On the Ulam type stability of several types of quadratic fuzzy set valued functional equations

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R E S E A R C H

Open Access

On the Ulam type stability of several types of

quadratic fuzzy set-valued functional

equations

Yonghong Shen

1,2*

_{, Wei Chen}

3

_{and Yaoyao Lan}

4

*_{Correspondence:}

shenyonghong2008@hotmail.com 1_{School of Mathematics and}

Statistics, Tianshui Normal University, Tianshui, 741001, P.R. China

2_{School of Mathematics and}

Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China

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

Abstract

LetYbe a real separable Banach space and (FKC(Y),d_∞) be the space of all normal fuzzy convex and upper semicontinuous fuzzy sets with compact support inY, where

d_∞stands for the supremum metric inFKC(Y). In the present paper, several types of quadratic fuzzy set-valued functional equations are introduced based on the space mentioned above. We prove the Hyers-Ulam stability of the standard quadratic fuzzy set-valued functional equation by using the ﬁxed point technique. Simultaneously, we also establish some Ulam type stability results of the Deeba and Appolonius type fuzzy set-valued functional equations by employing the direct method, respectively. The stability results of the corresponding single-valued and set-valued functional equations acting as special cases will be included in our results.

Keywords: Ulam type stability; Hausdorﬀ separation; supremum metric; quadratic fuzzy set-valued functional equation

1 Introduction

Nowadays, the Ulam type stability is gradually becoming one of the most active research topics in the theory of functional equations. The study of such stability problems of func-tional equations originated from a question of Ulam [] concerning the stability of group homomorphisms,i.e.:

Let G be a group and let G be a metric group with the metric d(·,·). Given> ,

does there exist aδ> such that if a function h:G→G satisﬁes the inequality

d(h(xy),h(x)h(y)) <δ for all x,y∈G, then there is a homomorphism H :G→G

with d(h(x),H(x)) <for all x∈G?

Afterwards, Hyers [] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Later, this result was generalized by Aoki [] for additive mappings and in-dependently by Rassias [] for linear mappings by considering an unbounded Cauchy def-erence. In , a further generalization of T.M. Rassias theorem was obtained by Gˇavruţa []. Since then, the Ulam type stability problems for different types of functional equations in various abstract spaces have been widely and extensively studied. For more details, we refer the readers to [–]. Moreover, it is worth noting that the study of Ulam stability

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for some quadratic functional equations is important and interesting. For instance,

f(x+y) +f(x–y) = f(x) + f(y) (Standard type),

f(x+y+z) +f(x) +f(y) +f(z) =f(x+y) +f(y+z) +f(z+x) (Deeba type),

f(x–z) +f(y–z) = 

f(x–y) + f

z–x+y 

(Appolonius type).

It should be noted that the functionf(x) =x_{is a solution of the three equations above.}

Naturally, we call these equationsquadratic functional equations and every solution a

quadratic function(mapping). In the case of single-valued equations, the corresponding Ulam type stability of these equations have been investigated by Czerwik [], Kim [] and Jung [], Jun and Kim [], respectively.

In , Mirmostafaee and Moslehian [] initiated the study of the Ulam stability prob-lems of functional equations in fuzzy setting. More precisely, they discussed the Ulam sta-bility of Cauchy type and then in the same year, together with Mirzavaziri [], Jensen type functional equations in a complete fuzzy normed linear space. Thereafter, the stability of various functional equations in a fuzzy abstract space has been investigated by various authors [–].

Summing up the above, the Ulam type stability problem is whether, for a given map-ping which satisfies approximately a functional equation (or which satisfies a functional inequality), there exists an exact solution of the corresponding functional equation such that the preceding mapping is sufficiently close to this solution. Usually, an appropriate metric associated with the corresponding space is chosen to characterize the error in functional inequalities. Certainly, a fuzzy metric is adopted in fuzzy environment. Un-like previous studies, Lu and Park [] first studied the stability of two types of additive set-valued functional equations, in which the inclusion relation is employed to measure the degree of approximation. Soon after, Parket al.[] further investigated the stability of quadratic, cubic and quartic set-valued functional equations. However, the disadvan-tage is that the obtained exact solution of these set-valued functional equations is just single-valued mapping. Recently, Kenaryet al.[] studied the stability of several types of set-valued functional equations by using the fixed point method. It should be pointed out that the same skill has been applied to solve the Ulam stability problems of single-valued and set-valued functional equations, respectively.

