Skip to main content

Functional equations and inequalities in matrix paranormed spaces

Abstract

In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality, the Cauchy additive functional equation and the quadratic functional equation in matrix paranormed spaces.

MSC:47L25, 39B82, 39B72, 46L07, 39B52, 39B62.

1 Introduction and preliminaries

The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and Steinhaus [2] independently, and since then several generalizations and applications of this notion have been investigated by various authors (see [37]). This notion was defined in normed spaces by Kolk [8].

We recall some basic facts concerning Fréchet spaces.

Definition 1.1 [9]

Let X be a vector space. A paranorm P():X[0,) is a function on X such that

  1. (1)

    P(0)=0;

  2. (2)

    P(x)=P(x);

  3. (3)

    P(x+y)P(x)+P(y) (triangle inequality);

  4. (4)

    If { t n } is a sequence of scalars with t n t and { x n }X with P( x n x)0, then P( t n x n tx)0 (continuity of multiplication).

The pair (X,P()) is called a paranormed space if P() is a paranorm on X.

The paranorm is called total if, in addition, we have

  1. (5)

    P(x)=0 implies x=0.

A Fréchet space is a total and complete paranormed space.

The stability problem of functional equations originated from the question of Ulam [10] 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 [11] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [12] for additive mappings and by Rassias [13] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [14] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

In 1990, during the 27th International Symposium on Functional Equations, Rassias [15] asked the question whether such a theorem can also be proved for p1. In 1991, Gajda [16], following the same approach as in Rassias [13], gave an affirmative solution to this question for p>1. It was shown by Gajda [16], as well as by Rassias and Šemrl [17], that one cannot prove a Rassias-type theorem when p=1 (cf. the books of Czerwik [18] and Hyers et al. [19]).

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 [20] for mappings f:XY, where X is a normed space and Y is a Banach space. Cholewa [21] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [22] proved the Hyers-Ulam stability of the quadratic functional equation. 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 [2327]).

In [28], Gilányi showed that if f satisfies the functional inequality

2 f ( x ) + 2 f ( y ) f ( x y 1 ) f ( x y ) ,
(1.1)

then f satisfies the Jordan-von Neumann functional equation

2f(x)+2f(y)=f(xy)+f ( x y 1 ) .

See also [29]. Gilányi [30] and Fechner [31] proved the Hyers-Ulam stability of functional inequality (1.1).

Park et al. [32] proved the Hyers-Ulam stability of the following functional inequality:

f ( x ) + f ( y ) + f ( z ) f ( x + y + z ) .
(1.2)

The abstract characterization given for linear spaces of bounded Hilbert space operators in terms of matricially normed spaces [33] 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 [34]).

The proof given in [33] appealed to the theory of ordered operator spaces [35]. Effros and Ruan [36] showed that one can give a purely metric proof of this important theorem by using the technique of Pisier [37] and Haagerup [38] (as modified in [39]).

We will use the following notations:

M n (X) is the set of all n×n-matrices in X;

e j M 1 , n (C) is that j th component is 1 and the other components are zero;

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

E i j x M n (X) is that (i,j)-component is x and the other components are zero.

For x M n (X), y M k (X),

xy= ( 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 AB 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.

Definition 1.2 Let (X,P()) be a paranormed space.

  1. (1)

    (X,{ P n ()}) is a matrix paranormed space if ( M n (X), P n ()) is a paranormed space for each positive integer n, P n ( E k l x)=P(x) for xX, and P( x k l ) P n ([ x i j ]) for [ x i j ] M n (X).

  2. (2)

    (X,{ P n ()}) is a matrix Fréchet space if X is a Fréchet space and (X,{ P n ()}) is a matrix paranormed space.

Let E, F be vector spaces. For a given mapping h:EF 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).

