Fuzzy stability of functional inequalities in matrix fuzzy normed spaces

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R E S E A R C H

Open Access

Fuzzy stability of functional inequalities in

matrix fuzzy normed spaces

Choonkil Park

1

_{, Dong Yun Shin}

2*

_{and Jung Rye Lee}

3

*_{Correspondence: [email protected]} 2_{Department of Mathematics,} Daejin University, Kyeonggi, 487-711, Korea

Full list of author information is available at the end of the article

Abstract

Using the ﬁxed point method, we prove the Hyers-Ulam stability of additive functional inequalities in matrix fuzzy normed spaces.

MSC: 47L25; 47H10; 46S40; 39B82; 46L07; 39B52; 26E50

Keywords: operator space; ﬁxed point; Hyers-Ulam stability; matrix fuzzy normed space; additive functional inequality; additive functional equation

1 Introduction and preliminaries

Katsaras [] deﬁned a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have deﬁned fuzzy norms on a vector space from various points of view [–]. In particular, Bag and Samanta [], following Cheng and Mordeson [], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type []. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [].

We use the deﬁnition of fuzzy normed spaces given in [, , ] to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy additive functional inequality and for the Cauchy-Jensen additive functional inequality in the fuzzy normed vector space setting.

Deﬁnition .[, –] LetXbe a real vector space. A functionN:X×R→[, ] is called afuzzy normonXif for allx,y∈Xand alls,t∈R,

(N) N(x,t) = fort≤;

(N) x= if and only ifN(x,t) = for allt> ; (N) N(cx,t) =N(x,_|t_c_|)ifc= ;

(N) N(x+y,s+t)≥min{N(x,s),N(y,t)};

(N) N(x,·)is a non-decreasing function ofRandlimt→∞N(x,t) = ;

(N) forx= ,N(x,·)is continuous onR.

The pair (X,N) is called afuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [, ].

Deﬁnition .[, –] Let (X,N) be a fuzzy normed vector space. A sequence{xn}inXis said tobe convergentorconvergeif there exists anx∈Xsuch thatlimn→∞N(xn–x,t) = 

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for allt> . In this case,xis called thelimit of the sequence{xn}and we denote it by N-limn→∞xn=x.

Deﬁnition .[, , ] Let (X,N) be a fuzzy normed vector space. A sequence{xn}in

Xis calledCauchyif for eachε>  and eacht>  there exists ann∈Nsuch that for all

n≥nand allp> , we haveN(xn+p–xn,t) >  –ε.

It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be com-pleteand the fuzzy normed vector space is called afuzzy Banach space.

We say that a mappingf:X→Ybetween fuzzy normed vector spacesXandYis con-tinuous at a pointx∈Xif for each sequence{xn} converging toxinX, the sequence

{f(xn)}converges tof(x). Iff :X→Yis continuous at eachx∈X, thenf:X→Y is said

to becontinuousonX(see []). We will use the following notations:

Mn(X)is the set of alln×n-matrices inX;

ej∈M,n(C)is thatjth component isand the other components are zero;

Eij∈Mn(C)is that(i,j)-component isand the other components are zero;

Eij⊗x∈Mn(X)is that(i,j)-component isxand the other components are zero. Forx∈Mn(X),y∈Mk(X),

x⊕y=

x 

 y

.

Note that (X,{ · n}) is a matrix normed space if and only if (Mn(X), · n) is a normed space for each positive integernandAxBk≤ ABxnholds forA∈Mk,n(C),x=

(xij)∈Mn(X) andB∈Mn,k(C), and that (X,{ · n}) is a matrix Banach space if and only if

Xis a Banach space and (X,{ · n}) is a matrix normed space.

A matrix normed space (X,{ · n}) is called anL∞-matrix normed spaceifx⊕yn+k= max{xn,yk}holds for allx∈Mn(X) and ally∈Mk(X).

LetE,Fbe vector spaces. For a given mappingh:E→Fand a given positive integern, deﬁnehn:Mn(E)→Mn(F) by

hn

[xij]

=h(xij)

for all [xij]∈Mn(E).

We introduce the concept of a matrix fuzzy normed space.

Deﬁnition . Let (X,N) be a fuzzy normed space.

() (X,{Nn})is called amatrix fuzzy normed spaceif for each positive integern,

(Mn(X),Nn)is a fuzzy normed space andNk(AxB,t)≥Nn(x,_A_·t_B)for allt> ,

A∈Mk,n(R),x= [xij]∈Mn(X)andB∈Mn,k(R)withA · B = .

() (X,{Nn})is called amatrix fuzzy Banach spaceif(X,N)is a fuzzy Banach space and

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Example . Let (X,{ · n}) be a matrix normed space. LetNn(x,t) :=_t₊t_x

n for allt> 

andx= [xij]∈Mn(X). Then

Nk(AxB,t) = t t+AxBk ≥

t

t+A · xn· B= t

A·B

t

A·B+xn

for allt> ,A∈Mk,n(R),x= [xij]∈Mn(X) andB∈Mn,k(R) withA·B = . So, (X,{Nn})

is a matrix fuzzy normed space.

The abstract characterization given for linear spaces of bounded Hilbert space opera-tors in terms ofmatricially normed spaces[] 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 signif-icant eﬀect on operator algebra theory (see []).

