Open Access

An AQCQ-functional equation in matrix Banach spaces

Advances in Difference Equations20132013:146

https://doi.org/10.1186/1687-1847-2013-146

Received: 2 April 2013

Accepted: 8 May 2013

Published: 23 May 2013

Abstract

Using the fixed point method, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix normed spaces.

MSC:47L25, 47H10, 39B82, 46L07, 39B52.

Keywords

operator spacefixed pointHyers-Ulam stabilityadditive-quadratic-cubic-quartic functional equation

1 Introduction and preliminaries

The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces [1] implies that quotients, mapping spaces and various tensor products of operator spaces may again be regarded as operator spaces. Owing in part to this result, the theory of operator spaces is having an increasingly significant effect on operator algebra theory (see [2]).

The proof given in [1] appealed to the theory of ordered operator spaces [3]. Effros and Ruan [4] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [5] and Haagerup [6] (as modified in [7]).

The stability problem of functional equations originated from a question of Ulam [8] concerning the stability of group homomorphisms.

The functional equation
f ( x + y ) = f ( x ) + f ( y )

is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [9] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [10] for additive mappings and by TM Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the TM Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of TM Rassias’ approach.

In 1990, TM Rassias [13] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p 1 . In 1991, Gajda [14], following the same approach as in TM Rassias [11], gave an affirmative solution to this question for p > 1 . It was shown by Gajda [14], as well as by TM Rassias and Šemrl [15], that one cannot prove a TM Rassias’ type theorem when p = 1 (cf. the books of Czerwik [16], Hyers et al. [17]).

In 1982, JM Rassias [18] followed the innovative approach of the TM Rassias’ theorem [11] in which he replaced the factor x p + y p by x p y q for p , q R with p + q 1 .

The functional equation
f ( x + y ) + f ( x y ) = 2 f ( x ) + 2 f ( 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 [19] for mappings f : X Y , where X is a normed space and Y is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [21] proved the Hyers-Ulam stability of the quadratic functional equation.

In [22], Jun and Kim considered the following cubic functional equation:
f ( 2 x + y ) + f ( 2 x y ) = 2 f ( x + y ) + 2 f ( x y ) + 12 f ( x ) .
(1.1)

It is easy to show that the function f ( x ) = x 3 satisfies functional equation (1.1), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping.

In [23], Lee et al. considered the following quartic functional equation:
f ( 2 x + y ) + f ( 2 x y ) = 4 f ( x + y ) + 4 f ( x y ) + 24 f ( x ) 6 f ( y ) .
(1.2)

It is easy to show that the function f ( x ) = x 4 satisfies functional equation (1.2), which is called a quartic functional equation, and every solution of the quartic functional equation is said to be a quartic mapping. 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 [2433]).

We will use the following notations:

e j = ( 0 , , 0 , 1 , 0 , , 0 ) ;

E i j is that ( i , j ) -component is 1 and the other components are zero;

E i j x is that ( i , j ) -component is x and the other components are zero;

For x M n ( X ) , y M k ( X ) ,
x y = ( x 0 0 y ) .

Note that ( X , { n } ) is a matrix normed space if and only if ( M n ( X ) , n ) is a normed space for each positive integer n and A x B k A B x n holds for A M k , n , x = ( x i j ) M n ( X ) and B M n , k , and that ( X , { n } ) is a matrix Banach space if and only if X is a Banach space and ( X , { n } ) is a matrix normed space.

Let E, F be vector spaces. For a given mapping h : E F and a given positive integer n, define h n : M n ( E ) M n ( F ) by
h n ( [ x i j ] ) = [ h ( x i j ) ]

for all [ x i j ] M n ( E ) .

Let X be a set. A function d : X × X [ 0 , ] is called a generalized metric on X 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 X ;

     
  3. (3)

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

     

We recall a fundamental result in fixed point theory.

Theorem 1.1 [34, 35]

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

     
  4. (4)

    d ( y , y ) 1 1 α d ( y , J y ) for all y Y .

     

In 1996, Isac and Rassias [36] 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 [3743]).

In this paper, we prove the Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation:
f ( x + 2 y ) + f ( x 2 y ) = 4 f ( x + y ) + 4 f ( x y ) 6 f ( x ) + f ( 2 y ) + f ( 2 y ) 4 f ( y ) 4 f ( y )
(1.3)

in matrix normed spaces by using the fixed point method.