The main objective of this paper is to further extend and establish some new Ulam type stability results of the quadratic functional equations mentioned above, in which the quadratic mapping is assumed to be a fuzzy set-valued mapping. More speciﬁcally, the standard quadratic, Deeba and Appolonius type fuzzy set-valued functional equations. In the mean time, the supremum metric will be used to estimate the error in fuzzy set-valued functional inequalities.

2 Preliminaries

In this section, we present some related concepts and results which are mainly derived from [–].

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LetAandBbe two nonempty subsets ofYand letλ∈R. The (Minkowski) addition and scalar multiplication can be deﬁned by

A+B={a+b|a∈A,b∈B}

and

λA={λa|a∈A}.

Notice that the setsK(Y) andKC(Y) are closed under the operations of addition and scalar multiplication. In fact, these two operations induce a linear structure onK(Y) and KC(Y) with zero element{}, respectively. It should be noted that this linear structure is just a cone rather than a vector space because, in general,A+ (–)A={}. Moreover, for allλ,μ∈R, it follows that

λ(A+B) =λA+λB, (λ+μ)A⊆λA+μA.

Note that ifAis convex andλμ≥, then (λ+μ)A=λA+μA. Deﬁne the Hausdorﬀ separation ofBfromAby

d_H∗(B,A) =inf{> |B⊆A+S},

where S denotes the closed unit ball in Y,i.e.,S={y∈Y| yY ≤}. The Hausdorﬀ separation ofAfromBcan also be deﬁned similarly.

Based on these two types of separations, the Hausdorﬀ distance between nonempty sub-setsAandBis deﬁned by

dH(A,B) =max

d∗_H(A,B),d∗_H(B,A).

In general, the Hausdorﬀ distance has the following properties:

(i) dH(λA,λB) =|λ|dH(A,B)for allλ∈R;

(ii) dH(A+C,B+C) =dH(A,B);

(iii) dH(A+C,B+D)≤dH(A,B) +dH(C,D)for allA,B,D∈K(Y)andC∈KC(Y). If we restrict our attention to the nonemtpy closed subsets C(Y) ofY, then one can obtain the fact from [] that (C(Y),dH) is a complete metric space. Clearly,K(Y) and KC(Y) are closed subsets ofC(Y). Hence, (K(Y),dH) and (KC(Y),dH) are also complete metric spaces.

In the s, the concept of the Banach space-valued fuzzy sets was introduced by Inoue [], who extended the usual fuzzy sets deﬁned onRorRn. In other words, the base space of a fuzzy set is replaced by a more general Banach space.

A fuzzy set deﬁned onY is a mappingu:Y→[, ]. We denote byF(Y) the set of all fuzzy sets deﬁned onY. LetFK(Y) denote the class of fuzzy setsu:Y→[, ] with the following properties:

(i) uis normal,i.e., there existsy∈Y, such thatu(y) = ;

(ii) uis upper semicontinuous;

(iii) [u]α₌_{_y_∈_Y_|_u₍_y₎_≥_α_}_{is compact for each}_α_∈_{(, ]}_;

(iv) [u]₌

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In fact, the conditions (ii) and (iv) imply that [u]_{is also compact. Moreover, we use the}

notationFKC(Y) to denote the subspace ofF(Y) whose members also satisfy

(v) [u]α_{is convex for each}_α_∈_{(, ]}_{, that is to say,}_u_{is fuzzy convex.}

A linear structure can be deﬁned inF(Y) in a similar way to fuzzy sets inRorRn_{, by}

(u⊕v)(y) = sup

x+z=y

minu(x),v(z),

(γu)(y) =

⎧ ⎨ ⎩

u(_γy), ifγ = ,

I(y), ifγ = 

foru,v∈F(Y) andγ∈R, whereI(y) =  ify=  andI() = . ThenF(Y) is closed under

these operations, and level setwise

[u⊕v]α= [u]α+ [v]α, [λu]α=λ[u]α

for eachα∈[, ] andλ∈R. Similar to the closeness ofKC(Y), it is easy to see thatFKC(Y) is also closed under these operations.

Based on the statement mentioned above, for anyu,v∈FKC(Y) andλ,μ∈R, the fol-lowing equalities hold:

(i) λ(u⊕v) =λu⊕λv; (ii) λ(μu) = (λμ)u;

(iii) (λ+μ)u=λu⊕μufor anyλ,μ≥.

Therefore, it is easy to check that the setFKC(Y) is just a cone deﬁned onYrather than a vector space.

As a generalization of the Hausdorﬀ metricdHinK(Y), the supremum metric is deﬁned by

d∞(u,v) = sup

α∈(,]

dH

[u]α, [v]α,

whereu,v∈FC(Y).

