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Approximation of a generalized additive mapping in multi-Banach modules and isomorphisms in multi- C -algebras: a fixed-point approach

Abstract

Let X, Y be vector spaces. It is shown that if an odd mapping f:XY satisfies the functional equation

(0.1)

then the odd mapping f:XY is additive, and we use a fixed-point method to prove the Hyers-Ulam stability of the functional equation (0.1) in multi-Banach modules over a unital multi- C -algebra. As an application, we show that every almost linear bijection h:AB of a unital multi- C -algebra A onto a unital multi- C -algebra B is a C -algebra isomorphism when h( 2 n r n uy)=h( 2 n r n u)h(y) for all unitaries uU(A), all yA, and n=0,1,2, .

MSC:39B52, 46L05, 47H10, 47B48.

1 Introduction

Throughout this paper we assume that r is a positive rational number and d, l are integers with 1<l< d 2 .

Let X and Y be Banach spaces. Consider a mapping f:XY such that f(tx) is continuous in tR for each fixed xX, and assume that there exist constants θ0 and p[0,1) with

f ( x + y ) f ( x ) f ( y ) θ ( x p + y p ) ,x,yX.

Rassias [1] showed that there exists a unique R-linear mapping T:XY such that

f ( x ) T ( x ) 2 θ 2 2 p x p ,xX.

Găvruta [2] extended the above theorem as follows: let G be an Abelian group, Y be a Banach space and put

φ ˜ (x,y)= j = 0 1 2 j φ ( 2 j x , 2 j y ) <,x,yG.

If f:GY is a mapping satisfying

f ( x + y ) f ( x ) f ( y ) φ(x,y),x,yG,

then there exists a unique additive mapping T:GY such that

f ( x ) T ( x ) 1 2 φ ˜ (x,x),xG.

Park [3] applied Găvruta’s result to linear functional equations in Banach modules over a C -algebra. Several functional equations have been investigated in [4, 5] and [6]. In 2006 Baak, Boo and Rassias [7] solved the following functional equation:

(1.1)

(any solution of (1.1) will be called a generalized additive mapping) and proved its Hyers-Ulam stability in Banach modules over a unital C -algebra via the direct method. These results were applied to investigate C -algebra isomorphisms in unital C -algebras.

In this paper, we prove the Hyers-Ulam stability of the functional equation (1.1) in multi-Banach modules over a unital multi- C -algebra via the fixed-point method. These results are applied to investigate C -algebra isomorphisms in unital multi- C -algebras.

2 Fixed-point theorems

We recall two fundamental results in the fixed-point theory.

Theorem 2.1 [8, 9]

Let (X,d) be a complete metric space and let J:XX be strictly contractive, i.e.,

d(Jx,Jy)Ld(x,y),x,yX

for a Lipschitz constant L<1. Then

  1. (1)

    the mapping J has a unique fixed point x X,

  2. (2)

    the fixed point x is globally attractive, i.e.,

    lim n J n x= x ,xX,
  3. (3)

    the following inequalities hold:

    d ( J n x , x ) L n d ( x , x ) , d ( J n x , x ) 1 1 L d ( J n x , J n + 1 x ) , d ( x , x ) 1 1 L d ( x , J x )

for all xX and nonnegative integers n.

Let X be a non-empty set. A function d:X×X[0,] is called a generalized metric on X if for any x,y,zX, we have:

  1. (1)

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

  2. (2)

    d(x,y)=d(y,x),

  3. (3)

    d(x,z)d(x,y)+d(y,z).

Theorem 2.2 [8, 10]

Let (X,d) be a complete generalized metric space and let J:XX be a strictly contractive mapping with a Lipschitz constant L<1. Then, for each xX, either

d ( J n x , J n + 1 x ) =,n0

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 Y={yX:d( J n 0 x,y)<},

  4. (4)

    d(y, y ) 1 1 L d(y,Jy), yY.

3 Multi-normed spaces

The notion of a multi-normed space was introduced by Dales and Polyakov [11]. This concept is somewhat similar to an operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples are given in [1113].

