Open Access

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

Journal of Inequalities and Applications20122012:251

https://doi.org/10.1186/1029-242X-2012-251

Received: 18 April 2012

Accepted: 15 October 2012

Published: 29 October 2012

Abstract

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

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

Keywords

additive functional equationfixed pointnon-Archimedean random spacehomomorphism in C -algebras and Lie C -algebrasgeneralized Hyers-Ulam stabilityderivation on C -algebras and Lie C -algebras

1 Introduction and preliminaries

By a non-Archimedean field we mean a field K equipped with a function (valuation) | | from K into [ 0 , ) such that | r | = 0 if and only if r = 0 , | r s | = | r | | s | and | r + s | max { | r | , | s | } for all r , s K . Clearly, | 1 | = | 1 | = 1 and | n | 1 for all n N . By the trivial valuation we mean the mapping | | taking everything but 0 into 1 and | 0 | = 0 . Let X be a vector space over a field K with a non-Archimedean non-trivial valuation | | . A function : X [ 0 , ) is called a non-Archimedean norm if it satisfies the following conditions:
  1. (i)

    x = 0 if and only if x = 0 ;

     
  2. (ii)

    for any r K , x X , r x = | r | x ;

     
  3. (iii)
    the strong triangle inequality (ultrametric) holds; namely
    x + y max { x , y } ( x , y X ) .
     
Then ( X , ) is called a non-Archimedean normed space. From the fact that
x n x m max { x j + 1 x j : m j n 1 } ( n > m )

holds, a sequence { x n } is Cauchy if and only if { x n + 1 x n } converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean the one in which every Cauchy sequence is convergent.

For any nonzero rational number x, there exists a unique integer n x Z such that x = a b p n x , where a and b are integers not divisible by p. Then | x | p : = p n x defines a non-Archimedean norm on . The completion of with respect to the metric d ( x , y ) = | x y | p is denoted by Q p , which is called the p-adic number field.

A non-Archimedean Banach algebra is a complete non-Archimedean algebra A which satisfies a b a b for all a , b A . For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [1, 2].

If U is a non-Archimedean Banach algebra, then an involution on U is a mapping t t from U into U which satisfies
  1. (i)

    t = t for t U ;

     
  2. (ii)

    ( α s + β t ) = α ¯ s + β ¯ t ;

     
  3. (iii)

    ( s t ) = t s for s , t U .

     

If, in addition, t t = t 2 for t U , then U is a non-Archimedean C -algebra.

The stability problem of functional equations was originated from a question of Ulam [3] concerning the stability of group homomorphisms. Let ( G 1 , ) be a group and let ( G 2 , , d ) be a metric group (a metric which is defined on a set with a group property) 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 would say that the equation of a homomorphism H ( x y ) = H ( x ) H ( y ) is stable (see also [46]).

We recall a fundamental result in fixed point theory. Let Ω be a set. A function d : Ω × Ω [ 0 , ] is called a generalized metric on Ω if d satisfies
  1. (1)

    d ( x , y ) = 0 if and only if x = y ;

     
  2. (2)

    d ( x , y ) = d ( y , x ) for all x , y Ω ;

     
  3. (3)

    d ( x , z ) d ( x , y ) + d ( y , z ) for all x , y , z Ω .

     

Theorem 1.1 [7]

Let ( Ω , d ) be a complete generalized metric space, and let J : Ω Ω be a contractive mapping with the Lipschitz constant L < 1 . Then for each given element x Ω , 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. (1)

    d ( J n x , J n + 1 x ) < , n n 0 ;

     
  2. (2)

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

     
  3. (3)

    y is the unique fixed point of J in the set Γ = { y Ω d ( J n 0 x , y ) < } ;

     
  4. (4)

    d ( y , y ) 1 1 L d ( y , J y ) for all y Γ .

     
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in non-Archimedean random C -algebras and non-Archimedean random Lie C -algebras for the following additive functional equation (see [8]):
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 ) ( m N , m 2 ) .
(1.1)

2 Random spaces

In the section, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces as in [921]. Throughout this paper, Δ + is the space of distribution functions, that is, the space of all mappings such that F is left-continuous and non-decreasing on , F ( 0 ) = 0 and F ( + ) = 1 . D + is a subset of Δ + consisting of all functions F Δ + for which l F ( + ) = 1 , where l f ( x ) denotes the left limit of the function f at the point x, that is, l f ( x ) = lim t x f ( t ) . The space Δ + is partially ordered by the usual point-wise ordering of functions, i.e., F G if and only if F ( t ) G ( t ) for all t in . The maximal element for Δ + in this order is the distribution function ε 0 given by
ε 0 ( t ) = { 0 if  t 0 , 1 if  t > 0 .

