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Approximation of linear mappings in Banach modules over C -algebras

Abstract

Let X, Y be Banach modules over a C -algebra and let r 1 ,, r n R be given. Using fixed-point methods, we prove the stability of the following functional equation in Banach modules over a unital C -algebra:

j = 1 n f ( 1 2 1 i n , i j r i x i 1 2 r j x j ) + i = 1 n r i f( x i )=nf ( 1 2 i = 1 n r i x i ) .

As an application, we investigate homomorphisms in unital C -algebras.

MSC:39B72, 46L05, 47H10, 46B03, 47B48.

1 Introduction and preliminaries

We say a functional equation (ζ) is stable if any function g satisfying the equation (ζ) approximately is near to the true solution of (ζ). We say that a functional equation is superstable if every approximate solution is an exact solution of it (see [1]). The stability problem of functional equations was originated from a question of Ulam [2] concerning the stability of group homomorphisms. Hyers [3] gave a first affirmative partial answer to the question of Ulam in Banach spaces. Hyers’ theorem was generalized by Aoki [4] for additive mappings and by T.M. Rassias [5] for linear mappings by considering an unbounded Cauchy difference. A generalization of the T.M. Rassias theorem was obtained by Găvruta [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of T.M. Rassias’ approach.

The functional equation

f(x+y)+f(xy)=2f(x)+2f(y)

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [7] for mappings f:XY, where X is a normed space and Y is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [9] proved the Hyers-Ulam stability of the quadratic functional equation. J.M. Rassias [10, 11] introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic functional equation

f( a 1 x 1 + a 2 x 2 )+f( a 2 x 1 a 1 x 2 )= ( a 1 2 + a 2 2 ) [ f ( x 1 ) + f ( x 2 ) ] .
(1.1)

Grabiec [12] has generalized these results mentioned above.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1343]).

Let X be a set. A function d:X×X[0,] is called a generalized metric on X if d satisfies the following conditions:

  1. (1)

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

  2. (2)

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

  3. (3)

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

We recall a fundamental result in fixed-point theory.

Theorem 1.1 [44, 45]

Let (X,d) be a complete generalized metric space and let J:XX be a strictly contractive mapping with Lipschitz constant L<1. Then, for each given element xX, 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)< for all 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={yXd( J n 0 x,y)<};

  4. (4)

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

In 1996, Isac and T.M. Rassias [46] 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 [4758]).

Recently, Park and Park [59] introduced and investigated the following additive functional equation of Euler-Lagrange type:

(1.2)

whose solution is said to be a generalized additive mapping of Euler-Lagrange type.

In this paper, we introduce the following additive functional equation of Euler-Lagrange type which is somewhat different from (1.2):

j = 1 n f ( 1 2 1 i n , i j r i x i 1 2 r j x j ) + i = 1 n r i f( x i )=nf ( 1 2 i = 1 n r i x i ) ,
(1.3)

where r 1 ,, r n R. Every solution of the functional equation (1.3) is said to be a generalized Euler-Lagrange type additive mapping.

Using fixed-point methods, we investigate the Hyers-Ulam stability of the functional equation (1.3) in Banach modules over a C -algebra. These results are applied to investigate C -algebra homomorphisms in unital C -algebras. Also, ones can get the super-stability results after all theorems by putting the product of powers of norms as the control functions (see for more details [60, 61]).

Throughout this paper, assume that A is a unital C -algebra with the norm A and the unit e, B is a unital C -algebra with the norm B , and X, Y are left Banach modules over a unital C -algebra A with the norms X and Y , respectively. Let U(A) be the group of unitary elements in A and let r 1 ,, r n R.

2 Hyers-Ulam stability of the functional equation (1.3) in Banach modules over a C -algebra

For any given mapping f:XY, uU(A) and μC, we define D u , r 1 , , r n f and D μ , r 1 , , r n f: X n Y by

D u , r 1 , , r n f ( x 1 , , x n ) : = j = 1 n f ( 1 2 1 i n , i j r i u x i 1 2 r j u x j ) + i = 1 n r i u f ( x i ) n f ( 1 2 i = 1 n r i u x i )

and

D μ , r 1 , , r n f ( x 1 , , x n ) : = j = 1 n f ( 1 2 1 i n , i j μ r i x i 1 2 μ r j x j ) + i = 1 n μ r i f ( x i ) n f ( 1 2 i = 1 n μ r i x i )

for all x 1 ,, x n X.

