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Approximation of homomorphisms and derivations on Lie C -algebras via fixed point method

Journal of Inequalities and Applications20132013:415

https://doi.org/10.1186/1029-242X-2013-415

  • Received: 19 December 2012
  • Accepted: 2 August 2013
  • Published:

Abstract

In this paper, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in C -algebras and Lie C -algebras and of derivations on C-algebras and Lie C-algebras for an m-variable additive functional equation.

MSC:39A10, 39B52, 39B72, 46L05, 47H10, 46B03.

Keywords

  • additive functional equation
  • fixed point
  • homomorphism in C -algebras and Lie C -algebras
  • generalized Hyers-Ulam stability
  • derivation on C -algebras and Lie C -algebras

1 Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms:

Let ( G 1 , ) be a group and let ( G 2 , , d ) be a metric group with the metric d ( , ) . Given ϵ > 0 , does there exist a δ ( ϵ ) > 0 such that if a mapping h : G 1 G 2 satisfies the inequality d ( h ( x y ) , h ( x ) h ( y ) ) < δ for all x , y G 1 , then there is a homomorphism H : G 1 G 2 with d ( h ( x ) , H ( x ) ) < ϵ for all x G 1 ?

If the answer is affirmative, we say that the equation of homomorphism H ( x y ) = H ( x ) H ( y ) is stable.

Since Ulam’s question, recently, many authors have given many answers and proved many kinds of functional equations in various spaces, for example, Banach algebras [2], random normed spaces [38], fuzzy normed spaces [9, 10], non-Archimedean Banach spaces [11], non-Archimedean lattice random spaces [12], inner product spaces [1315] and others [1621].

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in Lie C -algebras for the following additive functional equation (see [22]):
i = 1 m f ( m x i + j = 1 , j i m x j ) + f ( i = 1 m x i ) = 2 f ( i = 1 m m x i )
(1.1)

for all m N with m 2 .

2 Stability of homomorphisms and derivations in C -algebras

Throughout this section, assume that A is a C -algebra with a norm A and B is a C -algebra with a norm B .

For any mapping f : A B , we define
D μ f ( x 1 , , x m ) : = i = 1 m μ f ( m x i + j = 1 , j i m x j ) + f ( μ i = 1 m x i ) 2 f ( μ i = 1 m m x i )

for all μ T 1 : = { ν C : | ν | = 1 } and x 1 , , x m A .

Recall that a -linear mapping H : A B is called a homomorphism in C -algebras if H satisfies H ( x y ) = H ( x ) H ( y ) and H ( x ) = H ( x ) for all x , y A .

Recently, in [23], O’Regan et al. proved the generalized Hyers-Ulam stability of homomorphisms in C -algebras for the functional equation D μ f ( x 1 , , x m ) = 0 .

Theorem 2.1 [23]

Let f : A B be a mapping for which there are functions φ : A m [ 0 , ) , ψ : A 2 [ 0 , ) and η : A [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 , D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) , f ( x y ) f ( x ) f ( y ) B ψ ( x , y ) , lim j m 2 j ψ ( m j x , m j y ) = 0 , f ( x ) f ( x ) B η ( x ) , lim j m j η ( m j x ) = 0
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( m x , 0 , , 0 ) m L φ ( x , 0 , , 0 )
for all x A , then there exists a unique homomorphism H : A B such that
f ( x ) H ( x ) B 1 m m L φ ( x , 0 , , 0 )

for all x A .

Theorem 2.2 [23]

Let f : A B be a mapping for which there are functions φ : A m [ 0 , ) , ψ : A 2 [ 0 , ) and η : A [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 , D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) , f ( x y ) f ( x ) f ( y ) B ψ ( x , y ) , lim j m 2 j ψ ( m j x , m j y ) = 0 , f ( x ) f ( x ) B η ( x ) , lim j m j η ( m j x ) = 0
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( x , 0 , , 0 ) L m φ ( m x , 0 , , 0 )
for all x A , then there exists a unique homomorphism H : A B such that
f ( x ) H ( x ) B L m m L φ ( x , 0 , , 0 )

for all x A .

Recall that a -linear mapping δ : A A is called a derivation on A if δ satisfies δ ( x y ) = δ ( x ) y + x δ ( y ) for all x , y A .

In [23], also, O’Regan et al. proved the generalized Hyers-Ulam stability of derivations on C -algebras for the functional equation D μ f ( x 1 , , x m ) = 0 .

Theorem 2.3 [23]

Let f : A B be a mapping for which there are functions φ : A m [ 0 , ) and ψ : A 2 [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 , D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) , f ( x y ) f ( x ) y x f ( y ) B ψ ( x , y ) , lim j m 2 j ψ ( m j x , m j y ) = 0
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( m x , 0 , , 0 ) m L φ ( x , 0 , , 0 )
for all x A , then there exists a unique derivation δ : A A such that
f ( x ) δ ( x ) B 1 m m L φ ( x , 0 , , 0 )

for all x A .

