On stability of functional equations related to quadratic mappings in fuzzy Banach spaces

R E S E A R C H

Open Access

On stability of functional equations related to

quadratic mappings in fuzzy Banach spaces

Seong Sik Kim

_{, Nawab Hussain}

_{and Yeol Je Cho}

*_{Correspondence:}

[email protected]; [email protected]

2_{Department of Mathematics, King}

Abdulaziz University, Jeddah, 21589, Saudi Arabia

3_{Department of Mathematics}

Education and the RINS, Gyeongsang National University, Chinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract

In this paper, we establish the generalized Hyers-Ulam stability problem of radical quadratic functional equationsf(x2_+y2_{) =f(x) +f(y) in fuzzy Banach spaces via the} direct and fixed point methods.

MSC: 39B72; 39B82; 39B52; 47H09

Keywords: radical functional equations; stability; fuzzy Banach spaces

1 Introduction

The stability problem concerning the stability of group homomorphisms of functional equations was originally introduced by Ulam [] in . The famous Ulam stability prob-lem was partially solved by Hyers [] for a linear functional equation of Banach spaces. Hyers’ theorem was generalized by Aoki [] for additive mappings and by Rassias [] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has had a lot of influence in the development of what we call the generalized Hyers-Ulam stability of functional equations. A generalization of Rassias’ theorem was obtained by Gˇavruta [] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. Cădariu and Radu [] applied thefixed point methodto the investigation of the Jensen functional equation. They could present a short and sim-ple proof (different from thedirect methodinitiated by Hyers in ) for the generalized Hyers-Ulam stability of the Jensen functional and the quadratic functional equations.

The functional equation

f(x+y) +f(x–y) = f(x) + f(y) (.)

is called aquadratic functional equation. Quadratic functional equations were used to characterize inner product spaces. In particular, every solution of the quadratic equation is said to be aquadratic mapping. The generalized Hyers-Ulam stability problem for the quadratic functional equation (.) was proved by Skof []. Recently, the stability problem of the radical quadratic functional equations in various spaces was proved in the papers [–].

In , Katsaras [] defined a fuzzy norm on a linear 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 [–]. Cheng and Mordeson [] introduced a definition of fuzzy norm on a linear space in such a manner that the corresponding

induced fuzzy metric is of the Kramosil and Michálek type []. In , Bag and Samanta [] modified the definition of Cheng and Mordeson by removing a regular condition. Also, they investigated a decomposition theorem of a fuzzy norm into a family to crisp norms and gave some properties of fuzzy norm. The fuzzy stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning these problems [, –].

In the sequel, we use the definitions and some basic facts concerning fuzzy Banach spaces given in Bag and Samanta [].

Definition . LetXbe a real linear space. A functionN:X×R→[, ] is called afuzzy normonXif, for allx,y∈Xands,t∈R,Nsatisfies the following conditions:

(N) N(x,t) = for allt≤;

(N) x= if and only ifN(x,t) = for allt> ; (N) N(cx,t) =N(x,t/|c|)for allc∈Rwithc= ; (N) N(x+y,s+t)≥min{N(x,s),N(y,t)};

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

(N) for allx∈Xwithx= ,N(x,·)is continuous onR.

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

Example . Let (X, · ) be a normed linear space and letα,β> . Then

N(x,t) =

⎧ ⎨ ⎩

αt

αt+βx, t> ,x∈X;

, t≤,x∈X,

is a fuzzy norm onX.

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

() A sequence{xn}inXis said to beconvergentto a pointx∈Xif, for any> and

t> , there existsn∈Z+such thatN(xn–x,t) >  –for alln≥n. In this case,xis

called thelimitof the sequence{xn}, which is denoted byx=limn→∞xn. () A sequence{xn}inXis called aCauchy sequenceif, for any> andt> , there

existsn∈Z+such thatN(xn+p–xn,t) >  –for alln≥nandp∈Z+.

() If every Cauchy sequence is convergent, then the fuzzy norm is said to becomplete and the fuzzy normed linear space is called afuzzy Banach space.

A mappingf :X→Y between fuzzy normed linear spacesXandY is said to be

con-tinuousat a pointx∈Xif, for any sequence{xn}inXconverging to a pointx∈X, the

sequence{f(xn)}converges tof(x). Iff :X→Y is continuous at every pointx∈X, then f is said to becontinuousonX.