Definition 2.1 [20]

A mapping T : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] is a continuous triangular norm (briefly, a continuous t-norm) if T satisfies the following conditions:
  1. (a)

    T is commutative and associative;

     
  2. (b)

    T is continuous;

     
  3. (c)

    T ( a , 1 ) = a for all a [ 0 , 1 ] ;

     
  4. (d)

    T ( a , b ) T ( c , d ) whenever a c and b d for all a , b , c , d [ 0 , 1 ] .

     

Typical examples of continuous t-norms are T P ( a , b ) = a b , T M ( a , b ) = min ( a , b ) and T L ( a , b ) = max ( a + b 1 , 0 ) (the Lukasiewicz t-norm).

Definition 2.2 [21]

A non-Archimedean random normed space (briefly, NA-RN-space) is a triple ( X , μ , T ) , where X is a vector space, T is a continuous t-norm, and μ is a mapping from X into D + such that the following conditions hold:

(RN1) μ x ( t ) = ε 0 ( t ) for all t > 0 if and only if x = 0 ;

(RN2) μ α x ( t ) = μ x ( t | α | ) for all x X , α 0 ;

(RN3) μ x + y ( t ) T ( μ x ( t ) , μ y ( t ) ) for all x , y X and all t 0 .

Every normed space ( X , ) defines a non-Archimedean random normed space ( X , μ , T M ) , where
μ x ( t ) = t t + x

for all t > 0 , and T M is the minimum t-norm. This space is called the induced random normed space.

Definition 2.3 [22]

A non-Archimedean random normed algebra ( X , μ , T , T ) is a non-Archimedean random normed space ( X , μ , T ) with an algebraic structure such that

(RN-4) μ x y ( t ) T ( μ x ( t ) , μ y ( t ) ) for all x , y X and all t > 0 , in which T is a continuous t-norm.

Every non-Archimedean normed algebra ( X , ) defines a non-Archimedean random normed algebra ( X , μ , T M ) , where
μ x ( t ) = t t + x
for all t > 0 iff
x y x y + t y + t x ( x , y X ; t > 0 ) .

This space is called an induced non-Archimedean random normed algebra.

Definition 2.4
  1. (1)

    Let ( X , μ , T M ) and ( Y , μ , T M ) be non-Archimedean random normed algebras. An -linear mapping f : X Y is called a homomorphism if f ( x y ) = f ( x ) f ( y ) for all x , y X .

     
  2. (2)

    An -linear mapping f : X X is called a derivation if f ( x y ) = f ( x ) y + x f ( y ) for all x , y X .

     
Definition 2.5 Let ( U , μ , T , T ) be a non-Archimedean random Banach algebra, then an involution on U is a mapping u u from U into U which satisfies
  1. (i)

    u = u for u U ;

     
  2. (ii)

    ( α u + β v ) = α ¯ u + β ¯ v ;

     
  3. (iii)

    ( u v ) = v u for u , v U .

     

If, in addition, μ u u ( t ) = T ( μ u ( t ) , μ u ( t ) ) for u U and t > 0 , then U is a non-Archimedean random C -algebra.

Definition 2.6 Let ( X , μ , T ) be an NA-RN-space.
  1. (1)

    A sequence { x n } in X is said to be convergent to x in X if, for every ϵ > 0 and λ > 0 , there exists a positive integer N such that μ x n x ( ϵ ) > 1 λ whenever n N .

     
  2. (2)

    A sequence { x n } in X is called a Cauchy sequence if, for every ϵ > 0 and λ > 0 , there exists a positive integer N such that μ x n x n + 1 ( ϵ ) > 1 λ whenever n m N .

     
  3. (3)

    An RN-space ( X , μ , T ) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.

     

3 Stability of homomorphisms and derivations in non-Archimedean random C -algebras

Throughout this section, assume that A is a non-Archimedean random C -algebra with the norm μ A and that is a non-Archimedean random C -algebra with the norm μ B .