One can easily show that an odd mapping f : X Y satisfies (1.3) if and only if the odd mapping f : X Y is an additive-cubic mapping, i.e.,
f ( x + 2 y ) + f ( x 2 y ) = 4 f ( x + y ) + 4 f ( x y ) 6 f ( x ) .

It was shown in [[44], Lemma 2.2] that g ( x ) : = f ( 2 x ) 2 f ( x ) and h ( x ) : = f ( 2 x ) 8 f ( x ) are cubic and additive, respectively, and that f ( x ) = 1 6 g ( x ) 1 6 h ( x ) .

One can easily show that an even mapping f : X Y satisfies (1.3) if and only if the even mapping f : X Y is a quadratic-quartic mapping, i.e.,
f ( x + 2 y ) + f ( x 2 y ) = 4 f ( x + y ) + 4 f ( x y ) 6 f ( x ) + 2 f ( 2 y ) 8 f ( y ) .

It was shown in [[45], Lemma 2.1] that g ( x ) : = f ( 2 x ) 4 f ( x ) and h ( x ) : = f ( 2 x ) 16 f ( x ) are quartic and quadratic, respectively, and that f ( x ) = 1 12 g ( x ) 1 12 h ( x ) .

Throughout this paper, let ( X , { n } ) be a matrix normed space and ( Y , { n } ) be a matrix Banach space.

2 Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces: odd mapping case

In this section, we prove the Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces for an odd mapping case.

Lemma 2.1 Let ( X , { n } ) be a matrix normed space. Then:
  1. (1)

    E k l x n = x for x X .

     
  2. (2)

    x k l [ x i j ] n i , j = 1 n x i j for [ x i j ] M n ( X ) .

     
  3. (3)

    lim n x n = x if and only if lim n x i j n = x i j for x n = [ x i j n ] , x = [ x i j ] M k ( X ) .

     
Proof (1) Since E k l x = e k x e l and e k = e l = 1 , E k l x n x . Since e k ( E k l x ) e l = x , x E k l x n . So E k l x n = x .
  1. (2)
    Since e k x e l = x k l and e k = e l = 1 , x k l [ x i j ] n . Since [ x i j ] = i , j = 1 n E i j x i j ,
    [ x i j ] n = i , j = 1 n E i j x i j n i , j = 1 n E i j x i j n = i , j = 1 n x i j .
     
  2. (3)
    By
    x k l n x k l [ x i j n x i j ] n = [ x i j n ] [ x i j ] n i , j = 1 n x i j n x i j ,
     

we get the result. □

For a mapping f : X Y , define D f : X 2 Y and D f n : M n ( X 2 ) M n ( Y ) by
D f ( a , b ) : = f ( a + 2 b ) + f ( a 2 b ) 4 f ( a + b ) 4 f ( a b ) + 6 f ( a ) f ( 2 b ) f ( 2 b ) + 4 f ( b ) + 4 f ( b ) , D f n ( [ x i j ] , [ y i j ] ) : = f n ( [ x i j ] + 2 [ y i j ] ) + f n ( [ x i j ] 2 [ y i j ] ) 4 f n ( [ x i j ] + [ y i j ] ) 4 f n ( [ x i j ] [ y i j ] ) + 6 f n ( [ x i j ] ) f n ( 2 [ y i j ] ) f n ( 2 [ y i j ] ) + 4 f n ( [ y i j ] ) + 4 f n ( [ y i j ] )

for all a , b X and all x = [ x i j ] , y = [ y i j ] M n ( X ) .

Theorem 2.2 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) 2 α φ ( a 2 , b 2 )
(2.1)
for all a , b X . Let f : X Y be an odd mapping satisfying
D f n ( [ x i j ] , [ y i j ] ) n i , j = 1 n φ ( x i j , y i j )
(2.2)
for all x = [ x i j ] , y = [ y i j ] M n ( X ) . Then there exists a unique additive mapping A : X Y such that
f n ( 2 [ x i j ] ) 8 f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n 1 1 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )
(2.3)

for all x = [ x i j ] M n ( X ) .

Proof Let x i j = 0 and y i j = 0 except for ( i , j ) = ( s , t ) in (2.2).

Putting y s t = x s t in (2.2), we get
f ( 3 y s t ) 4 f ( 2 y s t ) + 5 f ( y s t ) φ ( y s t , y s t )
(2.4)

for all y s t X .