Example . LetN:R×R→[, ] be a fuzzy norm onRdefined by

N(x,t) =

⎧ ⎨ ⎩

t

t+x, t> ;

, t≤.

In this paper, we establish the generalized Hyers-Ulam stability problem of a radical quadratic functional equationf(x_+y_{) =f(x) +f(y) in fuzzy Banach spaces via the}

di-rect and fixed point methods.

2 Fuzzy stability of the radical quadratic functional equations

In this section, we study a fuzzy version of the generalized Hyers-Ulam stability of func-tional equation which approximate uniformly a radical quadratic mapping in fuzzy Banach spaces.

2.1 The direct method

Theorem . Let∈ {–, }be fixed, (Y,N)be a fuzzy Banach space andφ:R_→[,∞) be a mapping such that

(x,y) :=

∞

n=–



 nφ

n_{x, }n_y+φn+_{x, <}∞ _(.)

for all x,y∈R.Suppose that f :R→Yis a mapping with f() = such that,for all t> ,

lim

t→∞N

fx_+y_{–f(x) –f(y),tφ(x,y)=  (.)}

uniformly onR.Then there exists a unique quadratic mapping Q:R→Ysuch that,if there existδ> andα> such that

Nfx_+y_{–f(x) –f(y),δφ(x,y)}≥_{α (.)}

for all x,y∈R,then

N f(x) –Q(x),δ (x,x)

≥α (.)

for all x∈R.Furthermore,the quadratic mapping Q:R→Yis a unique mapping such that,for all t> ,

Nf(x) –Q(x),t(x,x)=  (.)

uniformly onR.

Proof Assume that= . For any> , by (.), we can find somet>  such that

Nfx_+y_{–f(x) –f(y),tφ(x,y)≥ – (.)}

for allx,y∈Randt≥t. Replacingxandyby x√+_yandx√–_yin (.), respectively, we have

N fx_+y_–f x_√+y



–f x√–y



,tφ x√+y

 ,

x–y

√ 

for allx,y∈Randt≥t. It follows from (.), (.), and (N) that

N f(x) +f(y) –f x√+y



–f x√–y



,t φ(x,y) +φ x√+y

 ,

x–y

√ 

≥ – (.)

for allx,y∈Randt≥t. Lettingy=xin (.), we have

Nf(x) –f_x,tφˆ_(x,x)≥_{ – (.)}

for allx∈Randt≥t, whereφˆ(x,x) =φ(x,x) +φ( 

_{x, ). By induction onn, we have}

N

nf(x) –fn_x,t

n–

k=

n–k–φˆk_{x, }k_x

≥ – (.)

for allx∈R,t≥tandn∈Z+. Lett=t. Replacingnandxbypand 

n

_{xin (.),}

respec-tively, we have

N

f(nx)

n –

f(n+px)

n+p ,

t

n+p p–

k=

p–k–_φˆ_n+k_{x, }n+k_x

≥ – (.)

for alln≥ andp> . It follows from (.) and the equality

p–

k=

 n+k+φˆ

n+k_{x, }n+k_x=

 n+p–

k=n  kφˆ

k_{x, }k_x

that, for anyδ> , there exists somen∈Z+such that t

 n+p–

k=n  kφˆ

k_{x, }k_x<δ

for alln≥nandp> . Now, it follows from (.) that

N f(

n

_x)

n –

f(n+p_x)

n+p ,δ

≥N

f(n_x)

n –

f(n+p_x)

n+p ,

t

n+p p–

k=

p–k–φˆn+k_{x, }n+k_x

≥ – (.)

for alln≥nandp> . Thus the sequence{f(

n

_x)

n }is a Cauchy sequence in a fuzzy Banach spaceYand so it converges to someQ(x)∈Y. We can define a mappingQ:R→Yby

Q(x) = lim

n→∞

f(n_x)

n ,

that is,limn→∞N(f(

n

x)

n –Q(x),t) =  for allx∈Randt> . Letx,y∈R,t>  and  << . Sincelimn→∞_nφˆ(

n

x, ny) = , there existsn_∈Z+withn_>n_such that

tφˆ

n_{x, }n_y<

n_t

for alln≥n. Then, by (N), we have

NQx_+y_{–Q(x) –Q(y),t}

≥min

N Q(x+y) –  nf

n_x_{+ }n_y_, t



,N Q(x) –  nf

n_x, t



,

N Q(y) –  nf

n_y,t



,N fn_x_{+ }n_y_–fn_x–fn_y,

n_t



(.)