Remark  Every ordinary crisp subsetAofYcan be identiﬁed with the fuzzy set onYby

its characteristic functionχA:Y→ {, }, that is withχA(y) =  ify∈AandχA(y) =  if

y∈/A. Therefore, ifA∈K(Y) (orA∈KC(Y)), thenχA∈FC(Y) (orFKC(Y)), andvice versa. From Remark , for anyA,B∈K(Y) (orFKC(Y)), it follows that

d∞(χA,χB) = sup α∈(,]

dH

[χA]α, [χB]α

=dH(A,B). ()

Especially, ifAandBare simpliﬁed into two singleton sets{a}and{b}, respectively, then

we can infer from equality () thatd∞(χ{a},χ{b}) =d(a,b), whereddenotes the usual metric

betweenaandb.

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(i) d∞(λu,λv) =λd∞(u,v)for anyλ≥;

(ii) d∞(u⊕w,v⊕w) =d∞(u,v);

(iii) d∞(u⊕v,u⊕v)≤d∞(u,u) +d∞(v,v)for allu,v,w,u,v∈FKC(Y).

Remark  If we restrict our attention to the setFKC(Y), we then can prove that (FKC(Y),

d∞) is a complete metric space by using the method analogous to that used in [,

Propo-sition ..].

Here we recall a ﬁxed point theorem in a complete generalized metric space which are useful in the next section.

Theorem .(Diaz and Margolis []) Let(X,d)be a complete generalized metric space,

i.e.,one for which d may assume inﬁnite values.Suppose that J:X→X be a strictly con-tractive mapping with Lipschitz constant L< .Then,for each given element x∈X,either

dJnx,Jn+x=∞

for all n≥or there exists an n∈Nsuch that

(i) d(Jnx,Jn+x) <∞for alln≥n;

(ii) the sequence{Jn_x_}_{converges to a ﬁxed point}_y∗_of_J_;

(iii) y∗is the unique ﬁxed point ofJin the setY={y∈X|d(Jn_x_,_y_{) <}∞}_;

(iv) d(y,y∗)≤_–_Ld(y,Jy)for ally∈Y.

3 Hyers-Ulam stability of the standard quadratic fuzzy set-valued functional equation

In this section, we shall establish the Hyers-Ulam stability of the standard quadratic fuzzy set-valued functional equation by using the ﬁxed point technique.

Deﬁnition . LetXbe a real vector space and letf :X→FKC(Y) be a fuzzy set-valued mapping. The quadratic fuzzy set-valued functional equation is deﬁned by

f(x+y)⊕f(x–y) = f(x)⊕f(y) ()

for all x,y∈X, where the equal sign means that the both sides of () (two fuzzy sets) are equal pointwise. Every solution of () is called a standard quadratic fuzzy set-valued mapping.

Example  LetX=Y=R. Suppose thatf :R→FKC(Y) is a triangular fuzzy set-valued

mapping,i.e., for everyt∈X,f(t) is a triangular fuzzy set inR, which is deﬁned by

f(t) =t–at,t,t+bt, t∈R,

wherea,bare two nonnegative real numbers. From the deﬁnition of anα-level set of a fuzzy set, it follows that

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for everyt∈Randα∈[, ]. Thus, for everyα∈[, ], it can easily be veriﬁed that

f(x+y)α+f(x–y)α= f(x)α+ f(y)α

for allx,y∈X. This implies thatf is a solution of () inR.

Remark  Letu∈FKC(Y) be a ﬁxed fuzzy set. Accordingly, it is easy to check thatf(t) =

t_u

is also a solution of () by the same method as shown in Example .

Theorem . Let j∈ {–, }be ﬁxed and letϕ:X×X→[,∞)be a function such that

there exists a positive constant L< satisfying

ϕ(x,y)≤j_L_ϕ_–j_x_{, }–j_y _()

for all x,y∈X.Assume that f :X→FKC(Y)is a mapping with f() =Iwhich satisﬁes

d∞f(x+y)⊕f(x–y), f(x)⊕f(y)≤ϕ(x,y) ()

for all x,y∈X,then

Q(x) = lim

n→∞

–jn_f_jn_x

exists for each x∈X and deﬁnes a unique quadratic fuzzy set-valued mapping Q:X→

FKC(Y)such that

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩



(–L)ϕ(x,x), if j= ,

L

(–L)ϕ(x,x), if j= –

()

for all x∈X.

