Skip to main content

Stability of functional inequalities in matrix random normed spaces

Abstract

In this paper, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality and the Cauchy-Jensen additive functional inequality in matrix random normed spaces by using the fixed point method.

MSC:47L25, 46S50, 47S50, 39B52, 54E70, 39B82.

1 Introduction

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. The proof given in [1] appealed to the theory of ordered operator spaces [2]. Effros and Ruan [3] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [4] and Haagerup [5]. The theory of operator spaces has an increasingly significant effect on operator algebra theory (see [6, 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 Rassias [11] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [12] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach.

In [13], Gilányi showed that if f satisfies the functional inequality

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

then f satisfies the Jordan-von Neumann functional equation

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

See also [14]. Gilányi [15] and Fechner [16] proved the Hyers-Ulam stability of the above functional inequality.

Park et al. [17] proved the Hyers-Ulam stability of the following functional inequalities:

f ( x ) + f ( y ) + f ( z ) f ( x + y + z ) ,
(1.1)
f ( x ) + f ( y ) + 2 f ( z ) 2 f ( x + y 2 + z ) .
(1.2)

In the sequel, we adopt the usual terminology, notations and conventions of the theory of random normed spaces, as in [1821]. Throughout this paper, Δ + is the space of distribution functions, that is, the space of all mappings F:R{,}[0,1] such that F is left-continuous and non-decreasing on , F(0)=0 and F(+)=1. D + is a subset of Δ + consisting of all functions F Δ + for which l F(+)=1, where l f(x) denotes the left limit of the function f at the point x, that is, l f(x)= lim t x f(t). The space Δ + is partially ordered by the usual point-wise ordering of functions, i.e., FG if and only if F(t)G(t) for all t in . The maximal element for Δ + in this order is the distribution function ε 0 given by

ε 0 (t)={ 0 if  t 0 , 1 if  t > 0 .

Definition 1.1 ([20])

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

  1. (a)

    T is commutative and associative;

  2. (b)

    T is continuous;

  3. (c)

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

  4. (d)

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

Definition 1.2 ([21])

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

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

  • (RN2) µ a x (t)= µ x ( t | a | ) for all xϵX, α0;

  • (RN3) µ x + y (t+s)=T( µ x (t), µ y (s)) for all x,yϵX and all t,s=0.

Every normed space (X,) defines a random normed space (X,μ, T M ), where

μ x (t)= t t + x

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

Definition 1.3 Let (X,μ,T) be an RN-space.

  1. (1)

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

  2. (2)

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

  3. (3)

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

Theorem 1.4 ([20])

If (X,μ,T) is an RN-space and { x n } is a sequence such that x n x, then lim n μ x n (t)= μ x (t) almost everywhere.

We introduce the concept of matrix random normed space.

Definition 1.5 Let (X,μ,T) be a random normed space. Then

  1. (1)

    (X,{ μ ( n ) },T) is called a matrix random normed space if for each positive integer n, ( M n (X), μ ( n ) ,T) is a random normed space and μ A x B ( k ) (t) μ x ( n ) ( t A B ) for all t>0, A M k , n (R), x=[ x i j ] M n (X) and B M n , k (R) with AB0.

  2. (2)

    (X,{ μ ( n ) },T) is called a matrix random Banach space if (X,μ,T) is a random Banach space and (X,{ μ ( n ) },T) is a matrix random normed 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).

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,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 1.6 ([22, 23])

Let (X,d) be a complete generalized metric space, and let J:XX be a strictly contractive mapping with a Lipschitz constant α<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)<, 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 α d(y,Jy) for all yY.

The stability problem in a random normed space was considered by Mihet and Radu [24]; next some authors proved some stability results in random normed spaces by different methods (see [2527]).

In 1996, Isac and Rassias [28] 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 [2934]).

Throughout this paper, let X be a normed space and (Y,{ μ ( n ) },T) be a matrix random Banach space. In Section 2, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality (1.1) in matrix normed spaces by using the direct method. In Section 3, we prove the Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (1.2) in matrix normed spaces by using the fixed point method.

2 Hyers-Ulam stability of the Cauchy additive functional inequality

In this section, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality (1.1) in matrix random normed spaces by using the fixed point method.

