q is C -linear and each x ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j   ( λ j ∈ C , u j ∈ U ( P ) ) ,

H ( x * ) = H ∑ j = 1 m λ j ¯ u j * = ∑ j = 1 m λ j ¯ H ( u j * ) = ∑ j = 1 m λ j ¯ H ( u j ) * = H ∑ j = 1 m λ j u j * = H ( x ) *

for all x ∈ P . So, the C -linear mapping H : P → Q is involutive.

It follows from (3.1) and (3.4) that

‖ H ( { u , v w } ) − { H ( u ) , H ( v ) H ( w ) } ‖ = lim n → ∞ 8 n g { u , v w } 8 n − g u 2 n , g v 2 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n = 0

and so,

H ( { u , v w } ) = { H ( u ) , H ( v ) H ( w ) }

for all u , v , w ∈ U ( P ) .

Since H : P → Q is C -linear and each x , y , z ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j , y = ∑ k = 1 l μ k v k , and z = ∑ s = 1 t ν s w s ( λ j , μ k , ν s ∈ C , u j , v k , w s ∈ U ( P ) ) ,

H ( { x , y z } ) = H ∑ j = 1 m λ j u j , ∑ k = 1 l μ k v k ∑ s = 1 t ν s w s = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s H ( { u j , v k w s } ) = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s { H ( u j ) , H ( v k ) H ( w s ) } = ∑ j = 1 m λ j H ( u j ) , ∑ k = 1 l μ k H ( v k ) ∑ s = 1 t ν s H ( w s ) = H ∑ j = 1 m λ j u j , H ∑ k = 1 l μ k v k H ∑ s = 1 t ν s w s = { H ( x ) , H ( y ) H ( z ) }

for all x , y , z ∈ P . So, the mapping H : P → Q satisfies H ( { x , y z } ) = { H ( x ) , H ( y ) H ( z ) } for all x , y , z ∈ P . Thus, the C -linear mapping H : P → Q is a Poisson C * -algebra homomorphism.□

Corollary 3.2

Let r > 3 and θ be nonnegative real numbers and g : P → Q be a mapping satisfying

(3.6) ‖ g ( λ ( 2 s u ) + 2 s v + 2 y ) − λ g ( 2 s u ) − g ( 2 s v ) − 2 g ( y ) ‖ ≤ θ ( 2 s + 1 + ‖ y ‖ r )

for all λ ∈ C , all u , v ∈ U ( P ) , all s ∈ Z with s ≤ 0 , and all y ∈ P . If the mapping g : P → Q satisfies

(3.7) ‖ g ( 8 s u v w ) − g ( 2 s u ) g ( 2 s v ) g ( 2 s w ) ‖ ≤ 3 ⋅ 2 r s θ ,

(3.8) ‖ g ( 2 s u * ) − g ( 2 s u ) * ‖ ≤ 3 ⋅ 2 r s θ ,

(3.9) ‖ g ( { 2 s u , 4 s v w } ) − { g ( 2 s u ) , g ( 2 s v ) g ( 2 s w ) } ‖ ≤ 3 ⋅ 2 r s θ

for all u , v , w ∈ U ( P ) and all s ∈ Z with s ≤ 0 , then there exists a unique Poisson C * -algebra homomorphism H : P → Q such that

(3.10) ‖ g ( y ) − H ( y ) ‖ ≤ θ 2 r − 2 ( 2 + ‖ y ‖ r )

for all y ∈ P .

Proof

The result follows from Theorem 3.1 by taking φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≤ 0 , and all y ∈ P .□

Theorem 3.3

Let φ : P 3 → [ 0 , ∞ ) be a function and g : P → Q be a mapping satisfying (2.9), (2.6), (3.2), (3.3) and (3.4) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra homomorphism H : P → Q such that

(3.11) ‖ g ( y ) − H ( y ) ‖ ≤ 1 2 ∑ j = 0 ∞ 1 2 j φ ( 2 j u , 2 j u , 2 j y )

for all u ∈ U ( P ) and all y ∈ P .

Proof

By Theorem 2.3, there exists a unique C -linear mapping H : P → Q satisfying (3.11). The mapping H : P → Q is given by

H ( y ) ≔ lim k → ∞ 1 2 k g ( 2 k y )

for all y ∈ P .