Let (E,) be a complex normed space and kN. We denote by E k the linear space EE consisting of k-tuples ( x 1 ,, x k ), where x 1 ,, x k E. The linear operations on E k are defined coordinate-wise. The zero element of either E or E k is denoted by 0. Finally, we denote by N k the set {1,,k} and by Σ k the group of permutations on k symbols.

Definition 3.1 [11, 14] A multi-norm on { E k :kN} is a sequence

( k ) = ( k : k N )

such that k is a norm on E k for kN, 1 =, and for any integer k2, we have

(A1) ( x σ ( 1 ) , , x σ ( k ) ) k = ( x 1 , , x k ) k , σ Σ k , x 1 ,, x k E,

(A2) ( α 1 x 1 , , α k x k ) k ( max i N k | α i |) ( x 1 , , x k ) k , α 1 ,, α k C, x 1 ,, x k E,

(A3) ( x 1 , , x k 1 , 0 ) k = ( x 1 , , x k 1 ) k 1 , x 1 ,, x k 1 E,

(A4) ( x 1 , , x k 1 , x k 1 ) k = ( x 1 , , x k 1 ) k 1 , x 1 ,, x k 1 E.

A sequence (( E k , k ):kN) is then said to be a multi-normed space.

Lemma 3.2 [11, 13]

Suppose that (( E k , k ):kN) is a multi-normed space. Then for any kN, we have

  1. (a)

    ( x , , x ) k =x, xE,

  2. (b)

    max i N k x i ( x 1 , , x k ) k i = 1 k x i k max i N k x i , x 1 ,, x k E.

From Lemma 3.2(b), it follows that if (E,) is a Banach space, then ( E k , k ) is a Banach space for each kN (in this case we say that (( E k , k ):kN) is a multi-Banach space).

Now, we recall two important examples of multi-norms (see [11, 12]).

Example 3.3 The sequence ( k :kN) on { E k :kN} defined by

( x 1 , , x k ) k := max i N k x i , x 1 ,, x k E

is a multi-norm called the minimum multi-norm. The terminology ‘minimum’ is justified by property (b) from Lemma 3.2.

Example 3.4 Let {( k α :kN):αA} be a (non-empty) family of all multi-norms on { E k :kN}. For kN, set

( x 1 ,, x k ) k := sup α A ( x 1 , , x k ) k α , x 1 ,, x k E.

Then ( k :kN) is a multi-norm on { E k :kN} called the maximum multi-norm.

Lemma 3.5 [14]

Suppose that kN and ( x 1 ,, x k ) E k . For each j{1,,k}, let ( x n j ) n N be a sequence in E such that lim n x n j = x j . Then for each ( y 1 ,, y k ) E k we have

lim n ( x n 1 y 1 , , x n k y k ) =( x 1 y 1 ,, x k y k ).

Definition 3.6 [12, 14]

Let (( E k , k ):kN) be a multi-normed space. A sequence ( x n ) n N in E is said to be a multi-null sequence if for each ε>0, there exists an n 0 N such that

sup k N ( x n , , x n + k 1 ) k <ε,n n 0 .

We say that the sequence ( x n ) n N is multi-convergent to xE and write lim n x n =x if ( x n x ) n N is a multi-null sequence.

Definition 3.7 [11, 14]

Let (A,) be a normed algebra such that (( A k , k ):kN) is a multi-normed space. Then (( A k , k ):kN) is called a multi-normed algebra if

( a 1 b 1 , , a k b k ) k ( a 1 , , a k ) k ( b 1 , , b k ) k ,kN, a 1 ,, a k , b 1 ,, b k A.

The multi-normed algebra (( A k , k ):kN) is said to be a multi-Banach algebra if (( A k , k ):kN) is a multi-Banach space.