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

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

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

We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean random C -algebras for the functional equation D λ f ( x 1 , , x m ) = 0 .

Theorem 3.1 Let f : A B be a mapping for which there are functions φ : A m D + , ψ : A 2 D + , and η : A D + such that | m | < 1 is far from zero and
(3.1)
(3.2)
(3.3)
for all λ T 1 , all x 1 , , x m , x , y A and t > 0 . If there exists an L < 1 such that
(3.4)
(3.5)
(3.6)
for all x , y , x 1 , , x m A and t > 0 , then there exists a unique random homomorphism H : A B such that
μ f ( x ) H ( x ) B ( t ) φ x , 0 , , 0 ( ( | m | | m | L ) t )
(3.7)

for all x A and t > 0 .

Proof It follows from (3.4), (3.5), (3.6), and L < 1 that
(3.8)
(3.9)
(3.10)

for all x , y , x 1 , , x m A and t > 0 .

Let us define Ω to be the set of all mappings g : A B and introduce a generalized metric on Ω as follows:
d ( g , h ) = inf { k ( 0 , ) : μ g ( x ) h ( x ) B ( k t ) > ϕ x , 0 , , 0 ( t ) , x A , t > 0 } .

It is easy to show that ( Ω , d ) is a generalized complete metric space (see [23]).

Now, we consider the function J : Ω Ω defined by J g ( x ) = 1 m g ( m x ) for all x A and g Ω . Note that for all g , h Ω , we have
d ( g , h ) < k μ g ( x ) h ( x ) B ( k t ) > ϕ x , 0 , , 0 ( t ) μ 1 m g ( m x ) 1 m h ( m x ) B ( k t ) > | m | ϕ m x , 0 , , 0 ( | m | t ) μ 1 m g ( m x ) 1 m h ( m x ) B ( k L t ) > ϕ m x , 0 , , 0 ( t ) d ( J g , J h ) < k L .

From this it is easy to see that d ( J g , J k ) L d ( g , h ) for all g , h Ω , that is, J is a self-function of Ω with the Lipschitz constant L.

Putting μ = 1 , x = x 1 and x 2 = x 3 = = x m = 0 in (3.1), we have
μ f ( m x ) m f ( x ) B ( t ) ϕ x , 0 , , 0 ( t )
for all x A and t > 0 . Then
μ f ( x ) 1 m f ( m x ) B ( t ) ϕ x , 0 , , 0 ( | m | t )
for all x A and t > 0 , that is, d ( J f , f ) 1 | m | < . Now, from the fixed point alternative, it follows that there exists a fixed point H of J in Ω such that
H ( x ) = lim n 1 | m | n f ( m n x )
(3.11)

for all x A since lim n d ( J n f , H ) = 0 .

On the other hand, it follows from (3.1), (3.8), and (3.11) that
μ D λ H ( x 1 , , x m ) B ( t ) = lim n μ 1 m n D f ( m n x 1 , , m n x m ) B ( t ) lim n ϕ m n x 1 , , m n x m ( | m | n t ) = 1 .

By a similar method to the above, we get λ H ( m x ) = H ( m λ x ) for all λ T 1 and all x A . Thus, one can show that the mapping H : A B is -linear.

It follows from (3.2), (3.9), and (3.11) that
μ H ( x y ) H ( x ) H ( y ) B ( t ) = lim n μ f ( m 2 n x y ) f ( m n x ) f ( m n y ) B ( | m | 2 n t ) lim n ψ m n x , m n y ( | m | 2 n t ) = 1

for all x , y A . So, H ( x y ) = H ( x ) H ( y ) for all x , y A . Thus, H : A B is a homomorphism satisfying (3.7) as desired.

Also by (3.3), (3.10), (3.11) and by a similar method, we have H ( x ) = H ( x ) . □

Corollary 3.2 Let r > 1 and θ be nonnegative real numbers, and let f : A B be a mapping such that
for all λ T 1 , all x 1 , , x m , x , y A and t > 0 . Then there exists a unique homomorphism H : A B such that
μ f ( x ) H ( x ) B ( t ) ( | m | | m | r ) t ( | m | | m | r ) t + θ | m | | m | r x A r

for all x A and t > 0 .

