Fixed points for modified fuzzy ψ-contractive set-valued mappings in fuzzy metric spaces

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R E S E A R C H

Open Access

Fixed points for modiﬁed fuzzy

ψ

-contractive

set-valued mappings in fuzzy metric spaces

Shihuang Hong

*

*_{Correspondence:}

[email protected] Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

Abstract

In this paper, we introduce a new concept of fuzzy

α

-

ψ

-contractive type set-valued mappings and establish ﬁxed-point theorems for such mappings in complete fuzzy metric spaces. Starting from the fuzzy version of the Banach contraction principle, the presented theorems extend, generalize and improve many existing results in the literature. Moreover, the results are supported by examples.

Keywords: ﬁxed point; fuzzy contraction; set-valued mapping; fuzzy metric spaces

1 Introduction

In metric fixed-point theory, the contractive conditions on underlying functions play an important role for ensuring the existence of fixed points. The Banach contraction principle is a remarkable result in metric fixed-point theory. Recently Gregori and Sapena [] have extended the Banach contraction principle to fuzzy contractive mappings on complete fuzzy metric spaces in some sense. Over the years, it has been generalized in different directions by several mathematicians (see [–]). In particular, Mihet [] introduced the concepts of fuzzyψ-contractive mappings which enlarge the class of fuzzy contractions in [], that is, the following implication takes place for the single-valued mappingT:

M(x,y,t) >  ⇒ M(Tx,Ty,t)≥ψM(x,y,t)

for any x,y∈Xandt> , whereψ is a function whose definition is given in Section . Moreover, some authors established fixed-point theorems for such mappings in complete fuzzy metric spaces. Afterwards, Hong and Peng [] modified the notion of the fuzzy ψ-contraction via a so-calledfw-distanceP instead of the fuzzy metricMand provided the sufficient conditions for the existence of fixed points for such contraction set-valued mappings.

Motivated by the works mentioned above, in this paper we will further modify the type of theψ-contraction and establish fixed-point theorems for such set-valued mappings on certain complete fuzzy metric spaces. Specifically, the main purpose is to extend the inequalityM(x,y,t) >  to a general functional inequality and introduce therefrom a new called fuzzyα-ψ-contraction which extends and improves the fuzzyψ-contraction of set-valued mappings; moreover, to formulate the conditions guaranteeing the convergence of fuzzyα-ψ-contractive sequences and the existence of fixed points of such a set-valued mapping. The present fixed-point theorem and a comprehensive set of its corollaries turn

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out to be generalizations of those of [, , ]. Some examples are given here to illustrate the usability of the results obtained.

Finally, the idea of present paper has originated from the study of an analogous problem examined by Salimiet al.[] and Sametet al.[] for single-valued contractive mappings and Hussainet al.[] for set-valued contractive mappings on complete determinacy met-ric spaces.

2 Preliminaries

Let us recall [] that a continuoust-norm is a binary operation∗: [, ]×[, ]→[, ] such that ([, ],≤,∗) is an ordered Abelian topological monoid with unit . In the sequel, we always assumea∗b≥abfor alla,b∈(, ].

For examples of a t-norm satisfying the above conditions, we enumerate a∗b=ab,

a∗b=min{a,b}anda∗b=ab/max{a,b,λ}for  <λ< , respectively.

Deﬁnition .[] A fuzzy metric space is an ordered triple (X,M,∗) such thatXis a (nonempty) set,∗is a continuoust-norm andMis a fuzzy set onX×X×(, +∞) that satisﬁes the following conditions, for allx,y,z∈X:

(F) M(x,y,t) > , for allt> ,

(F) M(x,x,t) = , for allt> , andM(x,y,t) = for somet> impliesx=y, (F) M(x,y,t) =M(y,x,t), for allt> ,

(F) M(x,y,t)∗M(y,z,s)≤M(x,z,t+s)for alls,t> and (F) M(x,y,·) : (, +∞)→[, ]is continuous.

Following the deﬁnition of Kramosil and Michálek [],Mis a fuzzy set onX×X×

[,∞) that satisﬁes (F) and (F), while (F), (F), (F) are replaced by (K), (K), (K), respectively, as follows:

(K) M(x,y, ) = ;

(K) M(x,y,t) = for allt> if and only ifx=y; (K) M(x,y,·) : [,∞)→[, ]is left continuous.

We refer to these spaces as KM-spaces and refer to the spaces given as in Deﬁnition . as GV-spaces. In addition, when Xis called a fuzzy metric space, it means it may be a GV-space or KM-space.

In these senses,Mis called a fuzzy metric onX. Some simple but useful facts are that

(I) M(·,·,t)is a continuous function onX×Xfort∈(,∞)and (II) M(x,y,·)is nondecreasing for allx,y∈X.

Indeed, let{xn}and{yn} be two sequences ofX withlimn→∞xn=xandlimn→∞yn=y. Then, for anyε>  andt> , we have

M(x,y,t)≥M(x,xn,ε)∗M(xn,yn,t– ε)∗M(yn,y,ε).