The proof given in [] appealed to the theory of ordered operator spaces []. Eﬀros and Ruan [] showed that one can give a purely metric proof of this important theorem by using a technique of Pisier [] and Haagerup [] (as modiﬁed in []).

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

In [], Gilányi showed that iff satisﬁes the functional inequality

_f_{(x) + f}_{(y) –}_f

xy–≤f(xy), (.)

thenf satisﬁes the Jordan-von Neumann functional equation

f(x) + f(y) =f(xy) +fxy–.

See also []. Gilányi [] and Fechner [] proved the Hyers-Ulam stability of the func-tional inequality (.).

Parket al.[] proved the Hyers-Ulam stability of the following functional inequalities:

f(x) +f(y) +f(z)≤f x+y+z 

,

f(x) +f(y) +f(z)≤f(x+y+z), (.)

f(x) +f(y) + f(z)≤f x+y  +z

. (.)

LetXbe a set. A functiond:X×X→[,∞] is called ageneralized metriconXifd satisﬁes

() d(x,y) = if and only ifx=y; () d(x,y) =d(y,x)for allx,y∈X;

() d(x,z)≤d(x,y) +d(y,z)for allx,y,z∈X.

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Theorem .[, ] Let(X,d)be a complete generalized metric space and let J:X→ X be a strictly contractive mapping with a Lipschitz constantα< .Then,for each given element x∈X,either

dJnx,Jn+x=∞

for all nonnegative integers n or there exists a positive integer nsuch that () d(Jnx,Jn+x) <∞,∀n≥n;

() the sequence{Jnx}converges to a ﬁxed pointy∗ofJ;

() y∗is the unique ﬁxed point ofJin the setY={y∈X|d(Jnx,y) <∞}; () d(y,y∗)≤_–_αd(y,Jy)for ally∈Y.

In , Isac and Rassias [] 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 [–]).

Throughout this paper, let (X, · n) be a matrix normed space, (Y, · n) be a matrix Banach space and letnbe a ﬁxed positive integer. Let (X,Nn) be a matrix fuzzy normed space and let (Y,Nn) be a matrix fuzzy Banach space.

In Section , we prove the Hyers-Ulam stability of the Cauchy additive functional in-equality (.) in fuzzy normed spaces by using the ﬁxed point method.

In Section , we prove the Hyers-Ulam stability of the Cauchy additive functional equa-tion in matrix fuzzy normed spaces by using the ﬁxed point method.

In Section , we prove the Hyers-Ulam stability of the Cauchy-Jensen additive functional inequality (.) in fuzzy normed spaces by using the ﬁxed point method.

In Section , we prove the Hyers-Ulam stability of the Cauchy additive functional in-equality (.) in matrix normed spaces by using the direct method and by using the ﬁxed point method.

2 Hyers-Ulam stability of the Cauchy functional inequality in fuzzy normed spaces

We need the following lemma to prove the main results.

Lemma .[, ] Let(Y,N)be a fuzzy normed vector space.Let f:X→Y be a mapping such that

Nf(x) +f(y) + f(z),t≥N f x+y  +z

,t 

for all x,y,z∈X and all t> .Then f is Cauchy additive,i.e.,f(x+y) =f(x) +f(y)for all x,y∈X.

In this section, using the ﬁxed point method, we prove the Hyers-Ulam stability of the Cauchy additive functional inequality (.) in fuzzy Banach spaces.

Theorem . Letϕ:X_→_[,_∞)_{be a function such that there exists an L}_{< }_with

ϕ(x,y,z)≤L

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for all x,y,z∈X.Let f :X→Y be an odd mapping satisfying

Nf(x) +f(y) +f(z),t

≥min

N f(x+y+z),t 

, t

t+ϕ(x,y,z)

(.)

for all x,y,z∈X and all t> .Then A(x) :=N-limn→∞nf(_xn)exists for each x∈X and

deﬁnes an additive mapping A:X→Y such that

Nf(x) –A(x),t≥ ( – L)t

( – L)t+Lϕ(x,x, –x) (.)

for all x∈X and all t> .

Proof Sincef is odd,f() = . So,N(f(),_t) = . Lettingy=xand replacingzby –xin (.), we get

Nf(x) – f(x),t≥ t

t+ϕ(x,x, –x) (.)

for allx∈X. Consider the set

S:={g:X→Y}

and introduce the generalized metric onS:

d(g,h) =inf

μ∈R+:N

g(x) –h(x),μt≥ t

t+ϕ(x,x, –x),∀x∈X,∀t> 

,

where, as usual,infφ= +∞. It is easy to show that (S,d) is complete. (See the proof of [, Lemma .].)

Now we consider the linear mappingJ:S→Ssuch that

Jg(x) := g x 

for allx∈X.

Letg,h∈Sbe given such thatd(g,h) =ε. Then

Ng(x) –h(x),εt≥ t t+ϕ(x,x, –x)

for allx∈Xand allt> . Hence

NJg(x) –Jh(x),Lεt=N g x 

– h x 

,Lεt

=N g x 

–h x 

,L εt

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≥ Lt

Lt

 +ϕ(

x

,

x

, –x)

≥ Lt

Lt

 +

L

ϕ(x,x, –x)

= t

t+ϕ(x,x, –x)

for allx∈Xand allt> . So,d(g,h) =εimplies thatd(Jg,Jh)≤Lε. This means that

d(Jg,Jh)≤Ld(g,h)

for allg,h∈S.

