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

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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.

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(xy),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(xy)=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 ) =2f ( i = 1 m m x i )
(1.1)

for all mN with m2.

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:AB, we define

D μ f( x 1 ,, x m ):= i = 1 m μf ( m x i + j = 1 , j i m x j ) +f ( μ i = 1 m x i ) 2f ( μ i = 1 m m x i )

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

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

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:AB 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,yA. If there exists 0<L<1 such that

φ(mx,0,,0)mLφ(x,0,,0)

for all xA, then there exists a unique homomorphism H:AB such that

f ( x ) H ( x ) B 1 m m L φ(x,0,,0)

for all xA.

Theorem 2.2 [23]

Let f:AB 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,yA. If there exists 0<L<1 such that

φ(x,0,,0) L m φ(mx,0,,0)

for all xA, then there exists a unique homomorphism H:AB such that

f ( x ) H ( x ) B L m m L φ(x,0,,0)

for all xA.

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

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:AB 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,yA. If there exists 0<L<1 such that

φ(mx,0,,0)mLφ(x,0,,0)

for all xA, then there exists a unique derivation δ:AA such that

f ( x ) δ ( x ) B 1 m m L φ(x,0,,0)

for all xA.

Theorem 2.4 [23]

Let f:AB 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,yA. If there exists 0<L<1 such that

φ(mx,0,,0) L m φ(x,0,,0)

for all xA, then there exists a unique derivation δ:AA such that

f ( x ) δ ( x ) B L m m L φ(x,0,,0)

for all xA.

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:AB is called a Lie C -algebra homomorphism if H([x,y])=[H(x),H(y)] for all x,yA.

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:AB 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,yA. If there exists 0<L<1 such that

φ(mx,0,,0)mLφ(x,0,,0)

for all xA, then there exists a unique Lie C -algebra homomorphism H:AB such that

f ( x ) H ( x ) B 1 m m L φ(x,0,,0)
(3.5)

for all xA.

Proof By the same method as in the proof of Theorem 2.1, we can find the mapping H:AB given by

H(x)= lim n f ( m n x ) m n

for all xA. 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,yA, and so

H ( [ x , y ] ) = [ H ( x ) , H ( y ) ]

for all x,yA. Therefore, H:AB 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:AB 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,yA, then there exists a unique Lie C -algebra homomorphism H:AB such that

f ( x ) H ( x ) B θ m m r x A r

for all xA.

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,yA and putting L= m r 1 . □

Theorem 3.4 Let f:AB 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,yA. If there exists 0<L<1 such that

φ(x,0,,0) L m φ(x,0,,0)

for all xA, then there exists a unique Lie C -algebra homomorphism H:AB such that

f ( x ) H ( x ) B L m m L φ(x,0,,0)

for all xA.

Corollary 3.5 Let r>1 and θ be nonnegative real numbers. If f:AB 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,yA, then there exists a unique Lie C -algebra homomorphism H:AB such that

f ( x ) H ( x ) B θ m r m x A r

for all xA.

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,yA 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:AA is called a Lie derivation if δ([x,y])=[δ(x),y]+[x,δ(y)] for all x,yA.

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:AA 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,yA. If there exists 0<L<1 such that

φ(mx,0,,0)mLφ(x,0,,0)

for all xA, then there exists a unique Lie derivation δ:AA such that

f ( x ) δ ( x ) B 1 m m L φ(x,0,,0)
(4.5)

for all xA.

Proof By the same method as in the proof of Theorem 2.3, there exists a unique -linear mapping δ:AA satisfying (3.5). Also, we can find the mapping δ:AA given by

δ(x)= lim n f ( m n x ) m n
(4.6)

for all xA. 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,yA, and so

δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ]

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

Corollary 4.3 Let 0<r<1 and θ be nonnegative real numbers. If f:AA 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,yA, then there exists a unique derivation δ:AA such that

f ( x ) δ ( x ) A θ m m r x A r

for all xA.

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,yA and putting L= m r 1 . □

Theorem 4.4 Let f:AA 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,yA. If there exists 0<L<1 such that

φ(mx,0,,0) L m φ(x,0,,0)

for all xA, then there exists a unique Lie derivation δ:AA such that

f ( x ) δ ( x ) B L m m L φ(x,0,,0)

for all xA.

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:AA 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,yA, then there exists a unique Lie derivation δ:AA such that

f ( x ) δ ( x ) A θ m r m x A r

for all xA.

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,yA and putting L= m 1 r . □

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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).

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Correspondence to Young-Oh Yang.

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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