It follows from (2.9) and (3.2) that

‖ H ( u v w ) − H ( u ) H ( v ) H ( w ) ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n u v w ) − g ( 2 n u ) g ( 2 n v ) g ( 2 n w ) ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ lim n → ∞ 1 2 n φ ( 2 n u , 2 n v , 2 n w ) = 0

and so H ( u v w ) = H ( u ) H ( v ) H ( w ) for all u , v , w ∈ U ( P ) .

It follows from (2.9) and (3.4) that

‖ H ( { u , v w } ) − { H ( u ) , H ( v ) H ( w ) } ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n { u , v w } ) − { g ( 2 n u ) , g ( 2 n v ) g ( 2 n w ) } ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ lim n → ∞ 1 2 n φ ( 2 n u , 2 n v , 2 n w ) = 0

and so H ( { u , v w } ) = { H ( u ) , H ( v ) H ( w ) } for all u , v , w ∈ U ( P ) .

The rest of the proof is similar to the proof of Theorem 3.1.□

Corollary 3.4

Let r < 1 and θ be nonnegative real numbers and g : P → Q be a mapping satisfying (3.6), (3.7), (3.8), and (3.9) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra homomorphism H : P → Q such that

(3.12) ‖ g ( x ) − H ( x ) ‖ ≤ θ 2 − 2 r ( 2 + ‖ y ‖ r )

for all y ∈ P .

Proof

The result follows from Theorem 3.3 by taking φ ( 2 s u , 2 s v , 2 j y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≥ 0 , and all y ∈ P .□

Now, we prove the Hyers-Ulam stability of Poisson C * -algebra derivations in unital Poisson C * -algebras associated with the additive functional equation (2.2) by using the direct method.

Theorem 3.5

Let φ : P 3 → [ 0 , ∞ ) be a function satisfying (3.1). If a mapping g : P → P satisfies (3.3), (2.6), and

(3.13) ‖ g ( 8 s u v w ) − g ( 2 s u ) 4 s v w − 4 s u g ( 2 s v ) w − 4 s u v g ( 2 s w ) ‖ ≤ φ ( 2 s u , 2 s v , 2 s w ) ,

(3.14) ‖ g ( { 2 s u , 4 s v w } ) − { g ( 2 s u ) , 4 s v w } − { 2 s u , g ( 2 s v ) 2 s w } − { 2 s u , 2 s v g ( 2 s w ) } ‖ ≤ φ ( 2 s u , 2 s v , 2 s w )

for all u , v , w ∈ U ( P ) and all s ∈ Z with s ≤ 0 , then there exists a unique Poisson C * -algebra derivation D : P → P such that

(3.15) ‖ g ( y ) − D ( y ) ‖ ≤ 1 2 ∑ j = 1 ∞ 2 j φ u 2 j , u 2 j , y 2 j

for all u ∈ U ( P ) and all y ∈ P .

Proof

By Theorem 2.2, there exists a unique C -linear mapping D : P → P satisfying (3.15). The mapping D : P → P is given by

D ( y ) ≔ lim k → ∞ 2 k g y 2 k

for all y ∈ P .

It follows from (3.1) and (3.13) that

‖ D ( u v w ) − D ( u ) v w − u D ( v ) w − u v D ( w ) ‖ = lim n → ∞ 8 n g u v w 8 n − g u 2 n v w 4 n − u 2 n g v 2 n w 2 n − u v 4 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n = 0

for all u , v , w ∈ U ( P ) . So,

D ( u v w ) = D ( u ) v w + u D ( v ) w + u v D ( w )

for all u , v , w ∈ U ( P ) .

Since D : P → P is C -linear and each x , y ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j , y = ∑ k = 1 l μ k v k , and z = ∑ s = 1 t ν s w s ( λ j , μ k , ν s ∈ C , u j , v k , w s ∈ U ( P ) ) ,

D ( x y z ) = D ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k u j v k w s = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s D ( u j v k w s ) = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s ( D ( u j ) v k w s + u j D ( v k ) w s + u j v k D ( w s ) ) = ∑ j = 1 m λ j D ( u j ) ∑ k = 1 l μ k v k ∑ s = 1 t ν s w s + ∑ j = 1 m λ j u j ∑ k = 1 l μ k D ( v k ) ∑ s = 1 t ν s w s + ∑ j = 1 m λ j u j ∑ k = 1 l μ k v k ( ∑ s = 1 t ν s D ( w s ) ) = D ∑ j = 1 m λ j u j y z + x D ∑ k = 1 l μ k v k z + x y D ∑ s = 1 t ν s w s = D ( x ) y z + x D ( y ) z + x y D ( z )

for all x , y , z ∈ P . So, the C -linear mapping D : P → P satisfies

D ( x y z ) = D ( x ) y z + x D ( y ) z + x y D ( z )

for all x , y , z ∈ P .