Lemma 2.1 Let X and Y be linear spaces and let r 1 ,, r n be real numbers with k = 1 n r k 0 and r i 0, r j 0 for some 1i<jn. Assume that a mapping L:XY satisfies the functional equation (1.3) for all x 1 ,, x n X. Then the mapping L is additive. Moreover, L( r k x)= r k L(x) for all xX and 1kn.

Proof One can find a complete proof at [62]. □

Lemma 2.2 Let X and Y be linear spaces and let r 1 ,, r n be real numbers with r i 0, r j 0 for some 1i<jn. Assume that a mapping L:XY with L(0)=0 satisfies the functional equation (1.3) for all x 1 ,, x n X. Then the mapping L is additive. Moreover, L( r k x)= r k L(x) for all xX and 1kn.

Proof One can find a complete proof at [62]. □

We investigate the Hyers-Ulam stability of a generalized Euler-Lagrange type additive mapping in Banach modules over a unital C -algebra. Throughout this paper, let r 1 ,, r n be real numbers such that r i 0, r j 0 for fixed 1i<jn.

Theorem 2.3 Let f:XY be a mapping satisfying f(0)=0 for which there is a function φ: X n [0,) such that

D e , r 1 , , r n f ( x 1 , , x n ) Y φ( x 1 ,, x n )
(2.1)

for all x 1 ,, x n X. Let

φ i j (x,y):=φ(0,,0, x i th ,0,,0, y j th ,0,,0)

for all x,yX and 1i<jn. If there exists 0<C<1 such that

φ(2 x 1 ,,2 x n )2Cφ( x 1 ,, x n )

for all x 1 ,, x n X, then there exists a unique generalized Euler-Lagrange type additive mapping L:XY such that

f ( x ) L ( x ) Y 1 4 4 C { φ i j ( 2 x r i , 2 x r j ) + 2 φ i j ( x r i , x r j ) + φ i j ( 2 x r i , 0 ) + 2 φ i j ( x r i , 0 ) + φ i j ( 0 , 2 x r j ) + 2 φ i j ( 0 , x r j ) }
(2.2)

for all xX. Moreover, L( r k x)= r k L(x) for all xX and 1kn.

Proof For each 1kn with ki,j, let x k =0 in (2.1). Then we get the following inequality:

f ( r i x i + r j x j 2 ) + f ( r i x i r j x j 2 ) 2 f ( r i x i + r j x j 2 ) + r i f ( x i ) + r j f ( x j ) Y φ ( 0 , , 0 , x i i th , 0 , , 0 , x j j th , 0 , , 0 )
(2.3)

for all x i , x j X. Letting x i =0 in (2.3), we get

f ( r j x j 2 ) f ( r j x j 2 ) + r j f ( x j ) Y φ i j (0, x j )
(2.4)

for all x j X. Similarly, letting x j =0 in (2.3), we get

f ( r i x i 2 ) f ( r i x i 2 ) + r i f ( x i ) Y φ i j ( x i ,0)
(2.5)

for all x i X. It follows from (2.3), (2.4) and (2.5) that

f ( r i x i + r j x j 2 ) + f ( r i x i r j x j 2 ) 2 f ( r i x i + r j x j 2 ) + f ( r i x i 2 ) + f ( r j x j 2 ) f ( r i x i 2 ) f ( r j x j 2 ) Y φ i j ( x i , x j ) + φ i j ( x i , 0 ) + φ i j ( 0 , x j )
(2.6)

for all x i , x j X. Replacing x i and x j by 2 x r i and 2 y r j in (2.6), we get

f ( x + y ) + f ( x y ) 2 f ( x + y ) + f ( x ) + f ( y ) f ( x ) f ( y ) Y φ i j ( 2 x r i , 2 y r j ) + φ i j ( 2 x r i , 0 ) + φ i j ( 0 , 2 y r j )
(2.7)

for all x,yX. Putting y=x in (2.7), we get

2 f ( x ) 2 f ( x ) 2 f ( 2 x ) Y φ i j ( 2 x r i , 2 x r j ) + φ i j ( 2 x r i , 0 ) + φ i j ( 0 , 2 x r j )
(2.8)