In Section 2, we prove the Hyers-Ulam stability of Cauchy additive functional inequality (1.2) in matrix paranormed spaces. In Section 3, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix paranormed spaces. In Section 4, we prove the Hyers-Ulam stability of the quadratic functional equation in matrix paranormed spaces.

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

2 Hyers-Ulam stability of additive functional inequality (1.2) in matrix paranormed spaces

In this section, we prove the Hyers-Ulam stability of additive functional inequality (1.2) in matrix paranormed spaces.

Lemma 2.1 Let (X,{ P n ()}) be a matrix paranormed space. Then

  1. (1)

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

  2. (2)

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

Proof (1) By Definition 1.2, P( x k l ) P n ([ x i j ]).

Since [ x i j ]= i , j = 1 n E i j x i j ,

P n ( [ x i j ] ) = P n ( i , j = 1 n E i j x i j ) i , j = 1 n P n ( E i j x i j )= i , j = 1 n P( x i j ).
  1. (2)

    By (1), we have

    P( x s k l x k l ) P n ( [ x s i j x i j ] ) = P n ( [ x s i j ] [ x i j ] ) i , j = 1 n P( x s i j x i j ).

So, we get the result. □

Lemma 2.2 Let (X,{ n }) be a matrix normed space. Then

(1) E k l x n =x for xX;

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

(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. □

We need the following result.

Lemma 2.3 Let f:XY be an odd mapping such that

P ( f ( a ) + f ( b ) + f ( c ) ) P ( f ( a + b + c ) )
(2.1)

for all a,b,cX. Then f:XY is additive.

Proof Letting c=ab in (2.1), we get P(f(a)+f(b)+f(ab))P(f(0))=0 for all a,bX. So,

f(a)+f(b)f(a+b)=f(a)+f(b)+f(ab)=0

for all a,bX. Thus f:XY is additive. □

Note that P(2x)2P(x) for all xY.

Theorem 2.4 Let r, θ be positive real numbers with r>1. Let f:XY be an odd mapping such that

P n ( f n ( [ x i j ] ) + f n ( [ y i j ] ) + f n ( [ z i j ] ) ) P n ( f n ( [ x i j ] + [ y i j ] + [ z i j ] ) ) + i , j = 1 n θ ( x i j r + y i j r + z i j r )
(2.2)

for all x=[ x i j ],y=[ y i j ],z=[ z i j ] M n (X). Then there exists a unique additive mapping A:XY such that

P n ( f n ( [ x i j ] ) A n ( [ x i j ] ) ) i , j = 1 n 2 r + 2 2 r 2 θ x i j r
(2.3)

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

Proof When n=1, (2.2) is equivalent to

P ( f ( a ) + f ( b ) + f ( c ) ) P ( f ( a + b + c ) ) +θ ( a r + b r + c r )
(2.4)

for all a,b,cX.

Letting b=a and c=2a in (2.4), we get

P ( f ( 2 a ) 2 f ( a ) ) ( 2 + 2 r ) θ a r ,

and so

P ( f ( a ) 2 f ( a 2 ) ) 2 + 2 r 2 r θ a r

for all a,bX.

One can easily show that

P ( 2 p f ( a 2 p ) 2 q f ( a 2 q ) ) l = p q 1 ( 2 + 2 r ) 2 l 2 ( l + 1 ) r θ a r
(2.5)

for all a,bX and nonnegative integers p, q with p<q. It follows from (2.5) that the sequence { 2 l f( a 2 l )} is Cauchy for all aX. Since Y is complete, the sequence { 2 l f( a 2 l )} converges. So, one can define the mapping A:XY by

A(a)= lim l 2 l f ( a 2 l )

for all aX.

Moreover, letting p=0 and passing the limit q in (2.5), we get

P ( f ( a ) A ( a ) ) 2 r + 2 2 r 2 θ a r
(2.6)

for all aX.