It follows from (3.1) and (3.3) that

‖ D ( u * ) − D ( u ) * ‖ = lim n → ∞ 2 n g u * 2 n − g u 2 n * ≤ lim n → ∞ 2 n φ u 2 n , u 2 n , u 2 n ≤ lim n → ∞ 8 n φ u 2 n , u 2 n , u 2 n = 0

and so,

D ( u * ) = D ( u ) *

for all u ∈ U ( P ) .

Since D : P → P is C -linear and each x ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j ( λ j ∈ C , u j ∈ U ( P ) ) ,

D ( x * ) = D ∑ j = 1 m λ j ¯ u j * = ∑ j = 1 m λ j ¯ D ( u j * ) = ∑ j = 1 m λ j ¯ D ( u j ) * = D ∑ j = 1 m λ j u j * = D ( x ) *

for all x ∈ P . So, the C -linear mapping D : P → P is involutive.

It follows from (3.1) and (3.14) that

‖ D ( { u , v w } ) − { D ( u ) , v w } − { u , D ( v ) w } − { u , v D ( w ) } ‖ = lim n → ∞ 8 n g { u , v w } 8 n − g u 2 n , v w 4 n − u 2 n , g v 2 n w 2 n − u v 4 n , g w 2 n ≤ lim n → ∞ 8 n φ ( 2 n u , 2 n v , 2 n w ) = 0

and so, D ( { u , v w } ) = { D ( u ) , v w } + { u , D ( v ) w } + { u , v D ( w ) } for all u , v , w ∈ U ( P ) .

Since D : P → P is C -linear and each x , y , z ∈ P is a finite linear combination of unitary elements [28], i.e., x = ∑ j = 1 m λ j u j , y = ∑ k = 1 l μ k v k , and z = ∑ s = 1 t ν s w s ( λ j , μ k , ν s ∈ C , u j , v k , w s ∈ U ( P ) ) ,

D ( { x , y z } ) = D ∑ j = 1 m λ j u j , ∑ k = 1 l μ k v k ∑ s = 1 t ν s w s = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s D ( { u j , v k w s } ) = ∑ j = 1 m ∑ k = 1 l ∑ s = 1 t λ j μ k ν s ( { D ( u j ) , v k w s } + { u j , D ( v k ) w s } + { u j v k , D ( w s ) } ) = ∑ j = 1 m λ j D ( u j ) , ∑ k = 1 l μ k v k ∑ s = 1 t ν s w s + ∑ j = 1 m λ j u j , ∑ k = 1 l μ k D ( v k ) ∑ s = 1 t ν s w s + ∑ j = 1 m λ j u j ∑ k = 1 l μ k v k , ∑ s = 1 t ν s D ( w s ) = { D ∑ j = 1 m λ j u j , y z } + { x , D ∑ k = 1 l μ k v k z } + { x y , D ( ∑ s = 1 t ν s w s ) } = { D ( x ) , y z } + { x , D ( y ) z } + { x y , D ( z ) }

for all x , y , z ∈ P . Thus, the mapping D : P → P is a Poisson C * -algebra derivation.□

Corollary 3.6

Let r > 3 and θ be nonnegative real numbers and g : P → P be a mapping satisfying (3.6), (3.8), and

(3.16) ‖ g ( 8 s u v w ) − g ( 2 s u ) 4 s v w − 4 s u g ( 2 s v ) w − 4 s u v g ( 2 s w ) ‖ ≤ 3 ⋅ 2 r s θ ,

(3.17) ‖ g ( { 2 s u , 4 s v w } ) − { g ( 2 s u ) , 4 s v w } − { 2 s u , g ( 2 s v ) 2 s w } − { 4 s u v , g ( 2 s w ) } ‖ ≤ 3 ⋅ 2 r s θ

for all u , v , w ∈ U ( P ) and all s ∈ Z with s ≤ 0 . Then, there exists a unique Poisson C * -algebra derivation D : P → P such that

(3.18) ‖ g ( y ) − D ( y ) ‖ ≤ θ 2 r − 2 ( 2 + ‖ y ‖ r )

for all y ∈ P .