for all xX. Replacing x and y by x 2 and x 2 in (2.7), respectively, we get

f ( x ) + f ( x ) Y φ i j ( x r i , x r j ) + φ i j ( x r i , 0 ) + φ i j ( 0 , x r j )
(2.9)

for all xX. It follows from (2.8) and (2.9) that

1 2 f ( 2 x ) f ( x ) Y 1 4 ψ(x)
(2.10)

for all xX, where

ψ ( x ) : = φ i j ( 2 x r i , 2 x r j ) + 2 φ i j ( x r i , x r j ) + φ i j ( 2 x r i , 0 ) + 2 φ i j ( x r i , 0 ) + φ i j ( 0 , 2 x r j ) + 2 φ i j ( 0 , x r j ) .

Consider the set W:={g:XY} and introduce the generalized metric on W:

d(g,h)=inf { C R + : g ( x ) h ( x ) Y C ψ ( x ) , x X } .

It is easy to show that (W,d) is complete.

Now, we consider the linear mapping J:WW such that

Jg(x):= 1 2 g(2x)
(2.11)

for all xX. By Theorem 3.1 of [44], d(Jg,Jh)Cd(g,h) for all g,hW. Hence, d(f,Jf) 1 4 .

By Theorem 1.1, there exists a mapping L:XY such that

  1. (1)

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

    L(2x)=2L(x)
    (2.12)

for all xX. The mapping L is a unique fixed point of J in the set

Z= { g W : d ( f , g ) < } .

This implies that L is a unique mapping satisfying (2.12) such that there exists C(0,) satisfying

L ( x ) f ( x ) Y Cψ(x)

for all xX.

  1. (2)

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

    lim n f ( 2 n x ) 2 n =L(x)

for all xX.

  1. (3)

    d(f,L) 1 1 C d(f,Jf), which implies the inequality d(f,L) 1 4 4 C . This implies that the inequality (2.2) holds.

Since φ(2 x 1 ,,2 x n )2Cφ( x 1 ,, x n ), it follows that

D e , r 1 , , r n L ( x 1 , , x n ) Y = lim k 1 2 k D e , r 1 , , r n f ( 2 k x 1 , , 2 k x n ) Y lim k 1 2 k φ ( 2 k x 1 , , 2 k x n ) lim k C k φ ( x 1 , , x n ) = 0

for all x 1 ,, x n X. Therefore, the mapping L:XY satisfies the equation (1.3) and L(0)=0. Hence, by Lemma 2.2, L is a generalized Euler-Lagrange type additive mapping and L( r k x)= r k L(x) for all xX and 1kn. This completes the proof. □

Theorem 2.4 Let f:XY be a mapping satisfying f(0)=0 for which there is a function φ: X n [0,) satisfying

D u , r 1 , , r n f ( x 1 , , x n ) φ( x 1 ,, x n )
(2.13)

for all x 1 ,, x n X and uU(A). If there exists 0<C<1 such that

φ(2 x 1 ,,2 x n )2Cφ( x 1 ,, x n )

for all x 1 ,, x n X, then there exists a unique A-linear generalized Euler-Lagrange type additive mapping L:XY satisfying (2.2) for all xX. Moreover, L( r k x)= r k L(x) for all xX and 1kn.

Proof By Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping L:XY satisfying (2.2), and moreover L( r k x)= r k L(x) for all xX and 1kn. By the assumption, for each uU(A), we get

D u , r 1 , , r n L ( 0 , , 0 , x i th , 0 , , 0 ) Y = lim k 1 2 k D u , r 1 , , r n f ( 0 , , 0 , 2 k x i th , 0 , , 0 ) Y lim k 1 2 k φ ( 0 , , 0 , 2 k x i th , 0 , , 0 ) lim k C k φ ( 0 , , 0 , x i th , 0 , , 0 ) = 0

for all xX. So, we have

r i uL(x)=L( r i ux)

for all uU(A) and xX. Since L( r i x)= r i L(x) for all xX and r i 0,

L(ux)=uL(x)

for all uU(A) and xX. By the same reasoning as in the proofs of [63] and [64],

L(ax+by)=L(ax)+L(by)=aL(x)+bL(y)

for all a,bA (a,b0) and x,yX. Since L(0x)=0=0L(x) for all xX, the unique generalized Euler-Lagrange type additive mapping L:XY is an A-linear mapping. This completes the proof. □