Proof Puttingy=xin (), we can obtain

d∞



f(x),f(x)

≤ 

ϕ(x,x) ()

for allx∈X, sincef() =Iandu⊕I=ufor anyu∈FKC(Y). Furthermore, it follows from () that

d∞

f(x), f

x



≤ϕ

x

,

x



≤L

ϕ(x,x) ()

for allx∈X.

Consider the setE={g|g:X→FKC(Y),g() =I}and introduce the generalized metric

DonE, which is deﬁned by

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where, as usual,inf∅=∞. In can easily be veriﬁed that (E,D) is a complete generalized metric space (see [], Theorem .).

Now, we deﬁne an operatorJ:E→Eby

Jg(x) = –jgjx

for allx∈X.

Letg,hbe given such thatD(g,h) =. Then

d∞g(x),h(x)≤ϕ(x,x) for allx∈X. Hence, we can obtain

d∞Jg(x),Jh(x)=d∞–jgjx, –jhjx

= –j_d

∞gjx,hjx

≤–jϕjx, jx

≤Lϕ(x,x)

for allx∈X. SoD(g,h) =implies thatD(Jg,Jh)≤L. This means that

D(Jg,Jh)≤LD(g,h)

for allg,h∈E. That is to say,Jis a strictly contractive self-mapping onEwith Lipschitz constantL< .

Moreover, we can infer that from () and () that

D(f,Jf)≤

⎧ ⎨ ⎩



, ifj= ,

L

, ifj= –.

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According to Theorem ., there exists a mappingQ:X→FKC(Y) satisfying the follow-ing:

(i)Qis a ﬁxed point ofJ,i.e.,

jQ(x) =Qjx ()

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

M=g∈E|D(f,g) <∞,

which implies thatQis the unique mapping satisfying () such that there exists anη∈(, ) satisfying

d∞f(x),Q(x)≤ηϕ(x,x)

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(ii)D(Jn_f_,_Q₎_→_{ as}_n_{→ ∞. This implies the equality}

lim

n→∞

–jn_g_jn_x₌_Q₍_x₎

for allx∈X.

(iii)D(f,Q)≤_–_LD(f,Jf), which implies the inequality

D(f,Q)≤

⎧ ⎨ ⎩



(–L), ifj= ,

L

(–L), ifj= –.

This shows that the inequality () holds.

Replacingxandyby jn_x_{and }jn_y_{in (), we can infer that}

d∞–jnfjn(x+y)⊕–jnfjn(x–y), ·–jnfjnx⊕·–jnfjny

≤–jnϕjnx, jny≤–jn·jnLnϕ(x,y) =Lnϕ(x,y),

which tends to zero asn→ ∞for allx,y∈X. Thus,

Q(x+y)⊕Q(x–y) = Q(x)⊕Q(y)

for allx,y∈Xand therefore the mappingQ:X→FKC(Y) is a quadratic fuzzy set-valued

mapping, as desired.

Corollary . Let X be a real normed space and let p, θ be positive real numbers with

p= .Assume that f :X→FKC(Y)is a mapping satisfying

d∞f(x+y)⊕f(x–y), f(x)⊕f(y)≤θxp+yp

for all x,y∈X.Then Q(x) = lim

n→∞

–jn_f_jn_x

FKC(Y)such that

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩

θxp

–p–, if j= ,p< , θxp

p–_–, if j= –,p> 

for all x∈X.

Proof In Theorem ., letϕ(x,y) =θ(xp₊_yp_{). Then we can choose}_L_{= }j(p–)_{and we}

get the desired result.

Corollary . Let X be a real normed space and let p, θ be positive real numbers with

p= .Assume that f :X→FKC(Y)is a mapping satisfying

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for all x,y∈X.Then

Q(x) = lim

n→∞

–jn_f_jn_x

FKC(Y)such that

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩

θxp

–p , if j= ,p< ,

θxp

p_–, if j= –,p> 

for all x∈X.

Proof In Theorem ., letϕ(x,y) =θxpyp. Then we can chooseL= j(p–)and we get

the desired result.

Corollary . Let X be a real normed space and let p,q,θ be positive real numbers with

p+q= .Assume that f :X→FKC(Y)is a mapping satisfying

d∞f(x+y)⊕f(x–y), f(x)⊕f(y)≤θxpyq+xp+q+yp+q

for all x,y∈X.Then

Q(x) = lim

n→∞

–jn_f_jn_x

FKC(Y)such that

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩

θxp+q

–p+q , if j= ,p+q< , θxp+q

p+q_– , if j= –,p+q> 

for all x∈X.