Replacing x s t by 2 y s t in (2.2), we get
f ( 4 y s t ) 4 f ( 3 y s t ) + 6 f ( 2 y s t ) 4 f ( y s t ) φ ( 2 y s t , y s t )
(2.5)

for all y s t X .

By (2.4) and (2.5),
f ( 4 y s t ) 10 f ( 2 y s t ) + 16 f ( y s t ) 4 ( f ( 3 y s t ) 4 f ( 2 y s t ) + 5 f ( y s t ) ) + f ( 4 y s t ) 4 f ( 3 y s t ) + 6 f ( 2 y s t ) 4 f ( y s t ) = 4 f ( 3 y s t ) 4 f ( 2 y s t ) + 5 f ( y s t ) + f ( 4 y s t ) 4 f ( 3 y s t ) + 6 f ( 2 y s t ) 4 f ( y s t ) 4 φ ( y s t , y s t ) + φ ( 2 y s t , y s t )
(2.6)
for all y s t X . Replacing y s t by x s t and letting g ( x s t ) : = f ( 2 x s t ) 8 f ( x s t ) in (2.6), we get
g ( 2 x s t ) 2 g ( x s t ) 4 φ ( x s t , x s t ) + φ ( 2 x s t , x s t )
for all x s t X . So
g ( x s t ) 1 2 g ( 2 x s t ) 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t )
(2.7)

for all x s t X .

Consider the set
S : = { h : X Y }
and introduce the generalized metric on S:
d ( g , h ) = inf { μ R + : g ( a ) h ( a ) μ ( 2 φ ( a , a ) + 1 2 φ ( 2 a , a ) ) , a X } ,

where, as usual, inf ϕ = + . It is easy to show that ( S , d ) is complete (see [46, 47]).

Now we consider the linear mapping J : S S such that
J g ( a ) : = 1 2 g ( 2 a )

for all a X .

Let g , h S be given such that d ( g , h ) = ε . Then
g ( a ) h ( a ) 2 φ ( a , a ) + 1 2 φ ( 2 a , a )
for all a X . Hence
J g ( a ) J h ( a ) = 1 2 g ( 2 a ) 1 2 h ( 2 a ) α ( 2 φ ( a , a ) + 1 2 φ ( 2 a , a ) )
for all a X . So d ( g , h ) = ε implies that d ( J g , J h ) α ε . This means that
d ( J g , J h ) α d ( g , h )

for all g , h S .

It follows from (2.7) that d ( g , J g ) 1 .

By Theorem 1.1, there exists a mapping A : X Y satisfying the following:
  1. (1)
    A is a fixed point of J, i.e.,
    A ( 2 a ) = 2 A ( a )
    (2.8)
     
for all a X . The mapping A is a unique fixed point of J in the set
M = { g S : d ( h , g ) < } .
This implies that A is a unique mapping satisfying (2.8) such that there exists a μ ( 0 , ) satisfying
g ( a ) A ( a ) μ ( 2 φ ( a , a ) + 1 2 φ ( 2 a , a ) )
for all a X ;
  1. (2)
    d ( J l g , A ) 0 as l . This implies the equality
    lim l 1 2 l g ( 2 l a ) = A ( a )
     
for all a X ;
  1. (3)
    d ( g , A ) 1 1 α d ( g , J g ) , which implies the inequality
    d ( g , A ) 1 1 α .
     
So
g ( a ) A ( a ) 1 1 α ( 2 φ ( a , a ) + 1 2 φ ( 2 a , a ) )
(2.9)

for all a X .

It follows from (2.1) and (2.2) that
D A ( a , b ) = lim l 1 2 l D g ( 2 l a , 2 l b ) lim l 1 2 l ( φ ( 2 l + 1 a , 2 l + 1 b ) + 8 φ ( 2 l a , 2 l b ) ) lim l 2 l α l 2 l ( φ ( 2 a , 2 b ) + 8 φ ( a , b ) ) = 0

for all a , b X . Hence D A ( a , b ) = 0 for all a, b. So A : X Y is additive.