It follows from (.) that

N f(x) – f x 

,L t

≥ t

t+ϕ(x,x, –x)

for allx∈Xand allt> . So,d(f,Jf)≤L_.

By Theorem ., there exists a mappingA:X→Ysatisfying the following: ()Ais a ﬁxed point ofJ,i.e.,

A x 

=

A(x) (.)

for allx∈X. Sincef :X→Y is odd,A:X→Yis an odd mapping. The mappingAis a unique ﬁxed point ofJin the set

M=g∈S:d(f,g) <∞.

This implies thatAis a unique mapping satisfying (.) such that there exists aμ∈(,∞) satisfying

Nf(x) –A(x),μt≥ t t+ϕ(x,x, –x)

for allx∈X;

()d(Jnf,A)→ asn→ ∞. This implies the equality

N- lim n→∞

n_f x n

=A(x)

for allx∈X; ()d(f,A)≤ 

–Ld(f,Jf), which implies the inequality

d(f,A)≤ L  – L.

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By (.),

N n f x n

+f y n

+f z n

, nt

≥min

N nf x+y+z n

, n–t

, t

t+ϕ(_xn,_yn,_zn)

for allx,y,z∈X, allt> , and alln∈N. So,

N n f x n

+f y n

+f z n ,t ≥min

N nf x+y+z n , t  , t n t

n+L n

nϕ(x,y,z)

for allx,y,z∈X, allt> , and alln∈N. Sincelimn→∞

t

n t

n+

Ln

nϕ(x,y,z)

=  for allx,y,z∈Xand

allt> ,

NA(x) +A(y) +A(z),t≥N A(x+y+z),t 

for allx,y,z∈Xand allt> . By [, Lemma .], the mappingA:X→Y is a Cauchy

additive, as desired.

Corollary . Letθ ≥and let p be a real number with p> .Let X be a normed vector space with the norm · .Let f :X→Y be an odd mapping satisfying

Nf(x) +f(y) +f(z),t≥min

N f(x+y+z),t 

, t

t+θ(xp₊_yp₊_zp₎

(.)

Nf(x) –A(x),t≥ ( p_{– )t}

(p_{– )t}_{+ ( + }p₎_θ_xp

Proof The proof follows from Theorem . by taking

ϕ(x,y) :=θxp+yp+zp

for allx,y,z∈X. Then we can chooseL= –p_{, and we get the desired result.}

Theorem . Letϕ:X_→_[,_∞)_{be a function such that there exists an L}_{< }_with

ϕ(x,y,z)≤Lϕ x

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for all x,y,z ∈ X. Let f : X →Y be an odd mapping satisfying (.). Then A(x) := N-limn→∞nf(nx)exists for each x∈X and deﬁnes an additive mapping A:X→Y

such that

Nf(x) –A(x),t≥ ( – L)t

( – L)t+ϕ(x,x, –x) (.)

Proof Let (S,d) be the generalized metric space deﬁned in the proof of Theorem .. Consider the linear mappingJ:S→Ssuch that

Jg(x) := g(x)

for allx∈X.

Ng(x) –h(x),εt≥ t t+ϕ(x,x, –x)

NJg(x) –Jh(x),Lεt=N  g(x) –



h(x),Lεt

=Ng(x) –h(x), Lεt

≥ Lt

Lt+ϕ(x, x, –x)≥

Lt Lt+ Lϕ(x,x, –x)

= t

t+ϕ(x,x, –x)

for allg,h∈S.

It follows from (.) that

N f(x) – f(x),

 t

≥ t

t+ϕ(x,x, –x)

for allx∈Xand allt> . So,d(f,Jf)≤ .

A(x) = A(x) (.)

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This implies thatAis a unique mapping satisfying (.) such that there exists aμ∈(,∞) satisfying

Nf(x) –A(x),μt≥ t t+ϕ(x,x, –x)

for allx∈X;

()d(Jn_f_,_A)_→_{ as}_n_{→ ∞}_{. This implies the equality}

N- lim n→∞

 nf

nx=A(x)

for allx∈X;

()d(f,A)≤_–_Ld(f,Jf), which implies the inequality

d(f,A)≤   – L.

This implies that the inequality (.) holds.

The rest of the proof is similar to the proof of Theorem ..

Corollary . Letθ ≥and let p be a real number with <p< .Let X be a normed

vector space with the norm · .Let f :X→Y be an odd mapping satisfying(.).Then A(x) :=N-limn→∞nf(nx)exists for each x∈X and deﬁnes an additive mapping A:X→

Y such that

Nf(x) –A(x),t≥ ( –  p_)t

( – p_)t_{+ ( + }p₎_θxp

Proof The proof follows from Theorem . by taking

ϕ(x,y) :=θxp+yp+zp

for allx,y,z∈X. Then we can chooseL= p–_{, and we get the desired result.}

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

Using a ﬁxed point method, we prove the Hyers-Ulam stability of the Cauchy additive functional equation in matrix fuzzy normed spaces.

