Fuzzy stability of functional inequalities in matrix fuzzy normed spaces

R E S E A R C H Open Access

Fuzzy stability of functional inequalities in matrix fuzzy normed spaces

Choonkil Park¹, Dong Yun Shin^2*and Jung Rye Lee³

*Correspondence: [email protected]

2Department 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,_|c|^t )ifc= ;

(N_) N(x+y,s+t)≥min{N(x,s),N(y,t)};

(N_) N(x,·)is a non-decreasing function ofRandlim_t→∞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 thatlim_n→∞N(xn–x,t) = 

©2013 Park et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion 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.

for allt> . In this case,xis called thelimit of the sequence{xn}and we denote it by N-lim_n→∞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 becom- 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{x_n} 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;

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

Forx∈M_n(X),y∈M_k(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∈M_n(X) and ally∈M_k(X).

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

h_n [x_ij]

= h(x_ij) 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·B^t )for allt> , A∈M_k,n(R),x= [x_ij]∈M_n(X)andB∈M_n,k(R)withA · B = .

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

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

n for allt>  andx= [xij]∈Mn(X). Then

Nk(AxB,t) = t t+AxBk

≥ t

t+A · xn· B=

A·Bt

A·Bt +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) +f xy^–

.

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.

We recall a fundamental result in fixed point theory.

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

d

Jⁿx,J^n+x

=∞

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

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

() y^∗is the unique fixed point ofJin the setY={y∈X|d(J^nx,y) <∞};

() d(y,y^∗)≤_–α^ d(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

N

f(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

ϕ(x, y, z)

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

f(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-lim_n→∞ⁿf(_^xn)exists for each x∈X and defines an additive mapping A:X→Y such that

N

f(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

N

f(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 N

g(x) –h(x),εt

≥ t t+ϕ(x,x, –x) for allx∈Xand allt> . Hence

N

Jg(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 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

N

f(x) –A(x),μt

≥ t t+ϕ(x,x, –x) for allx∈X;

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

n→∞ⁿf x

ⁿ

=A(x) for allx∈X;

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

d(f,A)≤ L

 – L.

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

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_ⁿnϕ(x,y,z)

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

t

n t

n+^Ln_nϕ(x,y,z)=  for allx,y,z∈Xand allt> ,

N

A(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

N

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

≥min

N f(x+y+z),t



, t

t+θ(x^p+y^p+z^p)

(.) for all x,y,z∈X and all t> .Then A(x) :=N-lim_n→∞ⁿf(_^xn)exists for each x∈X and defines an additive mapping A:X→Y such that

N

f(x) –A(x),t

≥ (^p– )t

(^p– )t+ ( + ^p)θx^p for all x∈X and all t> .

Proof The proof follows from Theorem . by taking ϕ(x,y) :=θ

x^p+y^p+z^p

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

N

f(x) –A(x),t

≥ ( – L)t

( – L)t+ϕ(x,x, –x) (.)

for all x∈X and all t> .

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.

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

g(x) –h(x),εt

≥ t t+ϕ(x,x, –x) for allx∈Xand allt> . Hence

N

Jg(x) –Jh(x),Lεt

=N 

g(x) – 

h(x),Lεt

=N

g(x) –h(x), Lεt

≥ Lt

Lt+ϕ(x, x, –x)≥ Lt

Lt+ Lϕ(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),

t

≥ t t+ϕ(x,x, –x) for allx∈Xand allt> . So,d(f,Jf)≤^_.

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

N

f(x) –A(x),μt

≥ t t+ϕ(x,x, –x) for allx∈X;

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

n→∞



ⁿf

ⁿx

=A(x) for allx∈X;

()d(f,A)≤_–L^ d(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-lim_n→∞^nf(ⁿx)exists for each x∈X and defines an additive mapping A:X→ Y such that

N

f(x) –A(x),t

≥ ( – ^p)t ( – ^p)t+ ( + ^p)θx^p for all x∈X and all t> .

Proof The proof follows from Theorem . by taking ϕ(x,y) :=θ

x^p+y^p+z^p

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:

M_n(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;

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

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

() Nn(Ekl⊗x,t) =N(x,t)for allt> andx∈X.

() For all[xij]∈Mn(X)andt=_n

i,j=tij, N(x_kl,t)≥Nn

[x_ij],t

≥min

N(x_ij,tij) :i,j= , , . . . ,n , N(x_kl,t)≥N_n

[x_ij],t

≥min

N x_ij, t n^

:i,j= , , . . . ,n

.

() lim_n→∞x_n=xif and only iflim_n→∞x_ijn=x_ijforx_n= [x_ijn],x= [x_ij]∈M_k(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).

N_n [x_ij],t

=N_{n n}

i,j=

E_ij⊗x_ij,t

≥min

N_n(E_ij⊗x_ij,t_ij) :i,j= , , . . . ,n

=min

N(xij,tij) :i,j= , , . . . ,n , wheret=_n

i,j=tij. So,Nn([xij],t)≥min{N(xij,_n^t) :i,j= , , . . . ,n}.

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

For a mappingf:X→Y, defineDf:X^→Y andDf_n:M_n(X^)→M_n(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= [x_ij],y= [y_ij]∈M_n(X).

Theorem . Letϕ:X^→[,∞)be a function such that there exists anα< with ϕ(a,b)≤α

ϕ(a, b) (.)