Proof

The result follows from Theorem 3.5 by taking φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≤ 0 and all y ∈ P .□

Theorem 3.7

Let φ : P 3 → [ 0 , ∞ ) be a function and g : P → P be a mapping satisfying (2.9), (2.6), (3.13), (3.14), and (3.3) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra derivation D : P → P satisfying

(3.19) ‖ g ( y ) − D ( y ) ‖ ≤ 1 2 Ψ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P , where Ψ ( u , v , y ) is given in the statement of Theorem 2.3.

Proof

By Theorem 2.3, there exists a unique C -linear mapping D : P → P satisfying (3.19). The mapping D : P → P is given by

D ( y ) ≔ lim k → ∞ 1 2 k g ( 2 k y )

for all y ∈ P .

It follows from (2.9) and (3.13) that

‖ D ( u v w ) − D ( u ) v w − u D ( v ) w − u v D ( w ) ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n u v w ) − g ( 2 n u ) ( 4 n v w ) − ( 2 n u ) g ( 2 n v ) ( 2 n w ) − ( 4 n u v ) g ( 2 n w ) ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ 1 2 n φ ( 2 n u , 2 n v , 2 n w ) = 0

and so, D ( u v w ) = D ( u ) v w + u D ( v ) w + u v D ( w ) for all u , v , w ∈ U ( P ) .

It follows from (2.9) and (3.14) that

‖ D ( { u , v w } ) − { D ( u ) , v w } − { u , D ( v ) w } − { u , v D ( w ) } ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n { u , v w } ) − { g ( 2 n u ) , 4 n v w } − { 2 n u , g ( 2 n v ) 2 n w } − { 2 n u , 2 n v g ( 2 n w ) } ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ 1 2 n φ ( 2 n u , 2 n v , 2 n w ) = 0

and so, D ( { u , v w } ) = { D ( u ) , v w } + { u , D ( v ) w } + { u , v D ( w ) } for all u , v , w ∈ U ( P ) .

The rest of the proof is similar to the proof of Theorem 3.5.□

Corollary 3.8

Let r < 1 and θ be nonnegative real numbers and g : P → P be a mapping satisfying (3.6), (3.16), (3.17), and (3.8) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra derivation D : P → P such that

(3.20) ‖ g ( x ) − D ( x ) ‖ ≤ θ 2 − 2 r ( 2 + ‖ y ‖ r )

for all y ∈ P .

Proof

The result follows from Theorem 3.7 by taking φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≥ 0 , and all y ∈ P .□

4 Hyers-Ulam stability of the functional equation (2.1): fixed point method

Using the fixed point method, we prove the Hyers-Ulam stability of the additive functional equation (2.1) in unital C * -algebras.

Theorem 4.1

[27, Theorem 8] Let φ : P 3 → [ 0 , ∞ ) be a function such that there exists an L < 1 with

φ u 2 , v 2 , y 2 ≤ L 2 φ ( u , v , y )

for all u , v ∈ U ( P ) and all y ∈ P . Let g : P → Q be a mapping satisfying (2.6) for all s ∈ Z with s ≤ 0 . Then, there exists a unique C -linear mapping H : P → Q such that

(4.1) ‖ g ( y ) − H ( y ) ‖ ≤ L 2 ( 1 − L ) φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P .

Proof

Consider the set

S ≔ { g : P → Q }

and introduce the generalized metric on S :

d ( g , h ) = inf { μ ∈ R + : ‖ g ( y ) − h ( y ) ‖ ≤ μ φ ( u , u , y ) , ∀ u ∈ U ( P ) , ∀ y ∈ P } ,

where, as usual, inf ϕ = + ∞ . It is easy to show that ( S , d ) is complete [30].

Now, we consider the linear mapping J : S → S such that

J g ( y ) ≔ 2 g y 2

for all y ∈ P .

Let g , h ∈ S be given such that d ( g , h ) = ε . Then,

‖ g ( y ) − h ( y ) ‖ ≤ ε φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P . Since

2 g y 2 − 2 h y 2 ≤ 2 ε φ u 2 , u 2 , y 2 ≤ 2 ε L 2 φ ( u , u , y ) = L ε φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P , d ( J g , J h ) ≤ L ε . This means that

d ( J g , J h ) ≤ L d ( g , h )

for all g , h ∈ S .