Proof In Theorem ., letϕ(x,y) =θ(xpyq+xp+q+yp+q). Then we can choose

L= j[(p+q)–]_{and we get the desired result.}

Remark  Due to the condition (), all results obtained in this section still hold true, even if the fuzzy set-valued mappingf :X→FKC(Y) degenerates into a set-valued mapping

f :X→KC(Y) or a single-valued mappingf :X→Y. Therefore, these results can be regarded as an important extension of the stability results of the ordinary single-valued and set-valued quadratic functional equations. In essence, our results extend some main results obtained by Cholewa [], Czerwik [] and Junget al.[].

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Deﬁnition . LetXbe a real vector space and letf :X→FKC(Y) be a fuzzy set-valued mapping. The Deeba type fuzzy set-valued functional equation is deﬁned by

f(x+y+z)⊕f(x)⊕f(y)⊕f(z) =f(x+y)⊕f(y+z)⊕f(z+x) () for allx,y,z∈X, where the equal sign means that the both sides of () (two fuzzy sets) are equal pointwise. Every solution of () is called a Deeba type fuzzy set-valued mapping.

Remark  In a similar way as shown before, it is easy to verify that the triangular fuzzy

set-valued mappingf constructed in Example  is a solution of (). Similarly,f(t) =tu

is also a solution of () for any ﬁxed fuzzy setu∈FKC(Y). Moreover, these two examples are suitable for () which is introduced in the next section.

Lemma . Let X be a real vector space.For a given≥,if a mapping f :X→FKC(Y)

satisﬁes the following inequality:

d∞f(x+y+z)⊕f(x)⊕f(y)⊕f(z),f(x+y)⊕f(y+z)⊕f(z+x)≤ ()

for all x,y,z∈X,then

d∞

f(x)⊕ n_{– }

n+f

–nx, n_{+ }

n+ f

nx≤ n

k=

–k ()

for all x∈X and n∈N+_.

Proof Lettingx=y=z=  in (), we getd∞(f(),I)≤. Puttingx=y= –zin (), we get

d∞f(x)⊕f(–x), f()⊕f(x)≤.

According to the properties of the supremum metric, we can obtain

d∞f(x)⊕f(–x),f(x)

≤d∞f(x)⊕f(–x), f()⊕f(x)+d∞f()⊕f(x),f(x)

=d∞f(x)⊕f(–x), f()⊕f(x)+ d∞f(),I

≤. ()

Furthermore, by substituting –xforxin (), we have

d∞f(–x)⊕f(x),f(–x)≤. ()

By () and (), we can infer that

d∞

f(x)⊕ f(–x),

 f(x)

=d∞

 f(x)⊕



f(–x)⊕  f(–x),

 f(x)⊕

 f(–x)⊕

 f(x)

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≤ d∞

f(x)⊕f(–x),f(x)+ d∞

f(–x), f(–x)⊕f(x) ≤ +  =  ,

which implies that the inequality () holds true forn= . By mathematical induction, we can assume that the inequality () is true for somen∈N+_{. Therefore, we have}

d∞

f(x)⊕ n+_{– }

n+ f

–n+x, n+_{+ }

n+ f

n+x

=d∞

f(x)⊕ n_{– }

n+ f

–nx⊕

n+_{+ }

n+

fnx⊕f–nx

⊕n+–  n+ f

–n+x, n+_{+ }

n+ f

n+x⊕

n+_{– }

n+

f–nx⊕fnx

⊕n+  n+f

nx

≤d∞

f(x)⊕ n_{– }

n+ f

–nx, n_{+ }

n+ f

nx+ n+_{+ }

n+ d∞

fnx

⊕f–nx,fn+x+ n+_{– }

n+ d∞

f–nx⊕fnx,f–n+x

≤

_n

k=

–k+ n+_{+ }

n+ +

n+–  n+

=  n+

k=

–k,

which proves the validity of the inequality () forn+ . This completes the proof.

Theorem . Let X be a real vector space.Assume that a mapping f :X→FKC(Y)satisﬁes

the inequality()and the following inequality:

d∞f(x),f(–x)≤θ ()

for someθ ≥and for all x∈X.Then there exists a unique Deeba quadratic fuzzy set-valued mapping Q:X→FKC(Y)such that

d∞f(x),Q(x)≤ ()

for all x∈X with Q(x) =Q(–x).