We will use the following notations: Mn(X)is the set of alln×n-matrices inX;

ej∈M,n(R)is thatjth component isand the other components are zero;

Eij∈Mn(R)is that(i,j)-component isand the other components are zero;

Eij⊗x∈Mn(X)is that(i,j)-component isxand the other components are zero.

Lemma . Let(X,{Nn})be a matrix fuzzy normed space.

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() For all[xij]∈Mn(X)andt=n_i_,_j_=tij,

N(xkl,t)≥Nn[xij],t≥minN(xij,tij) :i,j= , , . . . ,n,

N(xkl,t)≥Nn

[xij],t

≥min

N xij, t n

:i,j= , , . . . ,n

.

() lim_n_→∞xn=xif and only iflimn→∞xijn=xijforxn= [xijn],x= [xij]∈Mk(X). Proof () SinceEkl⊗x=e∗_kxelande_k∗=el= ,Nn(Ekl⊗x,t)≥N(x,t). Sinceek(Ekl⊗ x)e∗_l =x,Nn(Ekl⊗x,t)≤N(x,t). So,N(Ekl⊗x,t) =N(x,t).

()N(xkl,t) =N(ek[xij]e∗_l,t)≥Nn([xij],_e t

k·el) =Nn([xij],t).

Nn

[xij],t

=Nn

_n

i,j=

Eij⊗xij,t

≥minNn(Eij⊗xij,tij) :i,j= , , . . . ,n

=minN(xij,tij) :i,j= , , . . . ,n,

wheret=n_i_,_j_=tij. So,Nn([xij],t)≥min{N(xij, t

n) :i,j= , , . . . ,n}. () ByN(xkl,t)≥Nn([xij],t)≥min{N(xij, t

n) :i,j= , , . . . ,n}, we obtain the result.

For a mappingf:X→Y, deﬁneDf:X_→_Y _and_Df

n:Mn(X)→Mn(Y) by

Df(a,b) =f(a+b) –f(a) –f(b),

Dfn[xij], [yij]:=fn[xij+yij]–fn[xij]–fn[yij]

for alla,b∈Xand allx= [xij],y= [yij]∈Mn(X).

Theorem . Letϕ:X_→_[,_∞)_{be a function such that there exists an}_α_{< }_with

ϕ(a,b)≤α

ϕ(a, b) (.)

for all a,b∈X.Let f :X→Y be a mapping satisfying

NnDfn[xij], [yij],t≥ t

t+n_i_,_j_=ϕ(xij,yij)

(.)

for all t> and x= [xij],y= [yij]∈Mn(X).Then A(a) :=N-liml→∞lf(_al)exists for each

a∈X and deﬁnes an additive mapping A:X→Y such that

Nfn[xij]–An[xij],t≥ ( –α)t ( –α)t+n_αn

i,j=ϕ(xij,xij)

(.)

for all t> and x= [xij]∈Mn(X).

Proof Letn= . Then (.) is equivalent to

Nf(a+b) –f(a) –f(b),t≥ t

t+ϕ(a,b) (.)

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Lettingb=ain (.), we get

Nf(a) – f(a),t≥ t

t+ϕ(a,a) (.)

and so

N f(a) – f a 

,t

≥ t

t+ϕ(a_,a_)≥ t

t+α_ϕ(a,a) (.)

for allt>  anda∈X. Consider the set

S:={g:X→Y}

d(g,h) =inf

μ∈R+:N

g(a) –h(a),μt≥ t

t+ϕ(a,a),∀a∈X,∀t> 

,

where, as usual,infφ= +∞. It is easy to show that (S,d) is complete (see the proof of [, Lemma .]).

Jg(a) := g a 

for alla∈X.

Ng(a) –h(a),εt≥ t t+ϕ(a,a)

for alla∈Xandt> . Hence

NJg(a) –Jh(a),αεt=N g a 

– h a 

,αεt

=N g a 

–h a 

,α εt

≥

αt

 αt

 +ϕ(

a

,

a

)

≥

αt

 αt

 + α ϕ(a,a)

= t

t+ϕ(a,a)

for alla∈Xandt> . So,d(g,h) =εimplies thatd(Jg,Jh)≤αε. This means that

d(Jg,Jh)≤αd(g,h)

for allg,h∈S.

It follows from (.) thatd(f,Jf)≤α .

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()Ais a ﬁxed point ofJ,i.e., A a  = A(a)

for alla∈X. The mappingAis a unique ﬁxed point ofJin the set

M=g∈S:d(f,g) <∞.

()d(Jl_f_,_A)_→_{ as}_l_{→ ∞}_{. This implies the equality}

N-lim l→∞

l_f a l

=A(a)

for alla∈X.

()d(f,A)≤_–_αd(f,Jf), which implies the inequality

d(f,A)≤ α

 – α. (.)

By (.),

N lf a+b l

– lf a l

– lf b l

, lt

≥ t

t+ϕ(_al,_bl)

for alla,b∈Xandt> . So,

N lf a+b l

– lf a l

– lf b l ,t ≥ t l t

l + αl lϕ(a,b)

for alla,b∈Xandt> . Sinceliml→∞

t

l

t

l+

αl

lϕ(a,b)

=  for alla,b∈Xandt> ,

NA(a+b) –A(a) –A(b),t= 

for alla,b∈Xandt> . ThusA(a+b) –A(a) –A(b) = . So, the mappingA:X→Y is additive.