Theorem 2.5 Let f:XY be a mapping satisfying f(0)=0 for which there is a function φ: X n [0,) such that

D e , r 1 , , r n f ( x 1 , , x n ) Y φ( x 1 ,, x n )
(2.14)

for all x 1 ,, x n X. If there exists 0<C<1 such that

φ( x 1 ,, 2 n ) C 2 φ(2 x 1 ,,2 x n )

for all x 1 ,, x n X, then there exists a unique generalized Euler-Lagrange type additive mapping L:XY such that

f ( x ) L ( x ) Y C 4 4 C { φ i j ( 2 x r i , 2 x r j ) + 2 φ i j ( x r i , x r j ) + φ i j ( 2 x r i , 0 ) + 2 φ i j ( x r i , 0 ) + φ i j ( 0 , 2 x r j ) + 2 φ i j ( 0 , x r j ) }
(2.15)

for all xX, where φ i j is defined in the statement of Theorem  2.3. Moreover, L( r k x)= r k L(x) for all xX and 1kn.

Proof It follows from (2.10) that

f ( x ) f ( x 2 ) Y 1 2 ψ ( x 2 ) C 4 ψ(x)

for all xX, where ψ is defined in the proof of Theorem 2.3. The rest of the proof is similar to the proof of Theorem 2.3. □

Theorem 2.6 Let f:XY be a mapping with f(0)=0 for which there is a function φ: X n [0,) satisfying

D u , r 1 , , r n f ( x 1 , , x n ) φ( x 1 ,, x n )
(2.16)

for all x 1 ,, x n X and uU(A). If there exists 0<C<1 such that

φ( x 1 ,, 2 n ) C 2 φ(2 x 1 ,,2 x n )

for all x 1 ,, x n X, then there exists a unique A-linear generalized Euler-Lagrange type additive mapping L:XY satisfying (2.15) for all xX. Moreover, L( r k x)= r k L(x) for all xX and all 1kn.

Proof The proof is similar to the proof of Theorem 2.4. □

Remark 2.7 In Theorems 2.5 and 2.6, one can assume that k = 1 n r k 0 instead of f(0)=0.

3 Homomorphisms in unital C -algebras

In this section, we investigate C -algebra homomorphisms in unital C -algebras. We use the following lemma in the proof of the next theorem.

Lemma 3.1 [64]

Let f:AB be an additive mapping such that f(μx)=μf(x) for all xA and μ S 1 n o 1 :={ e i θ ;0θ2π n o }. Then the mapping f:AB is -linear.

Note 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.

Theorem 3.2 Let f:AB be a mapping with f(0)=0 for which there is a function φ: A n [0,) satisfying

(3.1)
(3.2)
(3.3)

for all x, x 1 ,, x n A, uU(A), kN and μ S 1 . If there exists 0<C<1 such that

φ(2 x 1 ,,2 x n )2Cφ( x 1 ,, x n )

for all x 1 ,, x n A, then the mapping f:AB is a C -algebra homomorphism.

Proof Since |J|3, letting μ=1 and x k =0 for all 1kn (ki,j) in (3.1), we get

f ( r i x i + r j x j 2 ) +f ( r i x i r j x j 2 ) + r i f( x i )+ r j f( x j )=2f ( r i x i + r j x j 2 )

for all x i , x j A. By the same reasoning as in the proof of Lemma 2.1, the mapping f is additive and f( r k x)= r k f(x) for all xA and k=i,j. So, by letting x i =x and x k =0 for all 1kn, ki, in (3.1), we get f(μx)=μf(x) for all xA and μ S 1 . Therefore, by Lemma 3.1, the mapping f is -linear. Hence, it follows from (3.2) and (3.3) that

f ( u ) f ( u ) B = lim k 1 2 k f ( 2 k u ) f ( 2 k u ) B lim k 1 2 k φ ( 2 k u , , 2 k u n times ) lim k C k φ ( u , , u n times ) = 0 , f ( u x ) f ( u ) f ( x ) B = lim k 1 2 k f ( 2 k u x ) f ( 2 k u ) f ( x ) B lim k 1 2 k φ ( 2 k u x , , 2 k u x n times ) lim k C k φ ( u x , , u x n times ) = 0