Proof According to Lemma . and the condition (), we can infer that

d∞f(x), –nfnx

=d∞

f(x)⊕ n_{– }

n+ f

–nx, n_{– }

n+f

–nx⊕–nfnx

≤d∞

f(x)⊕ n_{– }

n+ f

–nx, n_{+ }

n+ f

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+d∞

n_{+ } n+f

nx, n_{– }

n+ f

≤ n

k=

–k+ n_{– }

n+ d∞

fnx,f–nx

≤ n

k=

–k+ n_{– }

n+ θ. ()

Forn≥m, by (), we can obtain

d∞–nfnx, –mfmx

= –md∞–(n–m)fn–m·mx,fmx

≤–m

 n–m

k=

–k+ n–m_{– }

(n–m)+θ

. ()

It is easy to see that the sequence{–n_f_(n_x_)}_{is Cauchy sequence, since the last expression} of the inequality () tends to zero asm→ ∞. From the completeness of the metric space

(FKC(Y),d∞), the sequence{–nf(nx)}is convergent for anyx∈X. Set

Q(x) = lim

n→∞

–n_f_n_x_.

To show thatQsatisﬁes (), one can replacex,y,zin () by n_x_{, }n_y_{, }n_,_z_{, respectively,} it follows that

d∞fn(x+y+z)⊕fnx⊕fny⊕fnz,fn(x+y)

⊕fn(y+z)⊕fn(z+x)≤.

Dividing both sides by n_{in the last inequality, and by letting}_n_{→ ∞}_{, it is easy to see} thatQis a solution of (). In a similar manner, we can ﬁnd the condition () implies thatQ(x) =Q(–x). Moreover, by lettingn→ ∞in (), it can easily be veriﬁed that the inequality () holds true.

Now we show the uniqueness ofQ. LetQbe another Deeba quadratic mapping which satisﬁes the equality () and the inequality (). It is obvious thatQ(n_x_{) = }n_Q₍_x_{) and}

Q(n_x_{) = }n_Q₍_x_{) for all}_x_∈_X_and_n_∈_{N. Then it follows from () that}

d∞Q(x),Q(x)= –nd∞Qnx,Qnx

≤–nd∞Qnx,fnx+d∞fnx,Qnx

≤ n

for allx∈X. By lettingn→ ∞, we conclude thatQ(x)≡Q(x). The proof of the theorem

is now completed.

Theorem . Let X be a real vector space.Assume that a mapping f :X→FKC(Y)satisﬁes

the inequality()and the following inequality:

d∞f(x)⊕f(–x),I

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for someθ ≥and for all x∈X.Then there exists a unique Deeba type additive fuzzy set-valued mapping A:X→FKC(Y)such that

d∞f(x),A(x)≤ ()

for all x∈X with A(x)⊕A(–x) =I.

Proof By Lemma . and the condition (), we have

d∞f(x), –nfnx

=d∞

f(x)⊕ n_{– }

n+ f

–nx, n_{– }

n+f

≤d∞

f(x)⊕ n_{– }

n+ f

–nx, n_{+ }

n+ f

nx

+d∞

n_{+ } n+f

nx, n_{– }

n+ f

≤ n

k=

–k+ n_{– }

n+ d∞

I,f

nx⊕f–nx

≤ n

k=

–k+ n_{– }

n+ θ. ()

By the preceding inequality, forn≥m, we can obtain

d∞–nfnx, –mfmx

= –md∞–(n–m)fn–m·mx,fmx

≤–m

 n–m

k=

–k+ n–m_{– }

(n–m)+θ

, ()

which means that the sequence{–n_f_(n_x₎_}_{is a Cauchy sequence in the complete metric} space (FKC(Y),d∞). Therefore, we can deﬁne

A(x) = lim

n→∞

–n_f_n_x

for allx∈X. Using the same argument as in the proof of Theorem ., we conclude that the mappingAsatisﬁes the equality () withA(x)⊕A(–x) =I. Clearly, we getA() =I.

Puttingz= –yin (). According to the preceding properties ofA, letu=x+y,v=x–y, we can obtain

A

u+v



=A(u) +A(v).

That is to say, the mappingAis additive fuzzy set-valued mapping due to [].

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A(n_x_{) = }n_A₍_x_{), we get}

d∞A(x),A(x)= –nd∞Anx,Anx

≤–nd∞Anx,fnx+d∞fnx,Anx

≤ n

for allx∈X. By lettingn→ ∞, we conclude thatA(x)≡A(x). This completes the proof.

Remark  From Theorems . and ., the condition () ensures that the mapping is

quadratic, whereas the condition () implies that the mapping is additive.

Remark  The preceding theorems extend some stability results of the single-valued

Deeba type functional equation obtained by Jung [].