for alln≥n. Since the first three terms on the right-hand side of the above inequality

tend to  asn→ ∞and

Nfn_x_{+ }n_y_–fn_x–fn_y,tφˆ_n_{, }n_y≥_{ –,}

we have

NQx_+y_{–Q(x) –Q(y),t≥ –}

for allx,y∈R,t>  and  << . It follows from (N) thatQ(x_+y_{) =Q(x) +Q(y) for}

allx,y∈R. This means thatQis a quadratic mapping [].

Now, suppose that (.) holds for someδ>  andα> . Then assume that

ψn(x,y) = n–

k=

 k+φˆ

k_{x, }k_y

for allx,y∈R. For allx∈R, by a similar method to the beginning of the proof, we have

N

nf(x) –fn_x,δ

n–

k=

n–k–φˆk_{x, }k_x

≥α (.)

for alln∈Z+_{. Lett> . Then we have}

Nf(x) –Q(x),δψn(x,x) +t

≥min

N f(x) –f( n

_x)

n ,δψn(x,x)

,N f(

n

_x)

n –Q(x),t

. (.)

Combining (.) and (.) and using the factlimn→∞N(f(

n

x)

n –Q(x),t) = , we obtain

Nf(x) –Q(x),δψn(x,x) +t

≥α (.)

for large enoughn∈Z+_{. It follows from the continuity of the functionN(f(x) –Q(x),·) that}

N f(x) –Q(x),δ

(x,x) +t

≥α.

Next, assume that there exists another quadratic mappingT which satisfies (.). For any> , by applying (.) for the mappingsQandT, we can find somet>  such that

N f(x) –Q(x),t (x,x)

≥ –, N f(x) –T(x),t (x,x)

≥ –

for allx∈Randt≥t. Fixx∈Randc> . Then we find somen∈Z+such that

t ∞

k=n  kφˆ

k_{x, }k_y< c



for allx,y∈Randn≥n. It follows from ∞

k=n  kφˆ

k_{x, }k_y= 

n

∞

k=n  k–nφˆ

k–nn_{x, }k–n_n_y

=  n ∞ m=  mφˆ

mn_{x, }m_n_y

= 

n

n_{x, }n_y

that

NQ(x) –T(x),c

≥min

N f(

n

x)

n –Q(x),

c



,N T(x) – f( n

x)

n ,

c



=minNfn_x–Qn_{x, }n–_c,NTn_x–fn_{x, }n–_c

≥min

N

fn_x–Qn_{x, }n_t 

∞

k=n  kφˆ

k_{x, }k_x

,

N

Tn_x–fn_{x, }n_t ∞

k=n  kφˆ

k_{x, }k_x

≥minNfn_x–Qn_x,tn_{x, }n_x,

NTn_x–fn_x,tn_{x, }n_x

≥ –

for allx,y∈Randc> . Thus we haveN(Q(x) –T(x),c) =  for allc>  and soQ(x) =T(x) for allx∈R.

For the case= –, we can state the proof in the same method as in the first case. In the case, the mappingQis defined byQ(x) =limn→∞nf(–

n

_{x). This completes the proof.}

Corollary . Let(Y,N)be a fuzzy Banach space,θ and p∈Rwith p< be positive real numbers.Suppose that f :R→Yis a mapping with f() = such that,for all t> ,

lim

t→∞N

uniformly onR.Then the limit Q(x) =limn→∞f(

n

x)

n exists for all x∈X and there exists a

unique quadratic mapping Q:R→Ysuch that

lim

t→∞N f(x) –Q(x),

( + p₎

 – p θ|x|pt

=  (.)

uniformly onR.

Proof The proof follows from Theorem . by taking φ(x,y) = θ(|x|p _+|y|p_{) for all}

x,y∈R.

Corollary . Let(Y,N)be a fuzzy Banach space andψ: [,∞)→[,∞)be a mapping such that,for all s,t> ,

(a) ψ(ts) =ψ(t)ψ(s); (b) ψ(√) < .