It follows from (2.4) that

P ( 2 l ( f ( a 2 l ) + f ( b 2 l ) + f ( c 2 l ) ) ) 2 l P ( f ( a + b + c 2 l ) ) + 2 l 2 l r θ ( a r + b r + c r )

for all a,b,cX. Passing the limit l in the above inequality, we get

P ( A ( a ) + A ( b ) + A ( c ) ) P ( A ( a + b + c ) )

for all a,b,cX. Since f:XY is an odd mapping, the mapping A:XY is odd. By Lemma 2.3, A:XY is additive.

Now, let T:XY be another additive mapping satisfying (2.6). Then we have

P ( A ( a ) T ( a ) ) = P ( 2 q A ( a 2 q ) 2 q T ( a 2 q ) ) P ( 2 q ( A ( a 2 q ) g ( a 2 q ) ) ) + P ( 2 q ( T ( a 2 q ) g ( a 2 q ) ) ) 2 2 r + 2 2 r 2 2 q 2 q r θ a r ,

which tends to zero as q for all aX. So, we can conclude that A(a)=T(a) for all aX. This proves the uniqueness of A.

By Lemma 2.1 and (2.6),

P n ( f n ( [ x i j ] ) A n ( [ x i j ] ) ) i , j = 1 n P ( f ( x i j ) A ( x i j ) ) i , j = 1 n ( 2 + 2 r ) 2 r 2 θ x i j r

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

Theorem 2.5 Let r, θ be positive real numbers with r<1. Let f:YX be an odd mapping such that

f n ( [ x i j ] ) + f n ( [ y i j ] ) + f n ( [ z i j ] ) n f n ( [ x i j ] + [ y i j ] + [ z i j ] ) n + i , j = 1 n θ ( P ( x i j ) r + P ( y i j ) r + P ( z i j ) r )
(2.7)

for all x=[ x i j ],y=[ y i j ],z=[ z i j ] M n (Y). Then there exists a unique additive mapping A:YX such that

f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n 2 + 2 r 2 2 r θP ( x i j ) r
(2.8)

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

Proof Let n=1 in (2.7). Then (2.7) is equivalent to

f ( a ) + f ( b ) + f ( c ) f ( a + b + c ) +θ ( P ( a ) r + P ( b ) r + P ( c ) r )
(2.9)

for all a,b,cY.

Letting b=a and c=2a in (2.9), we get

f ( 2 a ) 2 f ( a ) ( 2 + 2 r ) θP ( a ) r ,

and so

f ( a ) 1 2 f ( 2 a ) 2 + 2 r 2 θP ( a ) r

for all aY.

One can easily show that

1 2 p f ( 2 p a ) 1 2 q f ( 2 q a ) l = p q 1 2 l r 2 l 2 + 2 r 2 θP ( a ) r
(2.10)

for all aY and nonnegative integers p, q with p<q. It follows from (2.10) that the sequence { 1 2 l f( 2 l a)} is Cauchy for all aY. Since X is complete, the sequence { 1 2 l f( 2 l a)} converges. So, one can define the mapping A:YX by

A(a)= lim l 1 2 l f ( 2 l a )

for all aY.

Moreover, letting p=0 and passing the limit q in (2.10), we get

f ( a ) A ( a ) 2 + 2 r 2 2 r θP ( a ) r
(2.11)

for all aY.

It follows from (2.9) that

1 2 l ( f ( 2 l a ) + f ( 2 l b ) + f ( 2 l c ) ) 1 2 l f ( 2 l ( a + b + c ) ) + 2 l r 2 l θ ( a r + b r + c r )

for all a,b,cY. Passing the limit l in the above inequality, we get

A ( a ) + A ( b ) + A ( c ) A ( a + b + c )

for all a,b,cY. By [[32], Lemma 3.1], the mapping A:YX is additive.