In view of Lemma ., for anyδ> , we haveM(x,xn,ε) >  –δandM(yn,y,ε) >  –δfor large enoughnand anyε> . Hence,

M(x,y,t)≥( –δ)∗M(xn,yn,t– ε)∗( –δ).

Letδ→; combining the arbitrariness ofεand the left continuity ofM(x,y,·), it follows that

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By taking the limit whenn→ ∞, we obtainlimn→∞M(xn,yn,t)≤M(x,y,t). By an anal-ogous inference, we have M(x,y,t)≥limn→∞M(xn,yn,t). Consequently, limn→∞M(xn, yn,t) =M(x,y,t),i.e., the ﬁrst fact is valid. To prove the second fact, by (F) we notice that

M(x,y,t)≥M(x,y,s)∗M(y,y,t–s) =M(x,y,s)∗ =M(x,y,s) fors,t∈[,∞) witht>s. Let (X,M,∗) be a fuzzy metric space. Fort>  andr∈(, ), the open ballB(x,t,r) with centerx∈Xis deﬁned by

B(x,t,r) =y∈X:M(x,y,t) >  –r.

A subsetA⊂Xis called open if for eachx∈A, there existt>  and  <r<  such that

B(x,t,r)⊂A. LetT denote the family of all open subsets ofX. ThenT is a topology on

Xinduced by the fuzzy metricM. This topology is metrizable []. Therefore, a closed subsetBofXis equivalent tox∈Bif and only if there exists a sequence{xn} ⊂Bsuch that

{xn}topologically converges tox. In fact, the topologically convergence of sequences can be indicated by the fuzzy metric as follows.

Lemma . A sequence{xn}in X is said to be convergent to a point x∈X,denoted by limn→∞xn=x,iflimn→∞M(xn,x,t) = for any t> .

Deﬁnition .[] Let (X,M,∗) be a fuzzy metric space.

(i) A sequence{xn}inXis called Cauchy sequence if for eachε> andt> , there existsn∈Nsuch thatM(xn,xm,t) >  –εfor anym,n≥n.

(ii) A fuzzy metric space(X,M,∗)in which every Cauchy sequence is convergent is said to be complete.

There exist two fuzzy versions of Cauchy sequences and completeness,i.e., besides the so-calledM-Cauchy sequence and M-completeness in the sense of Deﬁnition ., the

G-Cauchy sequence deﬁned by limn→∞M(xn+p,xn,t) =  for all t,p>  and the

corre-spondingG-completeness introduced by []. In [] the authors have pointed out that a

G-Cauchy sequence is not aM-Cauchy in general. It is clear that aM-Cauchy sequence is

G-Cauchy and hence a fuzzy metric space isM-complete if it isG-complete. From now on, by a Cauchy sequence and completeness we mean anM-Cauchy sequence andM -com-pleteness.

ByCB(X) we denote the collection consisting of all nonempty closed subsets ofX (obvi-ously, every closed subset ofXis bounded in the sense of fuzzy metric spaces). Motivated by [], we deﬁne a function onCB(X)×CB(X) as follows:

HM(A,B,t) =min

inf

a∈AM(a,B,t),binf∈BM(A,b,t)

for anyA,B∈CB(X) andt> , whereM(c,D,t) =M(D,c,t) =sup_d_∈_DM(c,d,t).

On the collection consisting of compact sunsets ofX, in [] the authors have shown thatHMsatisﬁes the conditions (F)-(F) given as in Deﬁnition .. Clearly,HM({x},{y},t) =

M(x,y,t) for allx,y∈Xandt> .

Lemma . If A⊂CB(X),then x∈A if and only if M(x,A,t) = for t> .

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Deﬁnition . Let (X,M,∗) be a fuzzy metric space. A subsetD⊂Xis said to be approx-imative if the set-valued mapping

PD(x) =

y∈D:M(x,y,t) =M(x,D,t),∀t> , ∀x∈X

has nonempty values. The set-valued mappingF:X→X_{is said to have approximative}

values ifF(x) is approximative for eachx∈X.

It is clear thatFhas approximative values if it has compact values.

3 Fixed-point theorems

Let (X,M,∗) be a fuzzy metric space andT :X→CB(X) be a set-valued mapping. An elementx∈Xis called a ﬁxed point ofTifx∈Tx.

Deﬁnition . LetT:X→X_{be a set-valued function, and let}_α,_η_:_X_×_X×_(,_∞₎_→_R +

be two functions, whereαis bounded. We say thatT is anα∗-η∗-admissible mapping if

α(x,y,t)≤η(x,y,t) implies that α∗(Tx,Ty,t)≤η∗(Tx,Ty,t), x,y∈X,t> ,

whereα∗(A,B,t) =sup_x_∈_A_,_y_∈_Bα(x,y,t) andη∗(A,B,t) =infx∈A,y∈Bη(x,y,t).

The following collectionof functions is described in [], that is,ψ∈implies that ψ from [, ] into itself is continuous, nondecreasing, andψ(s) >sfor eachs∈[, ).