By Lemma 2.1 and (2.9),
f n ( 2 [ x i j ] ) 8 f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n f ( 2 x i j ) 8 f ( x i j ) A ( x i j ) i , j = 1 n 1 1 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) . Thus A : X Y is a unique additive mapping satisfying (2.3), as desired. □

Corollary 2.3 Let r, θ be positive real numbers with r < 1 . Let f : X Y be an odd mapping such that
D f n ( [ x i j ] , [ y i j ] ) n i , j = 1 n θ ( x i j r + y i j r )
(2.10)
for all x = [ x i j ] , y = [ y i j ] M n ( X ) . Then there exists a unique additive mapping A : X Y such that
f n ( 2 [ x i j ] ) 8 f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n 9 + 2 r 2 2 r θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 2.2 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 r 1 and we get the desired result. □

Theorem 2.4 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) α 2 φ ( 2 a , 2 b )
for all a , b X . Let f : X Y be an odd mapping satisfying (2.2). Then there exists a unique additive mapping A : X Y such that
f n ( 2 [ x i j ] ) 8 f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n α 1 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

Now we consider the linear mapping J : S S such that
J g ( a ) : = 2 g ( a 2 )

for all a X .

It follows from (2.7) that
g ( x s t ) 2 g ( x s t 2 ) 4 φ ( x s t 2 , x s t 2 ) + φ ( x s t , x s t 2 ) α ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
for all x s t X . Thus d ( g , J g ) α . So
d ( g , A ) α 1 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.5 Let r, θ be positive real numbers with r > 1 . Let f : X Y be an odd mapping satisfying (2.10). Then there exists a unique additive mapping A : X Y such that
f n ( 2 [ x i j ] ) 8 f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n 2 r + 9 2 r 2 θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 2.4 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 1 r and we get the desired result. □

Theorem 2.6 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) 8 α φ ( a 2 , b 2 )
for all a , b X . Let f : X Y be an odd mapping satisfying (2.2). Then there exists a unique cubic mapping C : X Y such that
f n ( 2 [ x i j ] ) 2 f n ( [ x i j ] ) C n ( [ x i j ] ) n i , j = 1 n 1 4 4 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

Replacing y s t by x s t and letting g ( x s t ) : = f ( 2 x s t ) 2 f ( x s t ) in (2.6), we get
g ( 2 x s t ) 8 g ( x s t ) 4 φ ( x s t , x s t ) + φ ( 2 x s t , x s t )
for all x s t X . So
g ( x s t ) 1 8 g ( 2 x s t ) 1 4 ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
(2.11)
for all x s t X . Thus d ( g , J g ) 1 4 . So
d ( g , A ) 1 4 4 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.7 Let r, θ be positive real numbers with r < 3 . Let f : X Y be an odd mapping satisfying (2.10). Then there exists a unique cubic mapping C : X Y such that
f n ( 2 [ x i j ] ) 2 f n ( [ x i j ] ) C n ( [ x i j ] ) n i , j = 1 n 9 + 2 r 8 2 r θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 2.6 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 r 3 and we get the desired result. □

Theorem 2.8 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) α 8 φ ( 2 a , 2 b )
for all a , b X . Let f : X Y be an odd mapping satisfying (2.2). Then there exists a unique cubic mapping C : X Y such that
f n ( 2 [ x i j ] ) 2 f n ( [ x i j ] ) C n ( [ x i j ] ) n i , j = 1 n α 4 4 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (2.11) that
g ( x s t ) 8 g ( x s t 2 ) 4 φ ( x s t 2 , x s t 2 ) + φ ( x s t , x s t 2 ) α 4 ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
for all x s t X . Thus d ( g , J g ) α 4 . So
d ( g , A ) α 4 4 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 2.9 Let r, θ be positive real numbers with r > 3 . Let f : X Y be an odd mapping satisfying (2.10). Then there exists a unique cubic mapping C : X Y such that
f n ( 2 [ x i j ] ) 2 f n ( [ x i j ] ) C n ( [ x i j ] ) n i , j = 1 n 2 r + 9 2 r 8 θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 2.8 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 3 r and we get the desired result. □

3 Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces: even mapping case

In this section, we prove the Hyers-Ulam stability of AQCQ-functional equation (1.3) in matrix normed spaces for an even mapping case.

Theorem 3.1 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) 4 α φ ( a 2 , b 2 )
for all a , b X . Let f : X Y be an even mapping satisfying f ( 0 ) = 0 and (2.2). Then there exists a unique quadratic mapping Q : X Y such that
f n ( 2 [ x i j ] ) 16 f n ( [ x i j ] ) Q n ( [ x i j ] ) n i , j = 1 n 1 2 2 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

Let x i j = 0 and y i j = 0 except for ( i , j ) = ( s , t ) in (2.2).