By Lemma . and (.),

Nn

fn

[xij]

–An

[xij]

,t≥min

N f(xij) –A(xij), t n

:i,j= , , . . . ,n

≥min

( –α)t ( –α)t+n_αϕ_(x

ij,xij)

:i,j= , , . . . ,n

≥ ( –α)t ( –α)t+n_αn

for allx= [xij]∈Mn(X). ThusA:X→Y is a unique additive mapping satisfying (.), as

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Corollary . Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping

satisfying

NnDfn[xij], [yij],t≥ t

t+n_i_,_j_=θ(xijr₊_yijr₎ (.)

for all t> and x= [xij],y= [yij]∈Mn(X).Then A(a) :=N-liml→∞lf(_al)exists for each

a∈X and deﬁnes an additive mapping A:X→Y such that

Nfn[xij]–An[xij],t≥ ( –  r_)t

( – r_)t₊_n_·_rn

i,j=θxijr

Proof The proof follows from Theorem . by taking ϕ(a,b) = θ(ar₊_br_{) for all} a,b∈X. Then we can chooseα= r–, and we get the desired result.

Theorem . Let f :X→Y be a mapping satisfying(.)for which there exists a function

ϕ:X_→_[,_∞₎_{such that there exists an}_α_{< }_with

ϕ(a,b)≤αϕ a

, b 

for all a,b∈X.Then A(a) :=N-lim_l_→∞ 

lf(la)exists for each a∈X and deﬁnes an

addi-tive mapping A:X→Y such that

Nfn[xij]–An[xij],t≥ ( –α)t ( –α)t+nn

for all t> and x= [xij]∈Mn(X).

Proof Let (S,d) be the generalized metric space deﬁned in the proof of Theorem .. Now we consider the linear mappingJ:S→Ssuch that

Jg(a) := g a 

for alla∈X.

It follows from (.) thatd(f,Jf)≤ . So,

d(f,A)≤   – α.

The rest of the proof is similar to the proof of Theorem ..

Corollary . Let r,θ be positive real numbers with r> .Let f :X→Y be a mapping

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mapping A:X→Y such that

Nfn[xij]–An[xij],t≥ ( r_{– )t}

(r_{– )t}₊_n_·_rn

i,j=θxijr

Proof The proof follows from Theorem . by taking ϕ(a,b) = θ(ar+br) for all a,b∈X. Then we can chooseα= –r_{, and we get the desired result.}

4 Fuzzy stability of the Cauchy-Jensen additive functional inequality (1.3) in fuzzy normed spaces

In this section, using the ﬁxed point method, we prove the generalized Hyers-Ulam sta-bility of the Cauchy-Jensen additive functional inequality (.) in fuzzy Banach spaces.

Theorem . Letϕ:X_→_[,_∞₎_{be a function such that there exists an L}_{< }_with

ϕ(x,y,z)≤L

ϕ(x, y, z)

for all x,y,z∈X.Let f :X→Y be an odd mapping satisfying

Nf(x) +f(y) +f(z),t≥min

N f x+y  +z

,t 

, t

t+ϕ(x,y,z)

(.)

for all x,y,z∈X and all t> .Then A(x) :=N-limn→∞nf(xn)exists for each x∈X and

Nf(x) –A(x),t≥ ( – L)t

( – L)t+Lϕ(x,x, –x) (.)

Proof Lettingy=x= –zin (.), we get

Nf(x) – f(x),t≥ t

t+ϕ(x,x, –x) (.)

for allx∈X. Consider the set

S:={g:X→Y}

d(g,h) =inf

μ∈R+:N

g(x) –h(x),μt≥ t

t+ϕ(x,x, –x),∀x∈X,∀t> 

,

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Jg(x) := g x 

for allx∈X.

Ng(x) –h(x),εt≥ t t+ϕ(x,x, –x)

NJg(x) –Jh(x),Lεt=N g x 

– h x 

,Lεt

=N g x 

–h x 

,L εt

≥

Lt



Lt

 +ϕ(

x

,

x

, –

x

)

≥

Lt



Lt

 +

L

ϕ(x,x, –x)

= t

t+ϕ(x,x, –x)

for allg,h∈S.

It follows from (.) that

N f(x) – f x 

,L t

≥ t

t+ϕ(x,x, –x)

for allx∈Xand allt> . So,d(f,Jf)≤L_.

A x 

=

A(x) (.)

M=g∈S:d(f,g) <∞.

This implies thatAis a unique mapping satisfying (.) such that there exists aμ∈(,∞) satisfying

Nf(x) –A(x),μt≥ t t+ϕ(x,x, –x)

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()d(Jn_f_,_A)_→_{ as}_n_{→ ∞. This implies the equality}

N- lim n→∞

n_f x n

=A(x)

for allx∈X;

d(f,A)≤ L  – L.

This implies that the inequality (.) holds.

The rest of proof is similar to the proof of Theorem ..