Example 3.8 Let p, q be such that 1pq< and A= p . The algebra A is a Banach sequence algebra with respect to coordinate-wise multiplication of sequences (see Example 4.1.42 of [15]). Let ( k :kN) be the standard (p,q)-multi-norm on { A k :kN} (see [11]). Then (( A k , k ):kN) is a multi-Banach algebra.

Definition 3.9 Let (( A k , k ):kN) be a multi-Banach algebra and assume that A is a (unital) C -algebra. If the involution satisfies

( a 1 a 1 , , a k a k ) k = ( a 1 , , a k ) k 2 ,kN, a 1 ,, a k A,

then (( A k , k ):kN) is called a (unital) multi- C -algebra.

Definition 3.10 Let (( A k , k ):kN) be a multi-Banach algebra and (( X k , k ):kN) be a multi-Banach space. Assume also that X is a Banach left module over A. We say that (( X k , k ):kN) is a multi-Banach left module over (( A k , k ):kN) if there is an M0 such that

( a 1 x 1 , , a k x k ) k M ( a 1 , , a k ) k ( x 1 , , x k ) k

for all kN, a 1 ,, a k A, x 1 ,, x k X.

4 Stability of an odd functional equation in multi-Banach modules over a multi- C -algebra

Throughout this section, we assume that (( A k , k ):kN) is a unital multi- C -algebra, and (( X k , k ):kN) and (( Y k , k ):kN) are multi-Banach left modules over (( A k , k ):kN). Moreover, by U(A) we denote the unitary group of A.

Lemma 4.1 [7]

Let X and Y be vector spaces. An odd mapping f:XY satisfies (1.1) for all x 1 ,, x d X if and only if f is additive.

Corollary 4.2 [7]

Let X and Y be vector spaces. An odd mapping f:XY satisfies

rf ( x + y r ) =f(x)+f(y),x,yX

if and only if f is additive.

Given a mapping f:XY, we set

D u f ( x 1 , , x d ) : = r f ( j = 1 d u x j r ) + ι ( j ) = 0 , 1 j = 1 d ι ( j ) = l r f ( j = 1 d ( 1 ) ι ( j ) u x j r ) ( d 1 C l d 1 C l 1 + 1 ) j = 1 d u f ( x j )

for all uU(A) and x 1 ,, x d X.

Theorem 4.3 Let r2 and f:XY be an odd mapping such that for every kN there is a function φ k : X k d [0,) with

(4.1)
(4.2)

for all uU(A) and x 11 ,, x 1 d ,, x k 1 ,, x k d X. If there exists an L<1 such that

φ k ( 2 r x 11 , 2 r x 11 , , 0 d , , 2 r x k 1 , 2 r x k 1 , , 0 d ) 2 r L φ k ( x 11 , x 11 , , 0 d , , x k 1 , x k 1 , , 0 d )

for all kN and x 11 ,, x k 1 X, then there is a unique A-linear generalized additive mapping L:XY with

(4.3)

for all kN and x 1 ,, x k X.

Proof Put

X:={L:XY}

and

d ( L , h ) = inf { C R + : ( L ( x 1 ) h ( x 1 ) , , L ( x k ) h ( x k ) ) k C φ k ( x 1 , x 1 , 0 , , 0 d , , x k , x k , 0 , , 0 d ) , k N , x 1 , , x k X }

for all L,hX. It is easy to show that (X,d) is a complete generalized metric space.

Define a mapping J:XX by

JL(x):= r 2 L ( 2 r x ) ,LX,xX.

Analysis similar to that in the proof of Theorem 3.1 in [8] (see also the proof of Lemma 3.2 in [12]) shows that

d(JL,Jh)Ld(L,h),L,hX.