Putting y s t = x s t in (2.2), we get
f ( 3 y s t ) 6 f ( 2 y s t ) + 15 f ( y s t ) φ ( y s t , y s t )
(3.1)

for all y s t X .

Replacing x s t by 2 y s t in (2.2), we get
f ( 4 y s t ) 4 f ( 3 y s t ) + 4 f ( 2 y s t ) + 4 f ( y s t ) φ ( 2 y s t , y s t )
(3.2)

for all y s t X .

By (3.1) and (3.2),
f ( 4 y s t ) 20 f ( 2 y s t ) + 64 f ( y s t ) 4 ( f ( 3 y s t ) 6 f ( 2 y s t ) + 15 f ( y s t ) ) + f ( 4 y s t ) 4 f ( 3 y s t ) + 4 f ( 2 y s t ) + 4 f ( y s t ) = 4 f ( 3 y s t ) 6 f ( 2 y s t ) + 15 f ( y s t ) + f ( 4 y s t ) 4 f ( 3 y s t ) + 4 f ( 2 y s t ) + 4 f ( y s t ) 4 φ ( y s t , y s t ) + φ ( 2 y s t , y s t )
(3.3)
for all y s t X . Replacing y s t by x s t and letting g ( x s t ) : = f ( 2 x s t ) 16 f ( x s t ) in (3.3), we get
g ( 2 x s t ) 4 g ( x s t ) 4 φ ( x s t , x s t ) + φ ( 2 x s t , x s t )
for all x s t X . So
g ( x s t ) 1 4 g ( 2 x s t ) 1 2 ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
(3.4)
for all x s t X . Thus d ( g , J g ) 1 2 . So
d ( g , A ) 1 2 2 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.2 Let r, θ be positive real numbers with r < 2 . Let f : X Y be an even mapping satisfying (2.10). Then there exists a unique quadratic mapping Q : X Y such that
f n ( 2 [ x i j ] ) 16 f n ( [ x i j ] ) Q n ( [ x i j ] ) n i , j = 1 n 9 + 2 r 4 2 r θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 3.1 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 r 2 and we get the desired result. □

Theorem 3.3 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) α 4 φ ( 2 a , 2 b )
for all a , b X . Let f : X Y be an even mapping satisfying f ( 0 ) = 0 and (2.2). Then there exists a unique quadratic mapping Q : X Y such that
f n ( 2 [ x i j ] ) 16 f n ( [ x i j ] ) Q n ( [ x i j ] ) n i , j = 1 n α 2 2 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (3.4) that
g ( x s t ) 4 g ( x s t 2 ) 4 φ ( x s t 2 , x s t 2 ) + φ ( x s t , x s t 2 ) α 2 ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
for all x s t X . Thus d ( g , J g ) α 2 . So
d ( g , A ) α 2 2 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.4 Let r, θ be positive real numbers with r > 2 . Let f : X Y be an even mapping satisfying (2.10). Then there exists a unique quadratic mapping Q : X Y such that
f n ( 2 [ x i j ] ) 16 f n ( [ x i j ] ) Q n ( [ x i j ] ) n i , j = 1 n 2 r + 9 2 r 4 θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 3.3 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 2 r and we get the desired result. □

Theorem 3.5 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) 16 α φ ( a 2 , b 2 )
for all a , b X . Let f : X Y be an even mapping satisfying f ( 0 ) = 0 and (2.2). Then there exists a unique quartic mapping R : X Y such that
f n ( 2 [ x i j ] ) 4 f n ( [ x i j ] ) R n ( [ x i j ] ) n i , j = 1 n 1 8 8 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

Replacing y s t by x s t and letting g ( x s t ) : = f ( 2 x s t ) 4 f ( x s t ) in (3.3), we get
g ( 2 x s t ) 16 g ( x s t ) 4 φ ( x s t , x s t ) + φ ( 2 x s t , x s t )
for all x s t X . So
g ( x s t ) 1 16 g ( 2 x s t ) 1 8 ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
(3.5)
for all x s t X . Thus d ( g , J g ) 1 8 . So
d ( g , A ) 1 8 8 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.6 Let r, θ be positive real numbers with r < 4 . Let f : X Y be an even mapping satisfying (2.10). Then there exists a unique quartic mapping R : X Y such that
f n ( 2 [ x i j ] ) 4 f n ( [ x i j ] ) R n ( [ x i j ] ) n i , j = 1 n 9 + 2 r 16 2 r θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 3.5 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 r 4 and we get the desired result. □