Theorem 2.4 [23]

Let f : A B be a mapping for which there are functions φ : A m [ 0 , ) and ψ : A 2 [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 , D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) , f ( x y ) f ( x ) y x f ( y ) B ψ ( x , y ) , lim j m 2 j ψ ( m j x , m j y ) = 0
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( m x , 0 , , 0 ) L m φ ( x , 0 , , 0 )
for all x A , then there exists a unique derivation δ : A A such that
f ( x ) δ ( x ) B L m m L φ ( x , 0 , , 0 )

for all x A .

3 Stability of homomorphisms in Lie C -algebras

A C -algebra , endowed with the Lie product
[ x , y ] : = x y y x 2

on , is called a Lie C -algebra (see [16, 2426]).

Definition 3.1 Let A and B be Lie C -algebras. A -linear mapping H : A B is called a Lie C -algebra homomorphism if H ( [ x , y ] ) = [ H ( x ) , H ( y ) ] for all x , y A .

Throughout this section, assume that A is a Lie C -algebra with a norm A and B is a Lie C -algebra with a norm B .

Now, we prove the generalized Hyers-Ulam stability of homomorphisms in Lie C -algebras for the functional equation D μ f ( x 1 , , x m ) = 0 .

Theorem 3.2 Let f : A B be a mapping for which there are functions φ : A m [ 0 , ) and ψ : A 2 [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 ,
(3.1)
D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) ,
(3.2)
f ( [ x , y ] ) [ f ( x ) , f ( y ) ] B ψ ( x , y ) ,
(3.3)
lim j m 2 j ψ ( m j x , m j y ) = 0
(3.4)
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( m x , 0 , , 0 ) m L φ ( x , 0 , , 0 )
for all x A , then there exists a unique Lie C -algebra homomorphism H : A B such that
f ( x ) H ( x ) B 1 m m L φ ( x , 0 , , 0 )
(3.5)

for all x A .

Proof By the same method as in the proof of Theorem 2.1, we can find the mapping H : A B given by
H ( x ) = lim n f ( m n x ) m n
for all x A . Thus it follows from (3.3) that
H ( [ x , y ] ) [ H ( x ) , H ( y ) ] B = lim n 1 m 2 n f ( m 2 n [ x , y ] ) [ f ( m n x ) , f ( m n y ) ] B lim n 1 m 2 n ψ ( m n x , m n y ) = 0
for all x , y A , and so
H ( [ x , y ] ) = [ H ( x ) , H ( y ) ]

for all x , y A . Therefore, H : A B is a Lie C -algebra homomorphism satisfying (3.5). This completes the proof. □

Corollary 3.3 Let 0 < r < 1 and θ be nonnegative real numbers. If f : A B is a mapping such that
D μ f ( x 1 , , x m ) B θ ( x 1 A r + x 2 A r + + x m A r ) , f ( [ x , y ] ) [ f ( x ) , f ( y ) ] B θ x A r y A r
for all μ T 1 and x 1 , , x m , x , y A , then there exists a unique Lie C -algebra homomorphism H : A B such that
f ( x ) H ( x ) B θ m m r x A r

for all x A .

Proof The proof follows from Theorem 3.2 by taking
φ ( x 1 , , x m ) = θ ( x 1 A r + x 2 A r + + x m A r ) , ψ ( x , y ) : = θ x A r y A r

for all x 1 , , x m , x , y A and putting L = m r 1 . □

Theorem 3.4 Let f : A B be a mapping for which there are functions φ : A m [ 0 , ) and ψ : A 2 [ 0 , ) satisfying (3.1)-(3.4) for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( x , 0 , , 0 ) L m φ ( x , 0 , , 0 )
for all x A , then there exists a unique Lie C -algebra homomorphism H : A B such that
f ( x ) H ( x ) B L m m L φ ( x , 0 , , 0 )

for all x A .

Corollary 3.5 Let r > 1 and θ be nonnegative real numbers. If f : A B is a mapping such that
D μ f ( x 1 , , x m ) B θ ( x 1 A r + x 2 A r + + x m A r ) , f ( [ x , y ] ) [ f ( x ) , f ( y ) ] B θ x A r y A r
for all μ T 1 and x 1 , , x m , x , y A , then there exists a unique Lie C -algebra homomorphism H : A B such that
f ( x ) H ( x ) B θ m r m x A r

for all x A .