Proof The proof follows from Theorem 3.1. By taking

for all x 1 , , x m , x , y A , L = | m | r 1 and t > 0 , we get the desired result. □

We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean random C -algebras for the functional equation D λ f ( x 1 , , x m ) = 0 .

Theorem 3.3 Let f : A A be a mapping for which there are functions φ : A m D + , ψ : A 2 D + , and η : A D + such that | m | < 1 is far from zero and
for all λ T 1 and all x 1 , , x m , x , y A and t > 0 . If there exists an L < 1 such that (3.4), (3.5), and (3.6) hold, then there exists a unique derivation δ : A A such that
μ f ( x ) δ ( x ) A ( t ) φ x , 0 , , 0 ( ( | m | | m | L ) t )

for all x A and t > 0 .

4 Stability of homomorphisms and derivations in non-Archimedean Lie C -algebras

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

on C , is called a Lie non-Archimedean random C -algebra.

Definition 4.1 Let A and be random Lie C -algebras. A -linear mapping H : A B is called a non-Archimedean 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 non-Archimedean random Lie C -algebra with the norm μ A and that is a non-Archimedean random Lie C -algebra with the norm μ B .

We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean random Lie C -algebras for the functional equation D λ f ( x 1 , , x m ) = 0 .

Theorem 4.2 Let f : A B be a mapping for which there are functions φ : A m D + and ψ : A 2 D + such that (3.1) and (3.3) hold and
μ f ( [ x , y ] ) [ f ( x ) , f ( y ) ] B ( t ) ψ x , y ( t )
(4.1)

for all λ T 1 , all x , y A and t > 0 . If there exists an L < 1 and (3.4), (3.5), and (3.6) hold, then there exists a unique homomorphism H : A B such that (3.7) holds.

Proof By the same reasoning as in the proof of Theorem 3.1, we can find the mapping H : A B given by
H ( x ) = lim n f ( m n x ) | m | n
(4.2)
for all x A . It follows from (3.5) and (4.2) that
μ H ( [ x , y ] ) [ H ( x ) , H ( y ) ] B ( t ) = lim n μ f ( m 2 n [ x , y ] ) [ f ( m n x ) , f ( m n y ) ] B ( | m | 2 n t ) lim n ψ m n x , m n y ( | m | 2 n t ) = 1
for all x , y A and t > 0 , then
H ( [ x , y ] ) = [ H ( x ) , H ( y ) ]

for all x , y A . Thus, H : A B is a Lie C -algebra homomorphism satisfying (3.7), as desired. □

Corollary 4.3 Let r > 1 and θ be nonnegative real numbers, and let f : A B be a mapping such that
for all λ T 1 , all x 1 , , x m , x , y A and t > 0 . Then there exists a unique homomorphism H : A B such that
μ f ( x ) H ( x ) B ( t ) ( | m | | m | r ) t ( | m | | m | r ) t + θ x A r

for all x A and t > 0 .

Proof The proof follows from Theorem 4.2 and a method similar to Corollary 3.2. □

Definition 4.4 Let A be a non-Archimedean random 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 .

We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean random Lie C -algebras for the functional equation D λ f ( x 1 , , x m ) = 0 .

Theorem 4.5 Let f : A A be a mapping for which there are functions φ : A m D + and ψ : A 2 D + such that (3.1) and (3.3) hold and
μ f ( [ x , y ] ) [ f ( x ) , y ] [ x , f ( y ) ] A ( t ) ψ x , y ( t ) ,
(4.3)

for all x , y A . If there exists an L < 1 and (3.4), (3.5), and (3.6) hold, then there exists a unique Lie derivation δ : A A such that (3.7) holds.

Proof By the same reasoning as the proof of Theorem 4.2, there exists a unique -linear mapping δ : A A satisfying (3.7); the mapping δ : A A is given by
δ ( x ) = lim n f ( m n x ) | m | n
(4.4)

for all x A .

It follows from (3.5) and (4.4) that
for all x , y A and t > 0 , then
δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ]

for all x , y A . Thus, δ : A A is a Lie derivation satisfying (3.7). □

Declarations

Authors’ Affiliations

(1)
National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, Yuseong-Gu, Daejeon, Korea
(2)
Department of Mathematics, Iran University Science and Technology, Tehran, Iran

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© Kang and Saadati; licensee Springer 2012

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