Fix kN. Putting u=1U(A), x i 1 = x i 2 = x 1 , and x i 3 == x i d =0 for i{1,,k} in (4.2), we have

( r f ( 2 r x 1 ) 2 f ( x 1 ) , , r f ( 2 r x k ) 2 f ( x k ) ) k 1 C l d 2 d 2 C l 2 + 1 φ k ( x 1 , x 1 , 0 , , 0 d 2 times , , x k , x k , 0 , , 0 d 2 times ) ,

because f is odd and t: = d 2 C l d 2 C l 2 +1 = d 1 C l d 1 C l 1 +1. We thus get

( f ( x 1 ) r 2 f ( 2 r x 1 ) , , f ( x k ) r 2 f ( 2 r x k ) ) k 1 2 t φ k ( x 1 , x 1 , 0 , , 0 d 2 times , , x k , x k , 0 , , 0 d 2 times ) , x 1 , , x k X ,

and therefore,

d(f,Jf) 1 2 t .
(4.4)

Consequently, by Theorem 2.2, there exists a mapping L:XY such that

  1. (1)

    L is a fixed point of J, i.e.,

    L ( 2 r x ) = 2 r L(x),xX,
    (4.5)

and L is unique in the set

Y= { L X : d ( f , L ) < } .

This means that L is a unique mapping satisfying (4.5) such that there exists a C(0,) with

( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k C φ k ( x 1 , x 1 , 0 , , 0 d 2 times ,, x k , x k , 0 , , 0 d 2 times )

for all kN and x 1 ,, x k X.

  1. (2)

    d( J n f,L)0 as n. This implies the equality

    lim n r n 2 n f ( 2 n r n x ) =L(x)xX.
    (4.6)
  2. (3)

    d(f,L) 1 1 L d(f,Jf), which together with (4.4) gives

    d(f,L) 1 2 t 2 t L ,

and therefore, inequality (4.3) holds for all x 1 ,, x k X.

Next, note that the fact that the mapping f is odd and (4.6) imply that L is odd. Moreover, by (4.1) and (4.2), we get

( D 1 L ( x 11 , , x 1 d ) , , D 1 L ( x k 1 , , x k d ) ) k = lim n r n 2 n ( D 1 f ( 2 n r n x 11 , , 2 n r n x 1 d ) , , D 1 f ( 2 n r n x k 1 , , 2 n r n x k d ) ) k lim n r n 2 n φ k ( 2 n r n x 11 , , 2 n r n x 1 d , , 2 n r n x k 1 , , 2 n r n x k d ) = 0

for all kN and x 11 ,, x 1 d ,, x k 1 ,, x k d X, and therefore, L is a generalized additive mapping.

Fix uU(A) and xX. Using (4.1) and (4.2), we have

( D u L ( x , 0 , , 0 d 1 times ) , , D u L ( x , 0 , , 0 d 1 times ) k = lim n r n 2 n ( D u f ( 2 n r n x , 0 , , 0 d 1 times ) , , D u f ( 2 n r n x , 0 , , 0 d 1 times ) ) k lim n r n 2 n φ k ( 2 n r n x , 0 , , 0 d 1 times , , 2 n r n x , 0 , , 0 d 1 times ) = 0 ,

and consequently,

( d 1 C l d 1 C l 1 +1)rL ( u x r ) = ( d 1 C l d 1 C l 1 +1)uL(x).

Since L is a generalized additive mapping, from Lemma 4.1 it follows that L is additive, and therefore,

L(ux)=rL ( u x r ) =uL(x),uU(A),xX.

As in the proof of Theorem 3.1, in [7] one can now show that L is an A-linear mapping.  □

Corollary 4.4 Let r2 and θ,p(0,). Assume also that p>1 for r>2, and p<1 for r<2. If f:XY is an odd mapping such that

( D u f ( x 11 , , x 1 d ) , , D u f ( x k 1 , , x k d ) ) k θ ( j = 1 d x 1 j p + + j = 1 d x k j p )

for all uU(A), kN, and x 11 ,, x 1 d ,, x k 1 ,, x k d X, then there exists a unique A-linear generalized additive mapping L:XY with

( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k r p 1 θ ( r p 1 2 p 1 ) ( d 2 C l d 2 C l 2 + 1 ) ( x 1 p + + x k p )

for all kN and x 1 ,, x k X.