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

Dfn

[xij], [yij] ,t

≥ t

t+n

i,j=ϕ(x_ij,y_ij) (.)

for all t> and x= [xij],y= [yij]∈Mn(X).Then A(a) :=N-lim_l→∞^lf(_^al)exists for each a∈X and defines an additive mapping A:X→Y such that

N fn

[xij] –An

[xij] ,t

≥ ( –α)t

( –α)t+n^α_n

i,j=ϕ(xij,xij) (.)

for all t> and x= [xij]∈Mn(X).

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

f(a+b) –f(a) –f(b),t

≥ t

t+ϕ(a,b) (.)

for allt>  anda,b∈X.

Lettingb=ain (.), we get N

f(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}

and introduce the generalized metric onS:

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 .]).

Now we consider the linear mappingJ:S→Ssuch that Jg(a) := g a



for alla∈X.

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

g(a) –h(a),εt

≥ t t+ϕ(a,a) for alla∈Xandt> . Hence

N

Jg(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_.

By Theorem ., there exists a mappingA:X→Ysatisfying the following:

()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(J^lf,A)→ asl→ ∞. This implies the equality N-lim

l→∞^lf a

^l

=A(a) for alla∈X.

()d(f,A)≤_–α^ d(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> . Sincelim_l→∞

t

l t

l+^αl

lϕ(a,b)=  for alla,b∈Xandt> , N

A(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 (.), N_n

f_n [x_ij]

–A_n [x_ij]

,t

≥min

N f(x_ij) –A(x_ij), t n^

:i,j= , , . . . ,n

≥min

( –α)t

( –α)t+n^αϕ(x_ij,x_ij):i,j= , , . . . ,n

≥ ( –α)t

( –α)t+n^α_n

i,j=ϕ(xij,xij)

for allx= [x_ij]∈M_n(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 satisfying

Nn

Dfn

[xij], [yij] ,t

≥ t

t+n

i,j=θ(xij^r+yij^r) (.)

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

^l)exists for each a∈X and defines an additive mapping A:X→Y such that

N fn

[xij] –An

[xij] ,t

≥ ( – ^r)t

( – ^r)t+n^·^rn

i,j=θx_ij^r for all t> and x= [xij]∈Mn(X).

Proof The proof follows from Theorem . by taking ϕ(a,b) = θ(a^r+b^r) 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 anα< 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

N fn

[xij] –An

[xij] ,t

≥ ( –α)t

( –α)t+n^_n

i,j=ϕ(xij,xij) for all t> and x= [x_ij]∈M_n(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 satisfying(.).Then A(a) :=N-lim_l→∞^lf(_^al)exists for each a∈X and defines an additive

mapping A:X→Y such that N

fn

[xij] –An

[xij] ,t

≥ (^r– )t

(^r– )t+n^·^rn

i,j=θx_ij^r for all t> and x= [xij]∈Mn(X).

Proof The proof follows from Theorem . by taking ϕ(a,b) = θ(a^r+b^r) 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 N

f(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-lim_n→∞ⁿf(_^xn)exists for each x∈X and defines an additive mapping A:X→Y such that

N

f(x) –A(x),t

≥ ( – L)t

( – L)t+Lϕ(x,x, –x) (.)

for all x∈X and all t> .

Proof Lettingy=x= –zin (.), we get N

f(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 N

g(x) –h(x),εt

≥ t t+ϕ(x,x, –x) for allx∈Xand allt> . Hence

N

Jg(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 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

N

f(x) –A(x),μt

≥ t t+ϕ(x,x, –x) for allx∈X;

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

n→∞ⁿf x

ⁿ

=A(x) for allx∈X;

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

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

N

f(x) +f(y) +f(z),t

≥min

N f x+y

 +z

,t



, t

t+θ(x^p+y^p+z^p)

(.) for all x,y,z∈X and all t> .Then A(x) :=N-lim_n→∞ⁿf(_^xn)exists for each x∈X and defines an additive mapping A:X→Y such that

N

f(x) –A(x),t

≥ (^p– )t (^p– )t+ θx^p for all x∈X and all t> .

Proof The proof follows from Theorem . by taking ϕ(x,y) :=θ

x^p+y^p+z^p

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

N

f(x) –A(x),t

≥ ( – L)t

( – L)t+ϕ(x,x, –x) (.)

for all x∈X and all t> .

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.

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

g(x) –h(x),εt

≥ t t+ϕ(x,x, –x) for allx∈Xand allt> . Hence

N

Jg(x) –Jh(x),Lεt

=N 

g(x) – 

h(x),Lεt

=N

g(x) –h(x), Lεt

≥ Lt

Lt+ϕ(x, x, –x)≥ Lt

Lt+ Lϕ(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),

t

≥ t t+ϕ(x,x, –x) for allx∈Xand allt> . So,d(f,Jf)≤^_.

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

N

f(x) –A(x),μt

≥ t t+ϕ(x,x, –x) for allx∈X;

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

n→∞



ⁿf

ⁿx

=A(x) for allx∈X;

()d(f,A)≤_–L^ d(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-lim_n→∞ ^

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

N

f(x) –A(x),t

≥ ( – ^p)t ( – ^p)t+ θx^p for all x∈X and all t> .

Proof The proof follows from Theorem . by taking ϕ(x,y) :=θ

x^p+y^p+z^p

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;

() x_kl ≤ [x_ij]n≤n

i,j=x_ijfor[x_ij]∈M_n(X);

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

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

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

i,j=E_ij⊗x_ij, [xij]

n=

n i,j=

Eij⊗xij

n

≤ n

i,j=

Eij⊗xijn= n i,j=

xij.

() 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= [x_ij],y= [y_ij],z= [z_ij]∈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

^la