Now, let T:YX be another additive mapping satisfying (2.11). Let n=1. Then we have

A ( a ) T ( a ) = 1 2 q A ( 2 q a ) 1 2 q T ( 2 q a ) 1 2 q ( A ( 2 q a ) g ( 2 q a ) ) + 1 2 q ( T ( 2 q a ) g ( 2 q a ) ) 2 2 + 2 r 2 2 r 2 q r 2 q θ P ( a ) r ,

which tends to zero as q for all aY. So, we can conclude that A(a)=T(a) for all aY. This proves the uniqueness of A.

By Lemma 2.2 and (2.11),

f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n f ( x i j ) A ( x i j ) i , j = 1 n 2 + 2 r 2 r 2 θP ( x i j ) r

for all x=[ x i j ] M n (Y). Thus A:YX is a unique additive mapping satisfying (2.8), as desired. □

3 Hyers-Ulam stability of the Cauchy additive functional equation in matrix paranormed spaces

In this section, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix paranormed spaces.

For a mapping f:XY, define Df: X 2 Y and D f n : M n ( X 2 ) M n (Y) by

D f ( a , b ) = f ( a + b ) f ( a ) f ( b ) , D f n ( [ x i j ] , [ y i j ] ) : = f n ( [ x i j + y i j ] ) f n ( [ x i j ] ) f n ( [ y i j ] )

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

Theorem 3.1 Let r, θ be positive real numbers with r>1. Let f:XY be a mapping such that

P n ( D f n ( [ x i j ] , [ y i j ] ) ) i , j = 1 n θ ( x i j r + y i j r )
(3.1)

for all x=[ x i j ],y=[ y i j ] M n (X). Then there exists a unique additive mapping A:XY such that

P n ( f n ( [ x i j ] ) A n ( [ x i j ] ) ) i , j = 1 n 2 θ 2 r 2 x i j r
(3.2)

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

Proof Let n=1 in (3.1). Then (3.1) is equivalent to

P ( f ( a + b ) f ( a ) f ( b ) ) θ ( a r + b r )
(3.3)

for all a,bX.

Letting b=a in (3.3), we get

P ( f ( 2 a ) 2 f ( a ) ) 2θ a r ,

and so

P ( f ( a ) 2 f ( a 2 ) ) 2 2 r θ a r

for all a,bX.

One can easily show that

P ( 2 p f ( a 2 p ) 2 q f ( a 2 q ) ) l = p q 1 2 2 l 2 ( l + 1 ) r θ a r
(3.4)

for all a,bX and nonnegative integers p, q with p<q. It follows from (3.4) that the sequence { 2 l f( a 2 l )} is Cauchy for all aX. Since Y is complete, the sequence { 2 l f( a 2 l )} converges. So, one can define the mapping A:XY by

A(a)= lim l 2 l f ( a 2 l )

for all aX.

Moreover, letting p=0 and passing the limit q in (3.4), we get

P ( f ( a ) A ( a ) ) 2 θ 2 r 2 a r
(3.5)

for all aX.

It follows from (3.3) that

P ( 2 l ( f ( a + b 2 l ) f ( a 2 l ) f ( b 2 l ) ) ) 2 l P ( f ( a + b 2 l ) f ( a 2 l ) f ( b 2 l ) ) 2 l 2 l r θ ( a r + b r ) ,

which tends to zero as l. So, P(A(a+b)A(a)A(b))=0, i.e., A(a+b)=A(a)+A(b) for all a,bX. Hence A:XY is additive.

The proof of the uniqueness of A is similar to the proof of Theorem 2.4.