Theorem 2.1 Let φ: X 3 [0,) be a function such that there exists α<1 with

φ(a,b,c) α 2 φ(2a,2b,2c)

for all a,b,cX. Let f:XY be an odd mapping satisfying

μ f n ( [ x i j ] ) + f n ( [ y i j ] ) + f n ( [ z i j ] ) ( n ) (t)min { μ f n ( [ x i j + y i j + z i j ] ) ( n ) ( t 2 ) , t t + i , j = 1 n φ ( x i j , y i j , z i j ) }
(2.1)

for all t>0 and x=[ x i j ],y=[ y i j ],z=[ z i j ] M n (X). Then A(a):= lim l 2 l f( a 2 l ) exists for each aX and defines an additive mapping A:XY such that

μ f n ( [ x i j ] ) A n ( [ x i j ] ) (t) 2 ( 1 α ) t 2 ( 1 α ) t + n 2 α i , j = 1 n φ ( x i j , x i j , 2 x i j )
(2.2)

for all t>0 and x=[ x i j ] M n (X).

Proof Let n=1. Then (2.1) is equivalent to

μ f ( a ) + f ( b ) + f ( c ) (t)min { μ f ( a + b + c ) ( t 2 ) , t t + φ ( a , b , c ) }
(2.3)

for all t>0 and a,b,cX.

Letting b=a and c=2a in (2.3), we get

μ f ( 2 a ) 2 f ( a ) (t) t t + φ ( a , a , 2 a ) ,
(2.4)

and so

μ f ( a ) 2 f ( a 2 ) (t) t t + φ ( a 2 , a 2 , a ) t t + α 2 φ ( a , a , 2 a )
(2.5)

for all t>0 and aX.

Consider the set

S:={g:XY}

and introduce the generalized metric on S:

d(g,h)=inf { ν R + : μ g ( a ) h ( a ) ( ν t ) t t + φ ( a , a , 2 a ) , a X , t > 0 } ,

where, as usual, infϕ=+. It is easy to show that (S,d) is complete (see the proof of [[24], Lemma 2.1]).

Now we consider the linear mapping J:SS such that

Jg(a):=2g ( a 2 )

for all aX.

Let g,hS be given such that d(g,h)=ε. Then

μ g ( a ) h ( a ) (εt) t t + φ ( a , a )

for all aX and t>0. Hence

μ J g ( a ) J h ( a ) ( α ε t ) = μ 2 g ( a 2 ) 2 h ( a 2 ) ( α ε t ) = μ g ( a 2 ) h ( a 2 ) ( α 2 ε t ) α t 2 α t 2 + φ ( a 2 , a 2 , a ) α t 2 α t 2 + α 2 φ ( a , a , 2 a ) = t t + φ ( a , a , 2 a )

for all aX and t>0. So d(g,h)=ε implies that d(Jg,Jh)αε. This means that

d(Jg,Jh)αd(g,h)

for all g,hS.

It follows from (2.5) that d(f,Jf) α 2 .

By Theorem 1.6, there exists a mapping A:XY satisfying the following:

  1. (1)

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

    A ( a 2 ) = 1 2 A(a)

for all aX. The mapping A is a unique fixed point of J in the set

M= { g S : d ( f , g ) < } .
  1. (2)

    d( J l f,A)0 as l. This implies the equality

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

for all aX.

  1. (3)

    d(f,A) 1 1 α d(f,Jf), which implies the inequality

    d(f,A) α 2 2 α .
    (2.6)

By (2.3),

μ 2 l f ( a + b 2 l ) 2 l f ( a 2 l ) 2 l f ( b 2 l ) ( 2 l t ) t t + φ ( a 2 l , b 2 l , a + b 2 l )

for all a,bX and t>0. So

μ 2 l f ( a + b 2 l ) 2 l f ( a 2 l ) 2 l f ( b 2 l ) (t) t 2 l t 2 l + α l 2 l φ ( a , b , a b )

for all a,bX and t>0. Since lim l t 2 l t 2 l + α l 2 l φ ( a , b , a b ) =1 for all a,bX and t>0,

μ A ( a + b ) A ( a ) A ( b ) (t)=1

for all a,bX and t>0. Thus A(a+b)A(a)A(b)=0. So the mapping A:XY is additive.