Proof Putting L= 2 p 1 r p 1 and

φ k ( x 11 , , x 1 d , , x k 1 , , x k d ) = θ ( j = 1 d x 1 j p + + j = 1 d x k j p ) ,

for all kN and x 11 ,, x 1 d ,, x k 1 ,, x k d X, in Theorem 4.3, we get the desired assertion. □

Theorem 4.5 Let r2. Let f:XY be an odd mapping for which there is a function φ: X k d [0,) such that

(4.7)

for all uU(A) and all x 11 ,, x 1 d ,, x k 1 ,, x k d X. If there exists an L<1 such that

φ ( r 2 x 11 , r 2 x 11 , , 0 d , r 2 x 21 , r 2 x 21 , , 0 d , , r 2 x k 1 , r 2 x k 1 , , 0 d ) r 2 L φ ( x 11 , x 11 , , 0 d , x 21 , x 21 , , 0 d , , x k 1 , x k 1 , , 0 d )

for all x 11 , x 21 ,, x k 1 X. Then there exists a unique A-linear generalized additive mapping L:XY such that

sup k N ( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k sup k N L 2 ( d 2 C l d 2 C l 2 + 1 ) ( 1 L ) φ ( x 1 , x 1 , 0 , , 0 d 2 times , , x k , x k , 0 , , 0 d 2 times )

for all x 1 ,, x k X.

Proof Note that f(0)=0 and f(x)=f(x) for all xX since f is an odd mapping. Let u=1U(A). Putting x i 1 = x i 2 = x 1 and x i 3 == x i m =0, 1ik in (4.7), we have

( r f ( 2 r x 1 ) 2 f ( x 1 ) , , r f ( 2 r x k ) 2 f ( x k ) ) k 1 C l d 2 d 2 C l 2 + 1 φ ( x 1 , x 1 , 0 , , 0 d 2 times , , x k , x k , 0 , , 0 d 2 times ) .

Letting t: = d 2 C l d 2 C l 2 +1, we get

( f ( x 1 ) 2 r f ( r 2 x 1 ) , , f ( x k ) 2 r f ( r 2 x k ) ) k 1 r t φ ( r 2 x 1 , r 2 x 1 , 0 , , 0 d 2 times , , r 2 x k , r 2 x k , 0 , , 0 d 2 times ) L 2 t φ ( x 1 , x 1 , 0 , , 0 d 2 times , , x k , x k , 0 , , 0 d 2 times )

for all x 1 ,, x k X.

The rest of the proof is similar to the proof of Theorem 4.3. □

Corollary 4.6 Let r<2, and let θ and p>1 be positive real numbers, or let r>2, and let θ and p<1 be positive real numbers. Let f:XY be an odd mapping such that

( D u f ( x 11 , , x 1 d ) , , D u f ( x k 1 , , x k d ) ) k θ ( j = 1 d x 1 j p + + j = 1 d x k j p )

for all uU(A) and all x 11 ,, x 1 d ,, x k 1 ,, x k d X. Then there exists a unique A-linear generalized additive mapping L:XY such that

sup k N ( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k sup k N r p 1 θ ( 2 p 1 r p 1 ) ( d 2 C l d 2 C l 2 + 1 ) ( x 1 p + + x k p )

for all xX.

Proof Define

φ( x 11 ,, x 1 d ,, x k 1 ,, x k d )=θ ( j = 1 d x 1 j p + + j = 1 d x k j p ) .

Putting L= r p 1 2 p 1 in Theorem 4.5, we get the desired result. □

Now we investigate the Hyers-Ulam stability of linear mappings for the case d=2.

Theorem 4.7 Let r2. Let f:XY be an odd mapping for which there is a function φ: X 2 k [0,) such that

(4.8)

for all uU(A) and all x 1 , x k , y 1 , y k X. If there exists an L<1 such that

φ ( 2 r x 1 , 2 r x 1 , 2 r x 2 , 2 r x 2 , , 2 r x k , 2 r x k ) 2 r Lφ( x 1 , x 1 , x 2 , x 2 ,, x k , x k )

for all x 1 ,, x k X. Then there exists a unique A-linear generalized additive mapping L:XY such that

sup k N ( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k sup k N L 2 ( 1 L ) φ ( x 1 , x 1 , , x k , x k )

for all x 1 ,, x k X.