Suppose that f :R→Yis a mapping with f() = such that,for all t> ,

lim

t→∞N

fx_+y_{–f(x) –f(y),tθψ}|_x|_+ψ|_y|_{=  (.)}

uniformly onR_{,whereθ> is fixed. Then the limit Q(x) =lim}

n→∞f(

n

x)

n exists for all

x∈Rand defines a quadratic mapping Q:R→Ysuch that,for all t> ,

lim

t→∞N f(x) –Q(x),

( +ψ(√))  –ψ(√) θ ψ

|x|t

=  (.)

uniformly onR.

Proof The proof follows from Theorem . by takingφ(x,y) =θ(ψ(|x|) +ψ(|y|)) for all

x,y∈R.

2.2 The fixed point method

Recall that a mappingd:X_{→[, +∞] is called ageneralized metricon a nonempty set} Xif

() d(x,y) = if and only ifx=y; () d(x,y) =d(y,x);

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

A setXwith the generalized metricdis called ageneralized metric space.

In [], Diaz and Margolis proved the following fixed point theorem, which plays an important role for the main results in this section.

Theorem .[] Suppose that(,d)is a complete generalized metric space and T:→ is a strictly contractive mapping with Lipshitz constant L.Then,for any x∈,either d(Tn_x,Tn+_{x) =∞for all n≥or there exists a positive integer n}

such that

() d(Tn_x,Tn+_{x) <∞for alln≥n} ;

() the sequence{Tn_{x}is convergent to a fixed pointy}∗_ofT;

() y∗is the unique fixed point ofT in the set={y∈:d(Tn_{x,y) <}∞}_;

Theorem . Let(Y,N)be a fuzzy Banach space andφ:R_{→[,∞)be a mapping such} that there exists L< with

φ_{x, }_y≤_{Lφ(x,y) (.)}

for all x,y∈R.If f :R→Yis a mapping with f() = and

Nfx_+y_{–f(x) –f(y),t}≥ t

t+φ(x,y) (.)

for all x,y∈Rand t> ,then the limit Q(x) =limn→∞nf( n

_{x)exists for all x}∈R_{and a} unique quadratic mapping Q:R→Ysatisfies the inequality

Nf(x) –Q(x),t≥ ( –L)t

( –L)t+φˆ(x,x) (.)

for all x∈R,whereφˆ(x,x) =φ(x,x) +φ(x, ). Proof Lettingxandyby x√+y

 and

x_√–y

 in (.), respectively, we have

N fx_+y_–f x_√+y



–f x√–y



,t

≥ t

t+φ(x√+y ,

x_√–y

)

(.)

for allx,y∈Randt≥t. It follows from (.), (.), and (N) that

N f(x) +f(y) –f x√+y



–f x√–y



, t

≥min

Nf(x) +f(y) –fx_+y_,t,

N fx_+y_–f x_√+y



–f x√–y



,t

≥min

t t+φ(x,y),

t t+φ(x√+y

,

x_√–y

)

≥ t

t+φ(x,y) +φ(x√+y ,

x_√–y

)

(.)

for allx,y∈Randt≥t. Lettingy=xin (.), we have

N f(x) – f

_x,t

≥ t

t+φˆ(x,x) (.)

for allx∈Randt≥t, whereφˆ(x,y) =φ(x,y) +φ(  _{x, ).}

Letbe a set of all mapping fromRtoYand introduce a generalized metric onas

follows:

d(g,h) =inf

μ∈[,∞) :Ng(x) –h(x),μt≥ t

t+φˆ(x,x),∀x∈R,t> 

It is easy to show that (,d) is a generalized complete metric space []. We consider the

mappingT:→defined by

Tg(x) =  g

_x

for allg∈andx∈R. Letg,h∈such thatd(g,h)≤μ. Then we have

NTg(x) –Th(x),tμL=Ng_x–h_{x, tμL}≥ t t+φˆ(x,x)

for allx∈R, and so

d(Tg,Th)≤Ld(g,h)

for all g,h∈. This means thatT is a strictly contractive self-mapping of with the Lipschitz constantL.