We note that e j M 1 , n (R) is that j th component is 1 and the others are zero, E i j M n (R) is that (i,j)-component is 1 and the others are zero, and E i j x M n (X) is that (i,j)-component is x and the others are zero. Since μ E k l x ( n ) (t)= μ x (t), we have

μ [ x i j ] ( n ) ( t ) = μ i , j = 1 n E i j x i j ( n ) ( t ) min { μ E i j x i j ( n ) ( t i j ) : i , j = 1 , 2 , , n } = min { μ x i j ( t i j ) : i , j = 1 , 2 , , n } ,

where t= i , j = 1 n t i j . So μ [ x i j ] ( n ) (t)min{ μ x i j ( t n 2 ):i,j=1,2,,n}.

By (2.6),

μ f n ( [ x i j ] ) A n ( [ x i j ] ) ( n ) ( t ) min { μ f ( x i j ) A ( x i j ) ( t n 2 ) : i , j = 1 , 2 , , n } min { 2 ( 1 α ) t 2 ( 1 α ) t + n 2 α φ ( x i j , x i j , 2 x i j ) : i , j = 1 , 2 , , n } 2 ( 1 α ) t 2 ( 1 α ) t + n 2 α i , j = 1 n φ ( x i j , x i j , 2 x i j )

for all x=[ x i j ] M n (X). Thus A:XY is a unique additive mapping satisfying (2.2), as desired. □

Corollary 2.2 Let r, θ be positive real numbers with r>1. Let f:XY be an odd mapping satisfying

μ f n ( [ x i j ] ) + f n ( [ y i j ] ) + f n ( [ z i j ] ) ( n ) ( t ) min { μ f n ( [ x i j + y i j + z i j ] ) ( n ) ( t 2 ) , t t + i , j = 1 n θ ( x i j r + y i j r + z i j r ) }
(2.7)

for all t>0 and x=[ x i j ],y=[ y i j ],z=[ z i j ] M n (X). Then A(a):= lim l 2 l f( a 2 l ) exists for each aX and defines an additive mapping A:XY such that

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

for all t>0 and x=[ x i j ] M n (X).

Proof The proof follows from Theorem 2.1 by taking φ(a,b,c)=θ( a r + b r + c r ) for all a,b,cX. Then we can choose α= 2 1 r and we get the desired result. □

Theorem 2.3 Let f:XY be an odd mapping satisfying (2.1) for which there exists a function φ: X 3 [0,) such that there exists α<1 with

φ(a,b,c)2αφ ( a 2 , b 2 , c 2 )

for all a,b,cX. Then A(a):= lim l 1 2 l f( 2 l a) exists for each aX and defines an additive mapping A:XY such that

μ f n ( [ x i j ] ) A n ( [ x i j ] ) (t) 2 ( 1 α ) t 2 ( 1 α ) t + n 2 i , j = 1 n φ ( x i j , x i j , 2 x i j )

for all t>0 and x=[ x i j ] M n (X).

Proof Let (S,d) be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping J:SS such that

Jg(a):=2g ( a 2 )

for all aX.

It follows from (2.4) that d(f,Jf) 1 2 . So

d(f,A) 1 2 2 α .

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.4 Let r, θ be positive real numbers with r<1. Let f:XY be a mapping satisfying (2.7). Then A(a):= lim l 2 l f( a 2 l ) exists for each aX and defines an additive mapping A:XY such that

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

for all t>0 and x=[ x i j ] M n (X).

Proof The proof follows from Theorem 2.3 by taking φ(a,b,c)=θ( a r + b r + c r ) for all a,b,cX. Then we can choose α= 2 r 1 and we get the desired result. □

3 Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality

In this section, we prove the Hyers-Ulam stability of Cauchy-Jensen additive functional inequality (1.2) in matrix random normed spaces by using the fixed point method.

Theorem 3.1 Let φ: X 3 [0,) be a function such that there exists α<1 with

φ(a,b,c) α 2 φ(2a,2b,2c)

for all a,b,cX. Let f:XY be an odd mapping satisfying

μ f n ( [ x i j ] ) + f n ( [ y i j ] ) + f n ( [ 2 z i j ] ) ( n ) ( t ) min { μ 2 f n ( [ x i j + y i j 2 + z i j ] ) ( n ) ( 2 t 3 ) , t t + i , j = 1 n φ ( x i j , y i j , z i j ) }
(3.1)

for all t>0 and x=[ x i j ],y=[ y i j ],z=[ z i j ] M n (X). Then A(a):= lim l 2 l f( a 2 l ) exists for each aX and defines an additive mapping A:XY such that

μ f n ( [ x i j ] ) A n ( [ x i j ] ) (t) 2 ( 1 α ) t 2 ( 1 α ) t + n 2 α i , j = 1 n φ ( x i j , x i j , x i j )
(3.2)

for all t>0 and x=[ x i j ] M n (X).