It follows from (2) that

g ( y ) − 2 g y 2 ≤ φ u 2 , u 2 , y 2 ≤ L 2 φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P . So d ( g , J g ) ≤ L 2 .

By Theorem 1.1, there exists a mapping H : P → P satisfying the following:

  1. H is a fixed point of J , i.e.,

    (4.2) H ( y ) = 2 H y 2

    for all y ∈ P . The mapping H is a unique fixed point of J . This implies that H is a unique mapping satisfying (4.2) such that there exists a μ ∈ ( 0 , ∞ ) satisfying

    ‖ g ( y ) − H ( y ) ‖ ≤ μ φ ( u , u , y )

    for all u ∈ U ( P ) and all y ∈ P ;

  2. d ( J l g , H ) → 0 as l → ∞ . This implies the equality

    lim l → ∞ 2 l g y 2 l = H ( y )

    for all y ∈ P ;

  3. d ( g , H ) ≤ 1 1 − L d ( g , J g ) , which implies

    ∥ g ( x ) − H ( x ) ∥ ≤ L 2 ( 1 − L ) φ ( u , u , y )

    for all u ∈ U ( P ) and all y ∈ P . Thus, we obtain the inequality (4.1).

The rest of the proof is the same as in the proof of Theorem 2.2.□

Theorem 4.2

[27, Theorem 8] Let φ : P 3 → [ 0 , ∞ ) be a function such that there exists an L < 1 with

(4.3) φ ( u , v , y ) ≤ 2 L φ u 2 , v 2 , y 2

for all u , v ∈ U ( P ) and all y ∈ P . Let g : P → Q be a mapping satisfying (2.6) for all s ∈ Z with s ≥ 0 . Then, there exists a unique C -linear mapping H : P → Q such that

(4.4) ‖ g ( y ) − H ( y ) ‖ ≤ 1 2 ( 1 − L ) φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P .

Proof

Let S and d be given in the proof of Theorem 4.1.

Now, we consider the linear mapping J : S → S such that

J g ( y ) ≔ 1 2 g ( 2 y )

for all y ∈ P .

Letting λ = − 1 and replacing v by u in (2.6), we obtain

g ( y ) − 1 2 g ( 2 y ) ≤ 1 2 φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P . So d ( g , J g ) ≤ 1 2 .

The rest of the proof is similar to the proofs of Theorems 2.3 and 4.1.□

5 Hyers-Ulam stability of Poisson C * -algebra derivations and Poisson C * -algebra homomorphisms in Poisson C * -algebras: Fixed point method

Using the fixed point method, we prove the Hyers-Ulam stability of Poisson C * -algebra homomorphisms in unital Poisson C * -algebras associated with the additive functional equation (2.2).

Theorem 5.1

Let φ : P 3 → [ 0 , ∞ ) be a function such that

(5.1) φ u 2 , v 2 , y 2 ≤ L 8 φ ( u , v , y ) ≤ L 2 φ ( u , v , y )

for all u , v ∈ U ( P ) and all y ∈ P . Let g : P → Q be a mapping satisfying (2.6), (3.2), (3.3), and (3.4) for all s ∈ Z with s ≤ 0 . Then, there exists a unique Poisson C * -algebra homomorphism H : P → Q satisfying (4.1).

Proof

It follows from (5.1) and (3.2) that

‖ H ( u v w ) − H ( u ) H ( v ) H ( w ) ‖ = lim n → ∞ 8 n g u v w 8 n − g u 2 n g v 2 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n ≤ lim n → ∞ 8 n L n 8 n φ ( u , v , w ) = 0

and so,

H ( u v w ) = H ( u ) H ( v ) H ( w )

for all u , v , w ∈ U ( P ) .

It follows from (5.1) and (3.4) that

‖ H ( { u , v w } ) − { H ( u ) , H ( v ) H ( w ) } ‖ = lim n → ∞ 8 n g { u , v w } 8 n − g u 2 n , g v 2 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n ≤ lim n → ∞ 8 n L n 8 n φ ( u , v , w ) = 0

and so,

H ( { u , v w } ) = { H ( u ) , H ( v ) H ( w ) }

for all u , v , w ∈ U ( P ) .