It follows from (.) thatd(f,Tf)≤ <∞. Now, it follows from Theorem . that the sequence{Tn_{f}converges to a unique fixed pointQofT. So there exists a fixed pointQ}

ofTinsuch that

Q(x) = lim

n→∞

 nf

n_{x (.)}

for allx∈Rsincelim_n→∞d(Tn_{,Q) = . Again, using the fixed point method, sinceQis} the unique fixed point ofT in∗={g∈:d(f,g) <∞}, we have

d(f,Q)≤ 

 –Ld(f,Tf)≤

  –L,

which gives

Nf(x) –Q(x),t≥ ( –L)t

( –L)t+φˆ(x,x)

for allx∈Randt> . Further, we have

NQx_+y_{–Q(x) –Q(y),t}

≥ lim

n→∞N

fn_x_{+ }n_y_–fn_x–fn_{y, }n_t

≥ lim

n→∞

t

t+Ln_φˆ_(x,y)=  (.)

for allx,y∈Randt> . It follows from (N) andN(Q(x_+y_{) –Q(x) –Q(y),t)}≥_{ that} Q(x_+y_{) =Q(x) +Q(y) for allx,y}∈R_{. This means thatQis a quadratic mapping on}R_.

This completes the proof.

Theorem . Let(Y,N)be a fuzzy Banach space andφ:R_{→[,∞)be a mapping such} that there exists L< with

φ √x

,

y

√ 

≤L

for all x,y∈R.If f :R→Y is a mapping with f() = and(.),then the limit Q(x) =

limn→∞nf( x

n)exists for all x∈Rand there exists a unique quadratic mapping Q:R→Y satisfying the inequality

Nf(x) –Q(x),t≥ ( –L)t

( –L)t+Lφˆ(x,x) (.)

for all x∈Rand t> ,whereφˆ(x,x) =φ(x,x) +φ(x, ). Proof It follows from (.) that

N f(x) – f √x



,Lt

≥ t

t+φˆ(x,x) (.)

for allx∈Randt≥t, whereφˆ(x,y) =φ(x,y) +φ( 

_{x, ). Letanddbe as in the proof of}

Theorem .. Then (,d) becomes a generalized complete metric space and we consider

the mappingT:→defined by

(Tg)(x) = g √x



,

x∈R. So, we haved(Tg,Th)≤Ld(g,h) for allg,h∈. It follows from Theorem . that there exists a unique mappingQ:R→Yin the set{g∈:d(f,g) <∞}which is a unique fixed point ofT such that

Q(x) = lim

n→∞ n_f x

n

for allx∈R. Also, from (.) we haved(f,Tf)≤L. So, we can conclude that

d(f,Q)≤ 

 –Ld(f,Tf)≤ L

 –L,

which implies the inequality (.). The remaining assertion goes through in a similar way

to the corresponding part of Theorem .. This completes the proof.

Corollary . Let(Y,N)be a fuzzy Banach space andθ,p= be positive real numbers.

Suppose that f :R→Yis a mapping with f() = such that,for all t> ,

Nfx_+y_{–f(x) –f(y),t≥} t

t+θ(|x|p₊|_y|p₎ (.)

uniformly onR.Then there exists a unique quadratic mapping Q:R→Ysuch that

Nf(x) –Q(x),t≥ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(–

p

_)t (–

p

_{)t+(+}p₎θ|x|p

, p< ,

(–

p

)t

(–

p

)t+

p

(+

p

)θ|x|p, p> ,

(.)

Proof Takingφ(x,y) =θ(|x|p_+|y|p_{) for allx,y∈Rand choosingL= }p_{, we have the desired}

result.

Remark . The radical quadratic functional equationf(x_+y_{) =f(x) +f(y) is not}

sta-ble forp=  [].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Dongeui University, Busan, 614-714, Korea.}2_{Department of Mathematics, King Abdulaziz}

University, Jeddah, 21589, Saudi Arabia.3_{Department of Mathematics Education and the RINS, Gyeongsang National}

University, Chinju, 660-701, Korea.

Acknowledgements

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments and fruitful suggestions. The first author was supposed by Dongeui University (2013AA071) and the third author 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-2013053358). This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second author acknowledges with gratitude DSR, KAU for financial support.

Received: 14 February 2014 Accepted: 23 May 2014 Published:04 Jun 2014 References

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