Proof Let u=1U(A). Putting x=y in (4.8), we have

( r f ( 2 r x 1 ) 2 f ( x 1 ) , , r f ( 2 r x k ) 2 f ( x k ) ) k φ( x 1 , x 1 ,, x k , x k )

for all xX. So

( f ( x 1 ) r 2 f ( 2 r x 1 ) , , f ( x k ) r 2 f ( 2 r x k ) ) k 1 2 φ( x 1 , x 1 ,, x k , x k )

for all xX.

The rest of the proof is the same as in the proof of Theorem 4.3. □

Corollary 4.8 Let r>2, and let θ and p>1 be positive real numbers, or let r<2, and let θ and p<1 be positive real numbers. Let f:XY be an odd mapping such that

( r f ( u x 1 + u y 1 r ) u f ( x 1 ) u f ( y 1 ) , , r f ( u x k + u y k r ) u f ( x k ) u f ( y k ) ) k θ j = 1 k ( x j p + y j p )

for all uU(A) and for all x 1 ,, x k X. Then there exists a unique A-linear generalized additive mapping L:XY such that

sup k N ( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k sup k N r p 1 θ r p 1 2 p 1 j = 1 k x j p

for all x 1 ,, x k X.

Proof Define φ( x 1 , y 1 ,, x k , y k )=θ j = 1 k ( x j p + y j p ), and apply Theorem 4.7. Then we get the desired result. □

Theorem 4.9 Let r2. Let f:XY be an odd mapping for which there is a function φ: X 2 k [0,) such that

(4.9)

for all uU(A) and all x 1 ,, x k , y 1 ,, y k X. If there exists an L<1 such that

φ ( r 2 x 1 , r 2 x 1 , r 2 x 2 , r 2 x 2 , , r 2 x k , r 2 x k ) r 2 Lφ( x 1 , x 1 , x 2 , x 2 ,, x k , x k )

for all x 1 ,, x k X. Then there exists a unique A-linear generalized additive mapping L:XY such that

sup k N ( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k sup k N 1 2 ( 1 L ) φ( x 1 , x 1 ,, x k , x k )

for all x 1 ,, x k X.

Proof Let u=1U(A). Putting x=y in (4.9), we have

( r f ( 2 r x 1 ) 2 f ( x 1 ) , , r f ( 2 r x k ) 2 f ( x k ) ) k φ( x 1 , x 1 ,, x k , x k )

for all x 1 ,, x k X. So

( f ( x 1 ) 2 r f ( r 2 x 1 ) , , f ( x k ) 2 r f ( r 2 x k ) ) k 1 r φ ( r 2 x 1 , r 2 x 1 , , r 2 x k , r 2 x k ) 1 2 L φ ( x 1 , x 1 , , x k , x k )

for all x 1 ,, x k X.

The rest of the proof is similar to the proof of Theorem 4.3. □

Corollary 4.10 Let r>2, and let θ and p>1 be positive real numbers. Or let r<2, and let θ and p<1 be positive real numbers. Let f:XY be an odd mapping such that

( r f ( u x 1 + u y 1 r ) u f ( x 1 ) u f ( y 1 ) , , r f ( u x k + u y k r ) u f ( x k ) u f ( y k ) ) k θ j = 1 k ( x j p + y j p )

for all uU(A) and all x 1 ,, x k X. Then there exists a unique A-linear generalized additive mapping L:XY such that

sup k N ( L ( x 1 ) f ( x 1 ) , , L ( x k ) f ( x k ) ) k sup k N r p 1 θ 2 p 1 r p 1 j = 1 k x j p

for all x 1 ,, x k X.