The rest of the proof is similar to the proofs of Theorems 2.2, 3.1, and 4.1.□

Corollary 5.2

Let r > 3 and θ be nonnegative real numbers and g : P → Q be a mapping satisfying (3.6), (3.7), (3.8), and (3.9) for all s ∈ Z with s ≤ 0 . Then, there exists a unique Poisson C * -algebra homomorphism H : P → Q satisfying (3.10).

Proof

The result follows from Theorem 5.1 by taking L = 2 1 − r and φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≤ 0 , and all y ∈ P .□

Theorem 5.3

Let φ : P 3 → [ 0 , ∞ ) be a function and g : P → Q be a mapping satisfying (4.3), (2.6), (3.2), (3.3), and (3.4) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra homomorphism H : P → Q satisfying (4.4).

Proof

It follows from (2.9) and (3.2) that

‖ H ( u v w ) − H ( u ) H ( v ) H ( w ) ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n u v w ) − g ( 2 n u ) g ( 2 n v ) g ( 2 n w ) ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ lim n → ∞ 2 n L n 8 n φ ( u , v , w ) = 0

and so, H ( u v w ) = H ( u ) H ( v ) H ( w ) for all u , v , w ∈ U ( P ) .

It follows from (2.9) and (3.4) that

‖ H ( { u , v w } ) − { H ( u ) , H ( v ) H ( w ) } ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n { u , v w } ) − { g ( 2 n u ) , g ( 2 n v ) g ( 2 n w ) } ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ lim n → ∞ 2 n L n 8 n φ ( u , v , w ) = 0

and so, H ( { u , v w } ) = { H ( u ) , H ( v ) H ( w ) } for all u , v , w ∈ U ( P ) .

The rest of the proof is similar to the proofs of Theorems 3.1, 3.3, and 5.1.□

Corollary 5.4

Let r < 1 and θ be nonnegative real numbers and g : P → Q be a mapping satisfying (3.6), (3.7), (3.8), and (3.9) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra homomorphism H : P → Q satisfying (3.12).

Proof

The result follows from Theorem 5.3 by taking L = 2 r − 1 and φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≥ 0 and all y ∈ P .□

Now, we prove the Hyers-Ulam stability of Poisson C * -algebra derivations in unital Poisson C * -algebras associated with the additive functional equation (2.2) by using the fixed point method.

Theorem 5.5

Let φ : P 3 → [ 0 , ∞ ) be a function satisfying (5.1). If the mapping g : P → P satisfies (2.6), (3.3), (3.13), and (3.14) for all s ∈ Z with s ≤ 0 , then there exists a unique Poisson C * -algebra derivation D : P → P such that

‖ g ( y ) − D ( y ) ‖ ≤ L 2 ( 1 − L ) φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P .

Proof

It follows from (5.1) and (3.13) that

‖ D ( u v w ) − D ( u ) v w − u D ( v ) w − u v D ( w ) ‖ = lim n → ∞ 8 n g u v w 8 n − g u 2 n v w 4 n − u 2 n g v 2 n w 2 n − u v 4 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n ≤ lim n → ∞ 8 n L n 8 n φ ( u , v , w ) = 0

and so,

D ( u v w ) = D ( u ) v w + u D ( v ) w + u v D ( w )

for all u , v , w ∈ U ( P ) .

It follows from (5.1) and (3.14) that

‖ D ( { u , v w } ) − { D ( u ) , v w } − { u , D ( v ) w } − { u , v D ( w ) } ‖ = lim n → ∞ 8 n g { u , v w } 8 n − g u 2 n , v w 4 n − u 2 n , g v 2 n w 2 n − u 2 n , v 2 n g w 2 n ≤ lim n → ∞ 8 n φ u 2 n , v 2 n , w 2 n ≤ lim n → ∞ 8 n L n 8 n φ ( u , v , w ) = 0

and so,

D ( { u , v w } ) = { D ( u ) , v w } + { u , D ( v ) w } + { u , v D ( w ) }

for all u , v , w ∈ U ( P ) .

The rest of the proof is similar to the proofs of Theorems 3.5 and 5.1.□

Corollary 5.6

Let r > 3 and θ be nonnegative real numbers and g : P → P be a mapping satisfying (3.6), (3.8), (3.16), and (3.17) for all s ∈ Z with s ≤ 0 . Then, there exists a unique Poisson C * -algebra derivation D : P → P satisfying (3.18).