By Lemma 2.1 and (3.5),

P n ( f n ( [ x i j ] ) A n ( [ x i j ] ) ) i , j = 1 n P ( f ( x i j ) A ( x i j ) ) i , j = 1 n 2 θ 2 r 2 x i j r

for all x=[ x i j ] M n (X). Thus A:XY is a unique additive mapping satisfying (3.2), as desired. □

Theorem 3.2 Let r, θ be positive real numbers with r<1. Let f:YX be a mapping such that

D f n ( [ x i j ] , [ y i j ] ) n i , j = 1 n θ ( P ( x i j ) r + P ( y i j ) r )
(3.6)

for all x=[ x i j ],y=[ y i j ] M n (Y). Then there exists a unique additive mapping A:YX such that

f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n 2 θ 2 2 r P ( x i j ) r
(3.7)

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

Proof Let n=1 in (3.6). Then (3.6) is equivalent to

f ( a + b ) f ( a ) f ( b ) θ ( P ( a ) r + P ( b ) r )
(3.8)

for all a,bY.

Letting b=a in (3.8), we get

f ( 2 a ) 2 f ( a ) 2θP ( a ) r ,

and so

f ( a ) 1 2 f ( 2 a ) θP ( a ) r

for all aY.

One can easily show that

1 2 p f ( 2 p a ) 1 2 q f ( 2 q a ) l = p q 1 2 l r 2 l θP ( a ) r
(3.9)

for all aY and nonnegative integers p, q with p<q. It follows from (3.9) that the sequence { 1 2 l f( 2 l a)} is Cauchy for all aY. Since X is complete, the sequence { 1 2 l f( 2 l a)} converges. So, one can define the mapping A:YX by

A(a)= lim l 1 2 l f ( 2 l a )

for all aY.

Moreover, letting p=0 and passing the limit q in (3.9), we get

f ( a ) A ( a ) 2 θ 2 2 r P ( a ) r
(3.10)

for all aY.

It follows from (3.8) that

1 2 l ( f ( 2 l ( a + b ) ) f ( 2 l a ) f ( 2 l b ) ) 2 l r 2 l θ ( a r + b r ) ,

which tends to zero as l. So, A(a+b)A(a)A(b)=0, i.e., A(a+b)=A(a)+A(b) for all a,bY. Hence A:YX is additive.

The proof of the uniqueness of A is similar to the proof of Theorem 2.5.

By Lemma 2.2 and (3.10),

f n ( [ x i j ] ) A n ( [ x i j ] ) n i , j = 1 n f ( x i j ) A ( x i j ) i , j = 1 n 2 θ 2 r 2 P ( x i j ) r

for all x=[ x i j ] M n (Y). Thus A:YX is a unique additive mapping satisfying (3.7), as desired. □

4 Hyers-Ulam stability of the quadratic functional equation in matrix paranormed spaces

In this section, we prove the Hyers-Ulam stability of the quadratic functional equation in matrix paranormed spaces.

For a mapping f:XY, define Df: X 2 Y and D f n : M n ( X 2 ) M n (Y) by

D f ( a , b ) = f ( a + b ) + f ( a b ) 2 f ( a ) 2 f ( b ) , D f n ( [ x i j ] , [ y i j ] ) : = f n ( [ x i j + y i j ] ) + f n ( [ x i j y i j ] ) 2 f n ( [ x i j ] ) 2 f n ( [ y i j ] )

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

Theorem 4.1 Let r, θ be positive real numbers with r>2. Let f:XY be a mapping such that

P n ( D f n ( [ x i j ] , [ y i j ] ) ) i , j = 1 n θ ( x i j r + y i j r )
(4.1)

for all x=[ x i j ],y=[ y i j ] M n (X). Then there exists a unique quadratic mapping Q:XY such that

P n ( f n ( [ x i j ] ) Q n ( [ x i j ] ) ) i , j = 1 n 2 θ 2 r 4 x i j r

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

Proof Let n=1 in (4.1). Then (4.1) is equivalent to

P ( f ( a + b ) + f ( a b ) 2 f ( a ) 2 f ( b ) ) θ ( a r + b r )
(4.2)

for all a,bX.

Letting a=b=0 in (4.2), we get P(2f(0))0 and so f(0)=0.

Letting b=a in (4.2), we get

P ( f ( 2 a ) 4 f ( a ) ) 2θ a r ,

and so

P ( f ( a ) 4 f ( a 2 ) ) 2 2 r θ a r

for all a,bX.