Corollary . Letθ≥and let p be a real number with p> .Let X be a normed vector space with the norm · .Let f :X→Y be an odd mapping satisfying

Nf(x) +f(y) +f(z),t

≥min

N f x+y  +z

,t 

, t

t+θ(xp₊_yp₊_zp₎

(.)

Nf(x) –A(x),t≥ ( p_{– )t}

(p_{– )t}_{+ }_θ_xp

Proof The proof follows from Theorem . by taking

ϕ(x,y) :=θxp+yp+zp

for allx,y,z∈X. Then we can chooseL= –p_{, and we get the desired result.}

Theorem . Letϕ:X→[,∞)be a function such that there exists an L< with

ϕ(x,y,z)≤Lϕ x

, y ,

z 

for all x,y,z ∈ X. Let f :X → Y be an odd mapping satisfying (.). Then A(x) := N-lim_n_→∞ 

nf(nx)exists for each x∈X and deﬁnes an additive mapping A:X→Y

such that

Nf(x) –A(x),t≥ ( – L)t

( – L)t+ϕ(x,x, –x) (.)

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Proof Let (S,d) be the generalized metric space deﬁned in the proof of Theorem .. Consider the linear mappingJ:S→Ssuch that

Jg(x) := g(x)

for allx∈X.

Ng(x) –h(x),εt≥ t t+ϕ(x,x, –x)

NJg(x) –Jh(x),Lεt=N  g(x) –



h(x),Lεt

=Ng(x) –h(x), Lεt

≥ Lt

Lt+ϕ(x, x, –x)≥

Lt Lt+ Lϕ(x,x, –x)

= t

t+ϕ(x,x, –x)

for allg,h∈S.

It follows from (.) that

N f(x) – f(x),

 t

≥ t

t+ϕ(x,x, –x)

for allx∈Xand allt> . So,d(f,Jf)≤_.

A(x) = A(x) (.)

M=g∈S:d(f,g) <∞.

This implies thatAis a unique mapping satisfying (.) such that there exists aμ∈(,∞) satisfying

Nf(x) –A(x),μt≥ t t+ϕ(x,x, –x)

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()d(Jn_f_,_A)_→_{ as}_n_{→ ∞. This implies the equality}

N- lim n→∞

 nf

nx=A(x)

for allx∈X;

d(f,A)≤   – L.

This implies that the inequality (.) holds.

The rest of the proof is similar to the proof of Theorem ..

Corollary . Letθ≥and let p be a real number with <p< .Let X be a normed

vector space with the norm · .Let f :X→Y be an odd mapping satisfying(.).Then A(x) :=N-lim_n_→∞ 

nf(nx)exists for each x∈X and deﬁnes an additive mapping A:X→

Y such that

Nf(x) –A(x),t≥ ( –  p_)t

( – p_)t_{+ }_θ_xp

Proof The proof follows from Theorem . by taking

ϕ(x,y) :=θxp+yp+zp

for allx,y,z∈X. Then we can chooseL= p–_{, and we get the desired result.}

5 Hyers-Ulam stability of the additive functional inequality (1.2) in matrix normed spaces

In this section, we prove the Hyers-Ulam stability of the additive functional inequality (.) in matrix normed spaces by using the direct method and by using the ﬁxed point method.

Lemma . Let(X,{ · n})be a matrix normed space.

() Ekl⊗xn=xforx∈X; () xkl ≤ [xij]n≤

n

i,j=xijfor[xij]∈Mn(X);

() limn→∞xn=xif and only iflimn→∞xnij=xijforxn= [xnij],x= [xij]∈Mk(X).

Proof () SinceEkl⊗x=e∗_kxelande∗_k=el= ,Ekl⊗xn≤ x. Sinceek(Ekl⊗x)e∗_l =x, x ≤ Ekl⊗xn. So,Ekl⊗xn=x.

() Sinceekxe∗_l =xklandek=e∗_l= ,xkl ≤ [xij]n. Since [xij] =

n

i,j=Eij⊗xij,

[xij]_n=

n

i,j=

Eij⊗xij

n ≤

n

i,j=

Eij⊗xijn= n

i,j=

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() By (), we have

xnkl–xkl ≤[xnij–xij]_n=[xnij] – [xij]_n≤ n

i,j=

xnij–xij.

So, we get the result.

We need the following result.

Lemma .[, Proposition .] Let f :X→Y be a mapping such that

f(a) +f(b) +f(c)≤f(a+b+c)

for all a,b,c∈X.Then f :X→Y is additive.

Theorem . Let f :X→Y be a mapping and letφ:X_→_[,_∞)_{be a function such that}

(a,b,c) := 

∞

l=

 lφ

la, lb, lc< +∞, (.)

_fn

[xij]+fn[yij]+fn[zij]_n

≤fn[xij] + [yij] + [zij]_n+ n

i,j=

φ(xij,yij,zij) (.)

for all a,b,c∈X and all x= [xij],y= [yij],z= [zij]∈Mn(X).Then there exists a unique additive mapping A:X→Y such that

fn[xij]–An[xij]_n≤ n

i,j=

(xij,xij, –xij) (.)

for all x= [xij]∈Mn(X).