Proof The proof follows from Theorem 3.4 by taking
φ ( x 1 , , x m ) = θ ( x 1 A r + x 2 A r + + x m A r ) , ψ ( x , y ) : = θ x A r y A r

for all x 1 , , x m , x , y A and putting L = m 1 r . □

4 Stability of derivations in Lie C -algebras

Definition 4.1 Let A be a Lie C -algebra. A -linear mapping δ : A A is called a Lie derivation if δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ] for all x , y A .

Throughout this section, assume that A is a Lie C -algebra with a norm A .

Finally, we prove the generalized Hyers-Ulam stability of derivations on Lie C -algebras for the functional equation D μ f ( x 1 , , x m ) = 0 .

Theorem 4.2 Let f : A A be a mapping for which there are functions φ : A m [ 0 , ) and ψ : A 2 [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 ,
(4.1)
D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) ,
(4.2)
f ( [ x , y ] ) [ f ( x ) , y ] [ x , f ( y ) ] B ψ ( x , y ) ,
(4.3)
lim j m 2 j ψ ( m j x , m j y ) = 0
(4.4)
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( m x , 0 , , 0 ) m L φ ( x , 0 , , 0 )
for all x A , then there exists a unique Lie derivation δ : A A such that
f ( x ) δ ( x ) B 1 m m L φ ( x , 0 , , 0 )
(4.5)

for all x A .

Proof By the same method as in the proof of Theorem 2.3, there exists a unique -linear mapping δ : A A satisfying (3.5). Also, we can find the mapping δ : A A given by
δ ( x ) = lim n f ( m n x ) m n
(4.6)
for all x A . Thus it follows from (4.3), (4.4) and (4.6) that
δ ( [ x , y ] ) [ δ ( x ) , y ] [ x , δ ( y ) ] A = lim n 1 m 2 n f ( m 2 n [ x , y ] ) [ f ( m n x ) , m n y ] [ m n x , f ( m n y ) ] A lim n 1 m 2 n ψ ( m n x , m n y ) = 0
for all x , y A , and so
δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ]

for all x , y A . Thus δ : A A is a Lie derivation satisfying (4.5). □

Corollary 4.3 Let 0 < r < 1 and θ be nonnegative real numbers. If f : A A is a mapping such that
D μ f ( x 1 , , x m ) B θ ( x 1 A r + x m A r ) , f ( [ x , y ] ) [ f ( x ) , y ] [ x , f ( y ) ] A θ x A r y A r
for all μ T 1 and x 1 , , x m , x , y A , then there exists a unique derivation δ : A A such that
f ( x ) δ ( x ) A θ m m r x A r

for all x A .

Proof The proof follows from Theorem 4.2 by taking
φ ( x 1 , , x m ) : = θ ( x 1 A r + + x m A r )
and
ψ ( x , y ) : = θ x A r y A r

for all x 1 , , x m , x , y A and putting L = m r 1 . □

Theorem 4.4 Let f : A A be a mapping for which there are functions φ : A m [ 0 , ) and ψ : A 2 [ 0 , ) such that
lim j m j φ ( m j x 1 , , m j x m ) = 0 , D μ f ( x 1 , , x m ) B φ ( x 1 , , x m ) , f ( [ x , y ] ) [ f ( x ) , y ] [ x , f ( y ) ] B ψ ( x , y ) , lim j m 2 j ψ ( m j x , m j y ) = 0
for all μ T 1 and x 1 , , x m , x , y A . If there exists 0 < L < 1 such that
φ ( m x , 0 , , 0 ) L m φ ( x , 0 , , 0 )
for all x A , then there exists a unique Lie derivation δ : A A such that
f ( x ) δ ( x ) B L m m L φ ( x , 0 , , 0 )

for all x A .

Proof The proof is similar to the proof of Theorem 4.2. □

Corollary 4.5 Let r > 1 and θ be nonnegative real numbers. If f : A A is a mapping such that
D μ f ( x 1 , , x m ) B θ ( x 1 A r + x m A r ) , f ( [ x , y ] ) [ f ( x ) , y ] [ x , f ( y ) ] A θ x A r y A r
for all μ T 1 and x 1 , , x m , x , y A , then there exists a unique Lie derivation δ : A A such that
f ( x ) δ ( x ) A θ m r m x A r

for all x A .

Proof The proof follows from Theorem 4.4 by taking
φ ( x 1 , , x m ) : = θ ( x 1 A r + x m A r )
and
ψ ( x , y ) : = θ x A r y A r

for all x 1 , , x m , x , y A and putting L = m 1 r . □

Declarations

Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

Authors’ Affiliations

(1)
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, 660-701, Korea
(2)
Department of Mathematics and Computer Science, Iran University of Science and Technology, Tehran, Iran
(3)
Department of Mathematics, Jeju National University, Jeju, 690-756, Korea

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© Cho et al.; licensee Springer 2013

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