5 Hyers-Ulam stability of the Appolonius type quadratic fuzzy set-valued functional equation

Here we shall prove the Hyers-Ulam stability of the Appolonius type quadratic fuzzy set-valued functional equation by using the same method as employed in the preceding sec-tion.

Deﬁnition . LetXbe a real vector space and letf :X→FKC(Y) be a fuzzy set-valued mapping. The Appolonius type fuzzy set-valued functional equation is deﬁned by

f(x–z)⊕f(y–z) = 

f(x–y)⊕f

z–x+y 

()

for allx,y,z∈X, where the equal sign means that the both sides of () (two fuzzy sets) are equal pointwise. Every solution of () is called an Appolonius type fuzzy set-valued mapping.

Theorem . Let j∈ {–, }be ﬁxed and letϕ:X×X×X→[,∞)be a function such

that

(j)(x,y,z) =

∞

k=

–jkϕjkx, jky, jkz< +∞ ()

for all x,y,z∈X.Assume that the mapping f :X→FKC(Y)satisﬁes

d∞

f(x–z)⊕f(y–z),

f(x–y)⊕f

z–x+y 

≤ϕ(x,y,z) ()

for all x,y,z∈X.Moreover,f() =Iin the case j= .Then

Q(x) = lim

n→∞

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exists for each x∈X and deﬁnes a unique Appolonius type quadratic fuzzy set-valued map-ping Q:X→FKC(Y)such that

d∞f(x),Q(x)≤ 



(j)₍_x_{, –}_x_,_x_{) +} 



(j)₍_x_,_x_{, –}_x₎ _()

for all x∈X.

Proof Firstly, we claim thatf() =Iin the case j= –. Indeed, lettingx=y=z=  in

(), we can infer thatf() =Ibecause the condition(, , ) =

_∞

k=kϕ(, , ) < +∞

implies thatϕ(, , ) = . So we havef() =Iregardless ofj=  orj= –.

Replacingy,zbyx, –xin (), respectively, we have

d∞f(x),f(–x)≤ϕ(x,x, –x) () for allx∈X. Substitutingy,zby –x,xin (), respectively, we get

d∞

f(–x),

f(x)⊕f(x)

≤ϕ(x, –x,x) ()

for allx∈X. From the inequalities () and (), it follows that

d∞f(x), f(x)=d∞f(x)⊕f(x), f(x)

≤d∞f(x)⊕f(x), f(–x)+d∞f(–x), f(x)

≤ϕ(x,x, –x) + ϕ(x, –x,x) ()

for allx∈X. Dividing both sides in () by , we obtain

d∞



f(x),f(x)

≤ 

ϕ(x,x, –x) + 

ϕ(x, –x,x) ()

for allx∈X. Replacingxby n–_x_{and dividing by }n–_{in (), we conclude that}

d∞

 nf

nx,  n–f

n–x

≤ϕ(n–x, n–x, –n–x)

n +

ϕ(n–_x_{, –}n–_x_{, }n–_x₎

·n– ()

for allx∈Xand for alln∈N. Combining the inequalities () and () gives

d∞f(x), –nfnx≤

n–

i=

ϕ(ix, –ix, ix) ·i +

n–

i=

ϕ(ix, ix, –ix)

i+ ()

for allx∈Xand for alln∈N.

Moreover, substitutingxby x_ in (), we get

d∞

f(x), f

x



≤ϕ

x

,

x

, –

x



+ ϕ

x

, –

x

,

x



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for allx∈X. By using a similar argument, we can infer that

d∞

f(x), nf

x n ≤   n i=

iϕ

x

i,

x

i, –

x i +  n i=

iϕ

x

i, –

x

i,

x

i

()

for allx∈Xandn∈N. Here we claim that the sequence{–jn_f_(jn_x_)}_{is a Cauchy sequence} in the complete metric space (FKC(Y),d∞). Indeed, it follows from () and () that

d∞–j(n+m)fj(n+m)x, –jmfjmx

= –jmd∞–jnfjn·jmx,fjmx

≤

⎧ ⎨ ⎩

 m(

n–

i=

ϕ(i+mx,–i+mx,i+mx) ·i +

n–

i=

ϕ(i+mx,i+mx,–i+mx)

i+ ), j= ,

 

n

i=i+mϕ(ix+m,_ix+m, –_i+xm) +_

n

i=i+m+ϕ(ix+m, –_ix+m,_xi), j= –

for allx∈Xand for alln,m∈N. Whenj= , it is obvious that the last expression tends to zero asm→ ∞. Whenj= –, we can infer from the condition () that the preced-ing expression also tends to zero as m→ ∞. Thus, we have proved that the sequence {–jn_f_(jn_x₎_}_{is Cauchy. By the completeness of}_F

KC(Y), we can deﬁne

Q(x) = lim

n→∞

–jn_f_jn_x

for allx∈X.