for all xA and uU(A). So, we have f( u )=f ( u ) and f(ux)=f(u)f(x) for all xA and uU(A). Since f is -linear and each xA is a finite linear combination of unitary elements (see [65]), i.e., x= k = 1 m λ k u k , where λ k C and u k U(A) for all 1kn, we have

f ( x ) = f ( k = 1 m λ ¯ k u k ) = k = 1 m λ ¯ k f ( u k ) = k = 1 m λ ¯ k f ( u k ) = ( k = 1 m λ k f ( u k ) ) = f ( k = 1 m λ k u k ) = f ( x ) , f ( x y ) = f ( k = 1 m λ k u k y ) = k = 1 m λ k f ( u k y ) = k = 1 m λ k f ( u k ) f ( y ) = f ( k = 1 m λ k u k ) f ( y ) = f ( x ) f ( y )

for all x,yA. Therefore, the mapping f:AB is a C -algebra homomorphism. This completes the proof. □

The following theorem is an alternative result of Theorem 3.2.

Theorem 3.3 Let f:AB be a mapping with f(0)=0 for which there is a function φ: A n [0,) satisfying

(3.4)
(3.5)

for all x, x 1 ,, x n A, uU(A), kN and μ S 1 . If there exists 0<C<1 such that

φ( x 1 ,, 2 n ) C 2 φ(2 x 1 ,,2 x n )

for all x 1 ,, x n A, then the mapping f:AB is a C -algebra homomorphism.

Remark 3.4 In Theorems 3.2 and 3.3, one can assume that k = 1 n r k 0 instead of f(0)=0.

Theorem 3.5 Let f:AB be a mapping with f(0)=0 for which there is a function φ: A n [0,) satisfying (3.2), (3.3) and

D μ , r 1 , , r n f ( x 1 , , x n ) B φ( x 1 ,, x n )
(3.6)

for all x 1 ,, x n A and μ S 1 . Assume that lim k 1 2 k f( 2 k e) is invertible. If there exists 0<C<1 such that

φ(2 x 1 ,,2 x n )2Cφ( x 1 ,, x n )

for all x 1 ,, x n A, then the mapping f:AB is a C -algebra homomorphism.

Proof Consider the C -algebras A and B as left Banach modules over the unital C -algebra . By Theorem 2.4, there exists a unique -linear generalized Euler-Lagrange type additive mapping H:AB defined by

H(x)= lim k 1 2 k f ( 2 k x )

for all xA. By (3.2) and (3.3), we get

H ( u ) H ( u ) B = lim k 1 2 k f ( 2 k u ) f ( 2 k u ) B lim k 1 2 k φ ( 2 k u , , 2 k u n times ) = 0 , H ( u x ) H ( u ) f ( x ) B = lim k 1 2 k f ( 2 k u x ) f ( 2 k u ) f ( x ) B lim k 1 2 k φ ( 2 k u x , , 2 k u x n times ) = 0

for all uU(A) and xA. So, we have H( u )=H ( u ) and H(ux)=H(u)f(x) for all uU(A) and xA. Therefore, by the additivity of H, we have

H(ux)= lim k 1 2 k H ( 2 k u x ) =H(u) lim k 1 2 k f ( 2 k x ) =H(u)H(x)
(3.7)

for all uU(A) and all xA. Since H is -linear and each xA is a finite linear combination of unitary elements, i.e., x= k = 1 m λ k u k , where λ k C and u k U(A) for all 1kn, it follows from (3.7) that

H ( x y ) = H ( k = 1 m λ k u k y ) = k = 1 m λ k H ( u k y ) = k = 1 m λ k H ( u k ) H ( y ) = H ( k = 1 m λ k u k ) H ( y ) = H ( x ) H ( y ) , H ( x ) = H ( k = 1 m λ ¯ k u k ) = k = 1 m λ ¯ k H ( u k ) = k = 1 m λ ¯ k H ( u k ) = ( k = 1 m λ k H ( u k ) ) = H ( k = 1 m λ k u k ) = H ( x )

for all x,yA. Since H(e)= lim k 1 2 k f( 2 k e) is invertible and

H(e)H(y)=H(ey)=H(e)f(y)

for all yA, it follows that H(y)=f(y) for all yA. Therefore, the mapping f:AB is a C -algebra homomorphism. This completes the proof. □

The following theorem is an alternative result of Theorem 3.5.