Proof Whenn= , (.) is equivalent to

f(a) +f(b) +f(c)≤f(a+b+c)+φ(a,b,c)

for alla,b,c∈X. By the same reasoning as in the proof of [, Theorem .], one can show that there is a unique additive mappingA:X→Ysuch that

f(a) –A(a)≤(a,a, –a)

for alla∈X. The mappingA:X→Yis given by

A(a) = lim l→∞

 lf

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for alla∈X. By Lemma .,

_fn

[xij]–An[xij]_n≤ n

i,j=

_f_{(xij) –}_A(xij)_≤n i,j=

(xij,xij, –xij)

for allx= [xij]∈Mn(X). ThusA:X→Y is a unique additive mapping satisfying (.), as

desired.

Corollary . Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping such that

fn[xij]+fn[yij]+fn[zij]_n≤fn[xij] + [yij] + [zij]_n

+ n

i,j=

θxijr+yijr+zijr (.)

for all x= [xij],y= [yij],z= [zij]∈Mn(X).Then there exists a unique additive mapping A: X→Y such that

fn

[xij]

–An

[xij]_n≤ n

i,j=

 + r  – rθxij

r

Proof Lettingφ(a,b,c) =θ(ar₊_br₊_cr_{) in Theorem ., we obtain the result.}

Theorem . Let f :X→Y be a mapping and letφ:X_→_[,_∞₎_{be a function satisfying}

(.)and

(a,b,c) := 

∞

l=

lφ a

l, b l,

c l

< +∞, (.)

for all a,b,c∈X.Then there exists a unique additive mapping A:X→Y such that

i,j=

(xij,xij, –xij)

Proof The proof is similar to the proof of Theorem ..

Corollary . Let r,θ be positive real numbers with r> .Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

i,j=

r_{+ } r_{– }θxij

r

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Proof Lettingφ(a,b,c) =θ(ar₊_br₊_cr_{) in Theorem ., we obtain the result.}

We need the following result.

Lemma .[] If E is an L∞-matrix normed space,then[xij]n≤ [xij]nfor all[xij]∈ Mn(E).

Theorem . Let Y be an L∞-normed Banach space.Let f :X→Y be a mapping and let

φ:X_→_[,_∞₎_{be a function satisfying}_(.)_and

_fn

[xij]+fn[yij]+fn[zij]_n≤fn[xij] + [yij] + [zij]_n+φ(xij,yij,zij)_n (.)

f(xij) –A(xij)_n≤(xij,xij, –xij)_n (.)

for all x= [xij]∈Mn(X).Hereis given in Theorem..

Proof By the same reasoning as in the proof of Theorem ., there exists a unique additive mappingA:X→Ysuch that

f(a) –A(a)≤(a,a, –a)

for alla∈X. The mappingA:X→Yis given by

A(a) = lim l→∞

 lf

la

for alla∈X.

It is easy to show that if ≤aij≤bijfor alli,j, then

[aij]_n≤[bij]_n. (.)

By Lemma . and (.),

f(xij) –A(xij)_n≤f(xij) –A(xij)_n≤(xij,xij, –xij)_n

for allx= [xij]∈Mn(X). So, we obtain the inequality (.).

Corollary . Let Y be an L∞-normed Banach space.Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping such that

fn

[xij]

+fn

[yij]

+fn

[zij]_n≤fn

[xij] + [yij] + [zij]_n

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fn[xij]–An[xij]_n≤

 – r  – rθxij

r n

Proof Lettingφ(a,b,c) =θ(ar₊_br₊_cr_{) in Theorem ., we obtain the result.}

Theorem . Let Y be an L∞-normed Banach space.Let f:X→Y be a mapping and let

φ:X→[,∞)be a function satisfying(.)and(.).Then there exists a unique additive mapping A:X→Y such that

f(xij) –A(xij)_n≤(xij,xij, –xij)_n

for all x= [xij]∈Mn(X).Hereis given in Theorem..

Proof The proof is similar to the proof of Theorem ..

Corollary . Let Y be an L∞-normed Banach space.Let r,θ be positive real numbers with r> .Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

r+  r_{– }θxij

r n

Proof Lettingφ(a,b,c) =θ(ar₊_br₊_cr_{) in Theorem ., we obtain the result.}

Theorem . Letφ:X→[,∞)be a function such that there exists anα< with

φ(a,b,c)≤αφ a

, b ,

c 

(.)

for all a,b,c∈X.Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

i,j=



 – αφ(xij,xij, –xij) (.)

for all x= [xij]∈Mn(X).

Proof Whenn= , (.) is equivalent to

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for alla,b,c∈X. It follows from (.) that

f(a) –f(a)≤φ(a,a, –a) (.)

for alla∈X. So,

f(a) – 

f(a)

≤ 

φ(a,a, –a) (.)

for alla∈X. Consider the set

S:={h:X→Y}

d(g,h) =infμ∈R+:g(a) –h(a)≤μφ(a,a, –a),∀a∈X

,

where, as usual,inf{}= +∞. It is easy to show that (S,d) is complete (see []). Now we consider the linear mappingJ:S→Ssuch that

Jg(a) :=  g(a)

for alla∈X.

g(a) –h(a)≤φ(a,a, –a)

for alla∈X. Hence

Jg(a) –Jh(a)= g(a) –

 h(a)

≤αφ(a,a, –a)

for alla∈X. So,d(g,h) =εimplies thatd(Jg,Jh)≤αε. This means that

d(Jg,Jh)≤αd(g,h)

for allg,h∈S.