Proof Define φ( x 1 , y 1 ,, x k , y k )=θ j = 1 k ( x j p + y j p ), and apply Theorem 4.9. Then we get the desired result. □

5 Isomorphisms in unital multi- C -algebras

Throughout this section, assume that A and B are unital multi- C -algebras with unit e. Let U(A) be the set of unitary elements in A.

We investigate C -algebra isomorphisms in unital multi- C -algebras.

Theorem 5.1 Let r2. Let h:AB be an odd bijective mapping satisfying h( 2 n r n uy)=h( 2 n r n u)h(y) for all uU(A), all yA, and n=0,1,2,, for which there exists a function φ: A k d [0,) such that

lim j r j 2 j φ ( 2 j r j x 11 , , 2 j r j x 1 d , , 2 j r j x k 1 , , 2 j r j x k d ) = 0 , ( D μ h ( x 11 , , x 1 d ) , , D μ h ( x k 1 , , x k d ) ) k φ ( x 11 , , x 1 d , , x k 1 , , x k d ) , ( h ( 2 n r n u 1 ) h ( 2 n r n u 1 ) , , h ( 2 n r n u k ) h ( 2 n r n u k ) ) k φ ( 2 n r n u 1 , , 2 n r n u 1 d times , , 2 n r n u k , , 2 n r n u k d times )

for all μ S 1 :={λC|λ|=1}, all u 1 ,, u k U(A), n=0,1,2,, and all x 11 ,, x k d A. Assume that lim n r n 2 n h( 2 n r n e) is invertible. Then the odd bijective mapping h:AB is a C -algebra isomorphism.

Proof Consider the multi- C -algebras A and B as left Banach modules over the unital multi- C -algebra C. By Theorem 4.3, there exists a unique C-linear generalized additive mapping H:AB such that

sup k N ( h ( x 1 ) H ( x 1 ) , , h ( x k ) H ( x k ) k sup k N 1 2 ( d 2 C l d 2 C l 2 + 1 ) φ ( x 1 , x 1 , 0 , , 0 d 2 times , , x k , x k , 0 , , 0 d 2 times )

for all x 1 ,, x k A in which H:AB is given by

H(x)= lim n r n 2 n h ( 2 n r n x )

for all xA.

The rest of the proof is similar to the proof of Theorem 4.1 of [7]. □

Corollary 5.2 Let r>2, and let θ and p>1 be positive real numbers. Or let r<2, and let θ and p<1 be positive real numbers. Let h:AB be an odd bijective mapping satisfying h( 2 n r n uy)=h( 2 n r n u)h(y) for all uU(A), all yA, and all n=0,1,2,, such that

( D μ h ( x 11 , , x 1 d ) , , D μ h ( x k 1 , , x k d ) ) k θ j = 1 d ( x 1 j p + + x k j p ) , ( h ( 2 n r n u 1 ) h ( 2 n r n u 1 ) , , h ( 2 n r n u k ) h ( 2 n r n u k ) ) k k d 2 p n r p n θ

for all μ S 1 , all uU(A), n=0,1,2,, and all x 11 ,, x k d A. Assume that lim n r n 2 n h( 2 n r n e) is invertible. Then the odd bijective mapping h:AB is a C -algebra isomorphism.

Proof Define φ( x 11 ,, x 1 d ,, x k 1 ,, x k d )=θ j = 1 d ( x 1 j p ++ x k j p ), and apply Theorem 5.1. Then we get the desired result. □

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The authors are grateful to the reviewers for their valuable comments and suggestions.

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Correspondence to Reza Saadati.

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All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.

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Park, C., Saadati, R. Approximation of a generalized additive mapping in multi-Banach modules and isomorphisms in multi- C -algebras: a fixed-point approach. Adv Differ Equ 2012, 162 (2012). https://doi.org/10.1186/1687-1847-2012-162

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Keywords

  • C -algebra isomorphism
  • fixed point
  • generalized additive functional equation
  • generalized Hyers-Ulam stability
  • multi-Banach module over multi- C -algebra