Theorem 3.6 Let f:AB be a mapping with f(0)=0 for which there is a function φ: A n [0,) satisfying (3.4), (3.5) and

D μ , r 1 , , r n f ( x 1 , , x n ) B φ( x 1 ,, x n )

for all x 1 ,, x n A and μ S 1 . Assume that lim k 2 k f( e 2 k ) is invertible. If there exists 0<C<1 such that

φ( x 1 ,, 2 n ) C 2 φ(2 x 1 ,,2 x n )

for all x 1 ,, x n A, then the mapping f:AB is a C -algebra homomorphism.

Remark 3.7 In Theorem 3.6, one can assume that k = 1 n r k 0 instead of f(0)=0.

Theorem 3.8 Let f:AB be a mapping with f(0)=0 for which there is a function φ: A n [0,) satisfying (3.2), (3.3) and

D μ , r 1 , , r n f ( x 1 , , x n ) B φ( x 1 ,, x n )
(3.8)

for all x 1 ,, x n A and μ=i,1. Assume that lim k 1 2 k f( 2 k e) is invertible and for each fixed xA the mapping tf(tx) is continuous in tR. If there exists 0<C<1 such that

φ(2 x 1 ,,2 x n )2Cφ( x 1 ,, x n )

for all x 1 ,, x n A, then the mapping f:AB is a C -algebra homomorphism.

Proof Put μ=1 in (3.8). By the same reasoning as in the proof of Theorem 2.3, there exists a unique generalized Euler-Lagrange type additive mapping H:AB defined by

H(x)= lim k f ( 2 k x ) 2 k

for all xA. By the same reasoning as in the proof of [58], the generalized Euler-Lagrange type additive mapping H:AB is -linear. By the same method as in the proof of Theorem 2.4, we have

D μ , r 1 , , r n H ( 0 , , 0 , x j th , 0 , , 0 ) Y = lim k 1 2 k D μ , r 1 , , r n f ( 0 , , 0 , 2 k x j th , 0 , , 0 ) Y lim k 1 2 k φ ( 0 , , 0 , 2 k x j th , 0 , , 0 ) = 0

for all xA and so

r j μH(x)=H( r j μx)

for all xA. Since H( r j x)= r j H(x) for all xX and r j 0,

H(μx)=μH(x)

for all xA and μ=i,1. For each λC, we have λ=s+it, where s,tR. Thus, it follows that

H ( λ x ) = H ( s x + i t x ) = s H ( x ) + t H ( i x ) = s H ( x ) + i t H ( x ) = ( s + i t ) H ( x ) = λ H ( x )

for all λC and xA and so

H(ζx+ηy)=H(ζx)+H(ηy)=ζH(x)+ηH(y)

for all ζ,ηC and x,yA. Hence, the generalized Euler-Lagrange type additive mapping H:AB is -linear.

The rest of the proof is the same as in the proof of Theorem 3.5. This completes the proof. □

The following theorem is an alternative result of Theorem 3.8.

Theorem 3.9 Let f:AB be a mapping with f(0)=0 for which there is a function φ: A n [0,) satisfying (3.4), (3.5) and

D μ , r 1 , , r n f ( x 1 , , x n ) B φ( x 1 ,, x n ),

for all x, x 1 ,, x n A and μ=i,1. Assume that lim k 2 k f( e 2 k ) is invertible and for each fixed xA the mapping tf(tx) is continuous in tR. If there exists 0<C<1 such that

φ( x 1 ,, 2 n ) C 2 φ(2 x 1 ,,2 x n )

for all x 1 ,, x n A, then the mapping f:AB is a C -algebra homomorphism.

Proof We omit the proof because it is very similar to last theorem. □

Remark 3.10 In Theorem 3.9, one can assume that k = 1 n r k 0 instead of f(0)=0.

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Acknowledgements

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

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Correspondence to Yeol Je Cho or Reza Saadati.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors 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.

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Keywords

  • fixed point
  • Hyers-Ulam stability
  • super-stability
  • generalized Euler-Lagrange type additive mapping
  • homomorphism
  • C -algebra