Now we show thatQsatisﬁes (). Replacingx,y,zin () by jnx, jny, jnz, respectively, and dividing both sides by jn_{, we can obtain}

–jn_d

∞

fjn₍_x_–_z₎_⊕_f_jn₍_y_–_z₎_, f

jn₍_x_–_y₎_⊕__f

jn

z–x+y 

≤–jnϕjnx, jny, jnz.

Taking the limit in the preceding expression, it can easily be seen thatQsatisﬁes () since the right hand side tends to zero asn→ ∞.

By lettingn→ ∞in () and (), we can infer that

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩

 

()₍_x_{, –}_x_,_x_{) +}  

()₍_x_,_x_{, –}_x_), _j_{= ,} 

(–)(x, –x,x) + 

(–)(x,x, –x), j= –

()

for allx∈X.

Let us prove the uniqueness. LetQbe another Appolonius type quadratic fuzzy set-valued mapping satisfying the inequality (). SinceQ(jn_x_{) = }jn_Q₍_x_),_Q_(jn_x_{) = }jn_Q₍_x_),

we have

d∞Q(x),Q(x)= –jnd∞Qjnx,Qjnx

≤–jnd∞Qjnx,fjnx+d∞fjnx,Qjnx

≤–jn

 

(j)_jn_x_{, –}jn_x_{, }jn_x₊ 

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From the condition (), it follows that the last expression tends to zero asn→ ∞. Thus,

we conclude thatQ(x)≡Q(x). This completes the proof. Based on Theorem ., we can obtain the following corollaries.

Corollary . Let j∈ {–, }be ﬁxed.Suppose that X is a real normed space and p,θare positive real numbers with p= .Assume that the mapping f:X→FKC(Y)satisﬁes

d∞

f(x–y)⊕f

z–x+y 

≤θxp+yp+zp

for all x,y∈X.Moreover,f() =Iin the case j= .Then

Q(x) = lim

n→∞

–jn_f_jn_x

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩

θxp

–p , if j= ,p< , θxp

––p, if j= –,p> 

for all x∈X.

Remark  In Theorem . and Corollary ., if the fuzzy set-valued mappingf

degen-erates into a single-valued mapping, then the supremum metric d∞ will reduce to the

ordinary metric between two points. Thus, these results can be viewed as an extension of the main results obtained by Jun and Kim [].

Corollary . Let j∈ {–, }be ﬁxed.Suppose that X is a real normed space and p,θare positive real numbers with p=_.Assume that the mapping f :X→FKC(Y)satisﬁes

d∞

f(x–y)⊕f

z–x+y 

≤θxpypzp

Q(x) = lim

n→∞

–jn_f_jn_x

d∞f(x),Q(x)≤ ⎧ ⎨ ⎩

θxp

–p , if j= ,p<_, θxp

––p, if j= –,p>_

for all x∈X.

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satisﬁes

d∞

f(x–y)⊕f

z–x+y 

≤θxpyqzr+xp+q+r+yp+q+r+zp+q+r

Q(x) = lim

n→∞

–jn_f_jn_x

d∞f(x),Q(x)≤

⎧ ⎨ ⎩

θxp+q+r

–(p+q+r)–, if j= ,p+q+r< ,

θxp+q+r

–(p+q+r)–, if j= –,p+q+r> 

for all x∈X.

Corollary . Let X is a real normed space.For a givenθ > ,if the mapping f :X→

FKC(Y)satisﬁes

d∞

f(x–y)⊕f

z–x+y 

≤θ

for all x,y∈X and with f() =I,then

Q(x) = lim

n→∞

–n_f_n_x

d∞f(x),Q(x)≤θ

for all x∈X.

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_{School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, P.R. China.}2_{School of Mathematics}

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Acknowledgements

This work was supported by ?Qing Lan? Talent Engineering Funds by Tianshui Normal University, and the Research Project of Higher Learning of Gansu Province (No. 2014B-080). The second author acknowledges the support of the Humanity and Social Science Youth Foundation of Ministry of Education of China (No. 13YJC630012), and the Specially

Commissioned Project of the Capital University of Economics and Business. The third author acknowledges the support of the National Natural Science Foundation of China (No. 11226268).

Received: 27 September 2014 Accepted: 13 December 2014

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