It follows from (.) thatd(f,Jf)≤ .

A(a) = A(x) (.)

for allx∈X. The mappingAis a unique ﬁxed point ofJin the set

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This implies thatAis a unique mapping satisfying (.) such that there exists aμ∈(,∞) satisfying

f(a) –A(a)≤μφ(a,a, –a)

for alla∈X;

()d(Jl_f_,_A)_→_{ as}_l_{→ ∞. This implies the equality}

lim l→∞

 lf

la=A(a)

for alla∈X;

()d(f,A)≤_–_αd(f,Jf), which implies the inequality

d(f,A)≤   – α.

So,

f(a) –A(a)≤ 

 – αφ(a,a, –a) (.)

for alla∈X.

It follows from (.) and (.) that

lim l→∞

 lf

la+flb+flc

≤ lim l→∞

 lf

l(a+b+c)+  lφ

la, lb, lc (.)

for alla,b,c∈X.

It follows from (.) that

A(a) +A(b) +A(c)≤A(a+b+c)

for alla,b,c∈X. By Lemma .,A:X→Yis additive. By Lemma . and (.),

_fn

[xij]–An[xij]_n≤ n

i,j=

_f(xij) –A(xij)≤ n

i,j=



 – αφ(xij,xij, –xij)

for allx= [xij]∈Mn(X). ThusA:X→Yis a unique additive mapping satisfying (.), as

desired.

Corollary . Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

i,j=

 + r  – rθxij

r

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Proof The proof follows from Theorem . by takingφ(a,b,c) =θ(ar₊_br₊_cr_{) for} alla,b,c∈X. Then we can chooseα= r–, and we get the desired result.

Theorem . Let f :X→Y be a mapping satisfying(.)for which there exists a function

φ:X_→_[,_∞)_{such that there exists an}_α_{< }_with

φ(a,b,c)≤α

φ(a, b, c) (.)

for all a,b,c∈X.Then there exists a unique additive mapping A:X→Y such that

fn

[xij]

–An

[xij]_n≤ n

i,j= α

 – αφ(xij,xij, –xij)

Proof Let (S,d) be the generalized metric space deﬁned in the proof of Theorem .. Now we consider the linear mappingJ:S→Ssuch that

Jg(a) := g a 

for alla∈X.

It follows from (.) that

f(a) – f a 

≤φ a

, a , –a

≤α

φ(a,a, –a)

for alla∈X. Thusd(f,Jf)≤α . So,

d(f,A)≤ α  – α.

The rest of the proof is similar to the proof of Theorem ..

Corollary . Let r,θ be positive real numbers with r> .Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

i,j=

r_{+ } r_{– }θxij

r

Proof The proof follows from Theorem . by takingφ(a,b,c) =θ(ar₊_br₊_cr_{) for} alla,b,c∈X. Then we can chooseα= –r_{, and we get the desired result.}

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Theorem . Let f :X→Y be a mapping and letφ:X_→_[,_∞)_{be a function satisfying}

(.)and(.).Then there exists a unique additive mapping A:X→Y such that

f(xij) –A(xij)_n≤



 – αφ(xij,xij, –xij)

n

(.)

Proof By the same reasoning as in the proof of Theorem ., there exists a unique additive mappingA:X→Ysuch that

f(a) –A(a)≤ 

 – αφ(a,a, –a)

for alla∈X.

By Lemma . and (.),

f(xij) –A(xij)_n≤f(xij) –A(xij)_n≤



 – αφ(xij,xij)

n

for allx= [xij]∈Mn(X). So, we obtain the inequality (.).

Corollary . Let r,θ be positive real numbers with r< .Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

fn

[xij]

–An

[xij]_n≤

 – _{ – }rrθxij r

n

Proof The proof follows from Theorem . by takingφ(a,b,c) =θ(ar₊_br₊_cr_{) for} alla,b,c∈X. Then we can chooseα= r–_{, and we get the desired result.}

Theorem . Let f:X→Y be a mapping and letφ:X→[,∞)be a function satisfying (.)and(.).Then there exists a unique additive mapping A:X→Y such that

f(xij) –A(xij)_n≤

α

 – αφ(xij,xij, –xij)

n

Proof The proof is similar to the proof of Theorem ..

Corollary . Let r,θ be positive real numbers with r> .Let f :X→Y be a mapping satisfying(.).Then there exists a unique additive mapping A:X→Y such that

+  r_{– }θxij

r n

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Proof The proof follows from Theorem . by takingφ(a,b,c) =θ(ar₊_br₊_cr_{) for} alla,b,c∈X. Then we can chooseα= –r_{, and we get the desired result.}

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.

Author details

1_{Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, Korea.} 2_{Department of Mathematics, Daejin University, Kyeonggi, 487-711, Korea.}3_{Department of Mathematics, University of} Seoul, Seoul, 130-743, Korea.

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).

Received: 26 October 2012 Accepted: 22 April 2013 Published: 6 May 2013

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doi:10.1186/1029-242X-2013-224