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

μ f ( a ) + f ( b ) + f ( 2 c ) (t)min { μ 2 f ( a + b 2 + c ) ( 2 t 3 ) , t t + φ ( a , b , c ) }
(3.3)

for all t>0 and a,b,cX.

Letting b=a and c=a in (3.3), we get

μ f ( 2 a ) 2 f ( a ) (t) t t + φ ( a , a , a ) ,
(3.4)

and so

μ f ( a ) 2 f ( a 2 ) (t) t t + φ ( a 2 , a 2 , a 2 ) t t + α 2 φ ( a , a , a )
(3.5)

for all t>0 and aX.

Let (S,d) be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping J:SS such that

Jg(a):=2g ( a 2 )

for all aX.

Let g,hS be given such that d(g,h)=ε. Then

μ g ( a ) h ( a ) (εt) t t + φ ( a , a , a )

for all aX and t>0. Hence

μ J g ( a ) J h ( a ) ( α ε t ) = μ 2 g ( a 2 ) 2 h ( a 2 ) ( α ε t ) = μ g ( a 2 ) h ( a 2 ) ( α 2 ε t ) α t 2 α t 2 + φ ( a 2 , a 2 , a 2 ) α t 2 α t 2 + α 2 φ ( a , a , a ) = t t + φ ( a , a , a )

for all aX and t>0. So d(g,h)=ε implies that d(Jg,Jh)αε. This means that

d(Jg,Jh)αd(g,h)

for all g,hS.

It follows from (3.5) that d(f,Jf) α 2 .

By Theorem 1.6, there exists a mapping A:XY satisfying the following:

  1. (1)

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

    A ( a 2 ) = 1 2 A(a)

for all aX. The mapping A is a unique fixed point of J in the set

M= { g S : d ( f , g ) < } .
  1. (2)

    d( J l f,A)0 as l. This implies the equality

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

for all aX.

  1. (3)

    d(f,A) 1 1 α d(f,Jf), which implies the inequality

    d(f,A) α 2 2 α .
    (3.6)

By (3.3),

μ 2 l f ( a + b 2 l ) 2 l f ( a 2 l ) 2 l f ( b 2 l ) ( 2 l t ) t t + φ ( a 2 l , b 2 l , a + b 2 l + 1 )

for all a,bX and t>0. So

μ 2 l f ( a + b 2 l ) 2 l f ( a 2 l ) 2 l f ( b 2 l ) (t) t 2 l t 2 l + α l 2 l φ ( a , b , a + b 2 )

for all a,bX and t>0. Since lim l t 2 l t 2 l + α l 2 l φ ( a , b , a b ) =1 for all a,bX and t>0,

μ A ( a + b ) A ( a ) A ( b ) (t)=1

for all a,bX and t>0. Thus A(a+b)A(a)A(b)=0. So the mapping A:XY is additive.

By (2.6),

μ f n ( [ x i j ] ) A n ( [ x i j ] ) ( n ) ( t ) min { μ f ( x i j ) A ( x i j ) ( t n 2 ) : i , j = 1 , 2 , , n } min { 2 ( 1 α ) t 2 ( 1 α ) t + n 2 α φ ( x i j , x i j , x i j ) : i , j = 1 , 2 , , n } 2 ( 1 α ) t 2 ( 1 α ) t + n 2 α i , j = 1 n φ ( x i j , x i j , x i j )

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

Corollary 3.2 Let r, θ be positive real numbers with r>1. Let f:XY be an odd mapping satisfying

μ f n ( [ x i j ] ) + f n ( [ y i j ] ) + f n ( [ 2 z i j ] ) ( n ) ( t ) min { μ f n ( [ x i j + y i j 2 + z i j ] ) ( n ) ( 2 t 3 ) , t t + i , j = 1 n θ ( x i j r + y i j r + z i j r ) }
(3.7)

for all t>0 and x=[ x i j ],y=[ y i j ],z=[ z i j ] M n (X). Then A(a):= lim l 2 l f( a 2 l ) exists for each aX and defines an additive mapping A:XY such that

μ f n ( [ x i j ] ) A n ( [ x i j ] ) (t) ( 2 r 2 ) t ( 2 r 2 ) t + 3 n 2 i , j = 1 n θ x i j r

for all t>0 and x=[ x i j ] M n (X).