Proof

The result follows from Theorem 5.5 by taking L = 2 1 − r and φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≤ 0 and all y ∈ P .□

Theorem 5.7

Let φ : P 3 → [ 0 , ∞ ) be a function and g : P → P be a mapping satisfying (2.6), (4.3), (3.3), (3.13), and (3.14) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra derivation D : P → P such that

‖ g ( y ) − D ( y ) ‖ ≤ 1 2 ( 1 − L ) φ ( u , u , y )

for all u ∈ U ( P ) and all y ∈ P .

Proof

It follows from (4.3) and (3.2) that

‖ D ( u v w ) − D ( u ) v w − u D ( v ) w − u v D ( w ) ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n u v w ) − g ( 2 n u ) ( 4 n v w ) − ( 2 n u ) g ( 2 n v ) ( 2 n w ) − ( 4 n u v ) g ( 2 n w ) ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ lim n → ∞ 2 n L n 8 n φ ( u , v , w ) = 0

and so, D ( u v w ) = D ( u ) v w + u D ( v ) w + u v D ( w ) for all u , v , w ∈ U ( P ) .

It follows from (4.3) and (3.14) that

‖ D ( { u , v w } ) − { D ( u ) , v w } − { u , D ( v ) w } − { u , v D ( w ) } ‖ = lim n → ∞ 1 8 n ∥ g ( 8 n { u , v w } ) − { g ( 2 n u ) , 4 n w } − { 2 n u , g ( 2 n v ) 2 n w } − { 2 n u , 2 n v g ( 2 n w ) } ∥ ≤ lim n → ∞ 1 8 n φ ( 2 n u , 2 n v , 2 n w ) ≤ lim n → ∞ 2 n L n 8 n φ ( u , v , w ) = 0

and so, D ( { u , v w } ) = { D ( u ) , v w } + { u , D ( v ) w } + { u , v D ( w ) } for all u , v , w ∈ U ( P ) .

The rest of the proof is similar to the proofs of Theorems 3.5 and 5.1.□

Corollary 5.8

Let r < 1 and θ be nonnegative real numbers and g : P → P be a mapping satisfying (3.6), (3.8), (3.16), and (3.17) for all s ∈ Z with s ≥ 0 . Then, there exists a unique Poisson C * -algebra derivation D : P → P satisfying (3.20).

Proof

The result follows from Theorem 5.7 by taking L = 2 r − 1 and φ ( 2 s u , 2 s v , y ) = θ ( 2 r s + 1 + ‖ y ‖ r ) for all u , v ∈ U ( P ) , all s ∈ Z with s ≥ 0 , and all y ∈ P .□

6 Conclusion

We introduced the additive functional equation (2.1) in a unital Poisson C * -algebra P . Using the direct method and the fixed point method, we proved the Hyers-Ulam stability of the additive functional equation (2.1) in unital Poisson C * -algebras. Furthermore, we applied to study Poisson C * -algebra homomorphisms and Poisson C * -algebra derivations in unital Poisson C * -algebras.

Acknowledgements

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

  1. Funding information: Yongqiao Wang was supported by the National Natural Science Foundation of China (Grant No. 12001079), the Fundamental Research Funds for the Central Universities(Grant No. 3132024198), and the Scientific Research Foundation of Liaoning Education Department (Grant No. LJKZ0053). Yuan Chang was supported by the Scientific Research Foundation of Liaoning Education Department (Grant No. LJKMZ20221556).

  2. 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.

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

  4. Ethical approval: We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being.

  5. Data availability statement: Not applicable.

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Received: 2023-11-30
Revised: 2024-05-02
Accepted: 2024-08-12
Published Online: 2024-12-10

© 2024 the author(s), published by De Gruyter

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

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  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
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  7. Existence of projected solutions for quasi-variational hemivariational inequality
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  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
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  45. Hyers-Ulam stability of Davison functional equation on restricted domains
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https://doi.org/10.1515/dema-2024-0053
Schlagwörter für diesen Artikel
Hyers-Ulam stability; fixed point method; additive functional equation; Poisson C *-algebra derivation in Poisson C *-algebra; Poisson C *-algebra homomorphism in Poisson C *-algebra
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Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
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  76. Review Articles
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  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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Heruntergeladen am 15.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0053/html
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