The rest of the proof is similar to the proof of Theorem 3.1. □

Theorem 4.2 Let r, θ be positive real numbers with r<2. Let f:YX be a mapping such that

D f n ( [ x i j ] , [ y i j ] ) n i , j = 1 n θ ( P ( x i j ) r + P ( y i j ) r )
(4.3)

for all x=[ x i j ],y=[ y i j ] M n (Y). Then there exists a unique quadratic mapping Q:YX such that

f n ( [ x i j ] ) Q n ( [ x i j ] ) n i , j = 1 n 2 θ 4 2 r P ( x i j ) r

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

Proof Let n=1 in (4.3). Then (4.3) is equivalent to

f ( a + b ) + f ( a b ) 2 f ( a ) 2 f ( b ) θ ( P ( a ) r + P ( b ) r )
(4.4)

for all a,bY.

Letting a=b=0 in (4.4), we get 2f(0)0 and so f(0)=0.

Letting b=a in (4.4), we get

f ( 2 a ) 4 f ( a ) 2θP ( a ) r ,

and so

f ( a ) 1 4 f ( 2 a ) θ 2 P ( a ) r

for all a,bY.

The rest of the proof is similar to the proof of Theorem 3.2. □

References

  1. Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241–244.

    MATH  MathSciNet  Google Scholar 

  2. Steinhaus H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2: 73–74.

    MathSciNet  Google Scholar 

  3. Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.

    MATH  MathSciNet  Article  Google Scholar 

  4. Karakus S: Statistical convergence on probabilistic normed spaces. Math. Commun. 2007, 12: 11–23.

    MATH  MathSciNet  Google Scholar 

  5. Mursaleen M: λ -Statistical convergence. Math. Slovaca 2000, 50: 111–115.

    MATH  MathSciNet  Google Scholar 

  6. Mursaleen M, Mohiuddine SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2009, 233: 142–149. 10.1016/j.cam.2009.07.005

    MATH  MathSciNet  Article  Google Scholar 

  7. Šalát T: On the statistically convergent sequences of real numbers. Math. Slovaca 1980, 30: 139–150.

    MATH  MathSciNet  Google Scholar 

  8. Kolk E: The statistical convergence in Banach spaces. Tartu Ülik. Toim. 1991, 928: 41–52.

    MathSciNet  Google Scholar 

  9. Wilansky A: Modern Methods in Topological Vector Space. McGraw-Hill, New York; 1978.

    Google Scholar 

  10. Ulam SM: A Collection of the Mathematical Problems. Interscience, New York; 1960.

    Google Scholar 

  11. Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222–224. 10.1073/pnas.27.4.222

    MathSciNet  Article  Google Scholar 

  12. Aoki T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 1950, 2: 64–66. 10.2969/jmsj/00210064

    MATH  Article  Google Scholar 

  13. Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1

    MATH  Article  Google Scholar 

  14. Gǎvruta G: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184: 431–436. 10.1006/jmaa.1994.1211

    MathSciNet  Article  Google Scholar 

  15. Rassias TM: Problem 16; 2. Report of the 27th international symp. on functional equations. Aequ. Math. 1990, 39: 292–293. 309

    Google Scholar 

  16. Gajda Z: On stability of additive mappings. Int. J. Math. Math. Sci. 1991, 14: 431–434. 10.1155/S016117129100056X

    MATH  MathSciNet  Article  Google Scholar 

  17. 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/S0002-9939-1992-1059634-1

    MATH  Article  Google Scholar 

  18. Czerwik P: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore; 2002.

    MATH  Book  Google Scholar 

  19. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables. Birkhäuser, Basel; 1998.