Theorem 3.7 Let φ : X 2 [ 0 , ) be a function such that there exists an α < 1 with
φ ( a , b ) α 16 φ ( 2 a , 2 b )
for all a , b X . Let f : X Y be an even mapping satisfying f ( 0 ) = 0 and (2.2). Then there exists a unique quartic mapping R : X Y such that
f n ( 2 [ x i j ] ) 4 f n ( [ x i j ] ) R n ( [ x i j ] ) n i , j = 1 n α 8 8 α ( 2 φ ( x i j , x i j ) + 1 2 φ ( 2 x i j , x i j ) )

for all x = [ x i j ] M n ( X ) .

Proof Let ( S , d ) be the generalized metric space defined in the proof of Theorem 2.2.

It follows from (3.5) that
g ( x s t ) 16 g ( x s t 2 ) 4 φ ( x s t 2 , x s t 2 ) + φ ( x s t , x s t 2 ) α 8 ( 2 φ ( x s t , x s t ) + 1 2 φ ( 2 x s t , x s t ) )
for all x s t X . Thus d ( g , J g ) α 8 . So
d ( g , A ) α 8 8 α .

The rest of the proof is similar to the proof of Theorem 2.2. □

Corollary 3.8 Let r, θ be positive real numbers with r > 4 . Let f : X Y be an even mapping satisfying (2.10). Then there exists a unique quartic mapping R : X Y such that
f n ( 2 [ x i j ] ) 4 f n ( [ x i j ] ) R n ( [ x i j ] ) n i , j = 1 n 2 r + 9 2 r 16 θ x i j r

for all x = [ x i j ] M n ( X ) .

Proof The proof follows from Theorem 3.7 by taking φ ( a , b ) = θ ( a r + b r ) for all a , b X . Then we can choose α = 2 4 r and we get the desired result. □

Declarations

Acknowledgements

CP was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299), and DYS was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792).

Authors’ Affiliations

(1)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University
(2)
Department of Mathematics, Daejin University
(3)
Department of Mathematics, University of Seoul