Proof The proof follows from Theorem 3.1 by taking φ(a,b,c)=θ( a r + b r + c r ) for all a,b,cX. Then we can choose α= 2 1 r and we get the desired result. □

Theorem 3.3 Let f:XY be an odd mapping satisfying (3.1) for which there exists a function φ: X 3 [0,) such that there exists α<1 with

φ(a,b,c)2αφ ( a 2 , b 2 , c 2 )

for all a,b,cX. Then A(a):= lim l 1 2 l f( 2 l a) exists for each aX and defines an additive mapping A:XY such that

μ f n ( [ x i j ] ) A n ( [ x i j ] ) (t) 2 ( 1 α ) t 2 ( 1 α ) t + n 2 i , j = 1 n φ ( x i j , x i j , x i j )

for all t>0 and x=[ x i j ] M n (X).

Proof Let (S,d) be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping J:SS such that

Jg(a):=2g ( a 2 )

for all aX.

It follows from (3.4) that d(f,Jf) 1 2 . So

d(f,A) 1 2 2 α .

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

Corollary 3.4 Let r, θ be positive real numbers with r<1. Let f:XY be a mapping satisfying (3.7). Then A(a):= lim l 2 l f( a 2 l ) exists for each aX and defines an additive mapping A:XY such that

μ f n ( [ x i j ] ) A n ( [ x i j ] ) (t) ( 2 2 r ) t ( 2 2 r ) t + 3 n 2 i , j = 1 n θ x i j r

for all t>0 and x=[ x i j ] M n (X).

Proof The proof follows from Theorem 3.3 by taking φ(a,b,c)=θ( a r + b r + c r ) for all a,b,cX. Then we can choose α= 2 r 1 and we get the desired result. □

References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. 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

    Article  MathSciNet  Google Scholar 

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

  6. Effros E Contemp. Math. 62. In On Multilinear Completely Bounded Module Maps. Am. Math. Soc., Providence; 1987:479–501.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google 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/00210064

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  16. 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

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. Chang SS, Cho Y, Kang S: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Publ., New York; 2001.

    Google Scholar 

  19. Cho Y, Rassias TM, Saadati R Springer Optimization and Its Applications 86. In Stability of Functional Equations in Random Normed Spaces. Springer, Berlin; 2013.

    Chapter  Google Scholar 

  20. Schweizer B, Sklar A: Probabilistic Metric Spaces. North-Holand, New York; 1983.

    Google Scholar 

  21. Sherstnev AN: On the notion of a random normed space. Dokl. Akad. Nauk SSSR 1963, 149: 280–283. (in Russian)

    MathSciNet  Google Scholar 

  22. Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 2003., 4(1): Article ID 4

  23. 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/S0002-9904-1968-11933-0

    Article  MathSciNet  Google Scholar 

  24. 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.2008.01.100

    Article  MathSciNet  Google Scholar 

  25. Cho Y, Kang S, Saadati R: Nonlinear random stability via fixed-point method. J. Appl. Math. 2012., 2012: Article ID 902931

    Google Scholar 

  26. Miheţ D, Saadati R: On the stability of the additive Cauchy functional equation in random normed spaces. Appl. Math. Lett. 2011, 24: 2005–2009. 10.1016/j.aml.2011.05.033

    Article  MathSciNet  Google Scholar 

  27. Miheţ D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces. Acta Appl. Math. 2010, 110: 797–803. 10.1007/s10440-009-9476-7

    Article  MathSciNet  Google Scholar 

  28. Isac G, Rassias TM: Stability of ψ -additive mappings: applications to nonlinear analysis. Int. J. Math. Math. Sci. 1996, 19: 219–228. 10.1155/S0161171296000324

    Article  MathSciNet  Google Scholar 

  29. Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 2004, 346: 43–52.

    Google Scholar 

  30. 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 749392

    Google Scholar 

  31. 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-z

    Article  MathSciNet  Google Scholar 

  32. Park C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008., 2008: Article ID 493751

    Google Scholar 

  33. 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 50175

    Google Scholar 

  34. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4: 91–96.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jung Rye Lee.

Additional information

Competing interests

The author declares that she has no competing interests.

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

Lee, J.R. Stability of functional inequalities in matrix random normed spaces. J Inequal Appl 2013, 569 (2013). https://doi.org/10.1186/1029-242X-2013-569

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2013-569

Keywords