    MATH  Book  Google Scholar 

  20. Skof F: Proprieta’ locali e approssimazione di operatori. Rend. Semin. Mat. Fis. Milano 1983, 53: 113–129. 10.1007/BF02924890

    MATH  MathSciNet  Article  Google Scholar 

  21. Cholewa PW: Remarks on the stability of functional equations. Aequ. Math. 1984, 27: 76–86. 10.1007/BF02192660

    MATH  MathSciNet  Article  Google Scholar 

  22. Czerwik S: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hamb. 1992, 62: 59–64. 10.1007/BF02941618

    MATH  MathSciNet  Article  Google Scholar 

  23. Aczel J, Dhombres J: Functional Equations in Several Variables. Cambridge University Press, Cambridge; 1989.

    MATH  Book  Google Scholar 

  24. 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.2010.05.011

    MATH  MathSciNet  Article  Google Scholar 

  25. Isac G, Rassias TM: On the Hyers-Ulam stability of ψ -additive mappings. J. Approx. Theory 1993, 72: 131–137. 10.1006/jath.1993.1010

    MATH  MathSciNet  Article  Google Scholar 

  26. 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.2004.04.009

    MATH  MathSciNet  Article  Google Scholar 

  27. Park C: Homomorphisms between Poisson J C -algebras. Bull. Braz. Math. Soc. 2005, 36: 79–97. 10.1007/s00574-005-0029-z

    MATH  MathSciNet  Article  Google Scholar 

  28. Gilányi A: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequ. Math. 2001, 62: 303–309. 10.1007/PL00000156

    MATH  Article  Google Scholar 

  29. Rätz J: On inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 2003, 66: 191–200. 10.1007/s00010-003-2684-8

    MATH  Article  Google Scholar 

  30. Gilányi A: On a problem by K. Nikodem. Math. Inequal. Appl. 2002, 5: 707–710.

    MATH  MathSciNet  Google Scholar 

  31. Fechner W: Stability of a functional inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math. 2006, 71: 149–161. 10.1007/s00010-005-2775-9

    MATH  MathSciNet  Article  Google Scholar 

  32. Park C, Cho Y, Han M: Functional inequalities associated with Jordan-von Neumann-type additive functional equations. J. Inequal. Appl. 2007., 2007: Article ID 41820

    Google Scholar 

  33. Ruan ZJ: Subspaces of C -algebras. J. Funct. Anal. 1988, 76: 217–230. 10.1016/0022-1236(88)90057-2

    MATH  MathSciNet  Article  Google Scholar 

  34. Effros E, Ruan ZJ: On approximation properties for operator spaces. Int. J. Math. 1990, 1: 163–187. 10.1142/S0129167X90000113

    MATH  MathSciNet  Article  Google Scholar 

  35. Choi MD, Effros E: Injectivity and operator spaces. J. Funct. Anal. 1977, 24: 156–209. 10.1016/0022-1236(77)90052-0

    MATH  MathSciNet  Article  Google Scholar 

  36. Effros E, Ruan ZJ: On the abstract characterization of operator spaces. Proc. Am. Math. Soc. 1993, 119: 579–584. 10.1090/S0002-9939-1993-1163332-4

    MATH  MathSciNet  Article  Google Scholar 

  37. 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)90038-1

    MATH  MathSciNet  Article  Google Scholar 

  38. Haagerup, U: Decomp. of completely bounded maps. Unpublished manuscript

  39. Effros E: On multilinear completely bounded module maps. Contemp. Math. 62. In Operator Algebras and Mathematical Physics. Am. Math. Soc., Providence; 1987:479–501.

    Chapter  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dong Yun Shin.

Additional information

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.

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Park, C., Lee, J.R. & Shin, D.Y. Functional equations and inequalities in matrix paranormed spaces. J Inequal Appl 2013, 547 (2013). https://doi.org/10.1186/1029-242X-2013-547

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-547

Keywords

  • Jordan-von Neumann functional equation
  • operator space
  • matrix paranormed space
  • Hyers-Ulam stability
  • functional equation
  • functional inequality