References

  1. Ruan ZJ:Subspaces of C -algebras. J. Funct. Anal. 1988, 76: 217-230. 10.1016/0022-1236(88)90057-2MathSciNetView ArticleMATHGoogle Scholar
  2. Effros E, Ruan ZJ: On approximation properties for operator spaces. Int. J. Math. 1990, 1: 163-187. 10.1142/S0129167X90000113MathSciNetView ArticleMATHGoogle Scholar
  3. Choi MD, Effros E: Injectivity and operator spaces. J. Funct. Anal. 1977, 24: 156-209. 10.1016/0022-1236(77)90052-0MathSciNetView ArticleMATHGoogle Scholar
  4. Effros E, Ruan ZJ: On the abstract characterization of operator spaces. Proc. Am. Math. Soc. 1993, 119: 579-584. 10.1090/S0002-9939-1993-1163332-4MathSciNetView ArticleMATHGoogle Scholar
  5. Pisier G:Grothendieck’s theorem for non-commutative C -algebras with an appendix on Grothendieck’s constants. J. Funct. Anal. 1978, 29: 397-415. 10.1016/0022-1236(78)90038-1MathSciNetView ArticleMATHGoogle Scholar
  6. Haagerup, U: Decomp. of completely bounded maps (unpublished manuscript)Google Scholar
  7. Effros E Contemp. Math. 62. In On Multilinear Completely Bounded Module Maps. Am. Math. Soc., Providence; 1987:479-501.Google Scholar
  8. Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.MATHGoogle Scholar
  9. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222-224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar
  10. Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64-66. 10.2969/jmsj/00210064View ArticleMathSciNetMATHGoogle Scholar
  11. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297-300. 10.1090/S0002-9939-1978-0507327-1View ArticleMathSciNetMATHGoogle Scholar
  12. Gǎvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431-436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar
  13. Rassias TM: Problem 16; 2. Report of the 27th international symp. on functional equations. Aequ. Math. 1990, 39: 292-293. 309Google Scholar
  14. Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431-434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
  15. Rassias TM, Šemrl P: On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 1992, 114: 989-993. 10.1090/S0002-9939-1992-1059634-1View ArticleMATHGoogle Scholar
  16. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.View ArticleMATHGoogle Scholar
  17. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.View ArticleMATHGoogle Scholar
  18. Rassias JM: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 1982, 46: 126-130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar
  19. Skof F: Proprietà locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53: 113-129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
  20. Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76-86. 10.1007/BF02192660MathSciNetView ArticleMATHGoogle Scholar
  21. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59-64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar
  22. Jun K, Kim H: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 2002, 274: 867-878. 10.1016/S0022-247X(02)00415-8MathSciNetView ArticleMATHGoogle Scholar
  23. Lee S, Im S, Hwang I: Quartic functional equations. J. Math. Anal. Appl. 2005, 307: 387-394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleMATHGoogle Scholar
  24. Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.View ArticleMATHGoogle Scholar
  25. Amyari M, Park C, Moslehian MS: Nearly ternary derivations. Taiwan. J. Math. 2007, 11: 1417-1424.MathSciNetMATHGoogle Scholar
  26. Chou CY, Tzeng JH:On approximate isomorphisms between Banach -algebras or C -algebras. Taiwan. J. Math. 2006, 10: 219-231.MathSciNetMATHGoogle Scholar
  27. Eshaghi Gordji M, Savadkouhi MB: Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces. Appl. Math. Lett. 2010, 23: 1198-1202. 10.1016/j.aml.2010.05.011MathSciNetView ArticleMATHGoogle Scholar
  28. Isac G, Rassias TM: On the Hyers-Ulam stability of ψ -additive mappings. J. Approx. Theory 1993, 72: 131-137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar
  29. Jun K, Lee Y: A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations. J. Math. Anal. Appl. 2004, 297: 70-86. 10.1016/j.jmaa.2004.04.009MathSciNetView ArticleMATHGoogle Scholar
  30. Jung S: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor; 2001.MATHGoogle Scholar
  31. Park C:Homomorphisms between Poisson J C -algebras. Bull. Braz. Math. Soc. 2005, 36: 79-97. 10.1007/s00574-005-0029-zMathSciNetView ArticleMATHGoogle Scholar
  32. Park C, Ghaleh SG, Ghasemi K: n -Jordan -homomorphisms in C -algebras. Taiwan. J. Math. 2012, 16: 1803-1814.MathSciNetMATHGoogle Scholar
  33. Rassias JM: Solution of a problem of Ulam. J. Approx. Theory 1989, 57: 268-273. 10.1016/0021-9045(89)90041-5MathSciNetView ArticleMATHGoogle Scholar
  34. Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2003., 4(1): Article ID 4MATHGoogle Scholar
  35. Diaz J, Margolis B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74: 305-309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar
  36. Isac G, Rassias TM: Stability of ψ -additive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci. 1996, 19: 219-228. 10.1155/S0161171296000324MathSciNetView ArticleMATHGoogle Scholar
  37. Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 2004, 346: 43-52.MathSciNetMATHGoogle Scholar
  38. Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008., 2008: Article ID 749392Google Scholar
  39. Jung Y, Chang I: The stability of a cubic type functional equation with the fixed point alternative. J. Math. Anal. Appl. 2005, 306: 752-760. 10.1016/j.jmaa.2004.10.017MathSciNetView ArticleMATHGoogle Scholar
  40. Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bull. Braz. Math. Soc. 2006, 37: 361-376. 10.1007/s00574-006-0016-zMathSciNetView ArticleMATHGoogle Scholar
  41. Park C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007., 2007: Article ID 50175Google Scholar
  42. Park C: Generalized Hyers-Ulam stability of functional equations: a fixed point approach. Taiwan. J. Math. 2010, 14: 1591-1608.MathSciNetMATHGoogle Scholar
  43. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91-96.MathSciNetMATHGoogle Scholar
  44. Eshaghi Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghari S: Stability of an additive-cubic-quartic functional equation. Adv. Differ. Equ. 2009., 2009: Article ID 395693Google Scholar
  45. Eshaghi Gordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 153084Google Scholar
  46. Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 2008, 343: 567-572. 10.1016/j.jmaa.2008.01.100MathSciNetView ArticleMATHGoogle Scholar
  47. Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008., 2008: Article ID 493751Google Scholar

Copyright

© Park et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.