Lemma . Letψ∈.For every s> ,ψ(s) >s if and only iflimn→∞ψn(s) = uniformly

for s∈[, ),whereψnis the nth iterate ofψ.

Proof Necessity. Sinceψ is nondecreasing, the sequence{ψn(s)}is also nondecreasing and hence its limit exists. Let limn→∞ψn(s) =c. Thusc≥s> . Ifc< , then from the

continuity ofψit follows that

c= lim n→∞ψ

n+₍_s_{) =}_ψ_lim n→∞ψ

n₍_s₎ ₌_ψ(_c_{) >}_c_,

a contradiction. Therefore,c= .

On the other hand, if the limit is not uniform, then there exists  <ε<  such that, for

everyn∈N, we can ﬁndtn∈(, ) andkn≥nwithkn+≥knsatisfying

 >ψkn₍_t n) +ε.

We can assume thattn+≤tnforn∈N. In fact, ift<t, then, by means of our assumptions,

we have  >ε+ψk(t)≥ε+ψk(t)≥ε+ψk(t). Without loss of generality, we put t=t. Inductively, lettj+≤tjforj> . Iftj+>tifor somei= , , . . . ,j+ , then we have  >

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limn→∞ψkn(t) =  it follows thatlimn→∞ψkn(tn) = . Hence, there existsk∈Nsuch that

ψkn₍_tn_{) >  –}_ε

/, for alln≥k, that is,

 –ε>ψkn(tn) >  –ε/,

a contradiction. Therefore,limn→∞ψn(s) =  uniformly fors∈[, ).

Suﬃciency. Assume that there existst∈(, ) such thatψ(t)≤t. Thenψn(t)≤tfor

alln∈Nsinceψis nondecreasing. Thus  =limn→∞ψn(t)≤t< , a contradiction.

Deﬁnition . Letψ ∈. The set-valued mappingT is called a fuzzyα-ψ-contractive mapping if the following implication takes place:

x,y∈X,t> , α(x,y,t)≤η(x,y,t) ⇒ HM(Tx,Ty,t)≥ψN(x,y,t), () whereψ∈and

N(x,y,t) =minM(x,y,t),M(x,Tx,t)M(y,Ty,t).

Theorem . Let(X,M,∗)be a complete fuzzy metric space and T:X→CB(X)be a fuzzy

α-ψ-contractive andα∗-η∗-admissible set-valued mapping.Suppose that the following

as-sertions hold:

(i) there existx∈Xandx∈PTx(x)such thatα(x,x,t)≤η(x,x,t)for eacht> ;

(ii) for any sequence{xn} ⊂Xconverging tox∈Xandα(xn,xn+,t)≤η(xn,xn+,t),for all n∈Nandt> ,we haveα(xn,x,t)≤η(xn,x,t)for alln∈N.

Then T has a ﬁxed point.

Proof Our assumptions guarantee that there exist x∈X andx∈PTx(x) such that

α(x,x,t)≤η(x,x,t) and M(x,x,t) =M(x,Tx,t) for eacht> . By the contractive

condition () we have

HM(Tx,Tx,t)≥ψ

N(x,x,t)

()

for allt> . Noting thatTis anα∗-η∗-admissible mapping, we have

α∗(Tx,Tx,t)≤η∗(Tx,Tx,t).

Forx∈Tx, there existsx∈PTx(x) such thatα(x,x,t)≤η(x,x,t). Applying again

the contractive condition () we have

HM(Tx,Tx,t)≥ψ

N(x,x,t)

.

Continuing this process, we can deﬁne a sequence{xn}inXbyxn∈PTxn–(xn–) satisfying,

for alln∈Nandt> ,

HM(Txn,Txn+,t)≥ψ

N(xn,xn+,t)

, ()

α(xn,xn+,t)≤η(xn,xn+,t), ()

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Ifxn+=xnfor somen∈N, thenx=xnis a ﬁxed point ofTand the result is proved. Hence,

we suppose thatxn+=xn,i.e.,xn∈/Txnfor alln∈N. From equation () and the deﬁnition

ofHMit follows that

M(xn,xn+,t)≥HM(Txn–,Txn,t)

for alln∈Nandt> . By means of equation () we have

M(xn,xn+,t)≥ψ

N(xn–,xn,t)

for alln∈Nandt> . ()

On the other hand, by equation () we get

N(xn–,xn,t) =min

M(xn–,xn,t),

M(xn–,Txn–,t)M(xn,Txn,t)

=minM(xn–,xn,t),

M(xn–,xn,t)M(xn,xn+,t)

.

By equation (), this implies that

M(xn,xn+,t)≥ψ

minM(xn–,xn,t),

for alln∈Nandt> . We claim that

M(xn–,xn,t)≤M(xn,xn+,t) for alln∈Nandt> . ()

Suppose the contrary; then

minM(xn–,xn,t),

=M(xn–,xn,t)M(xn,xn+,t).

By virtue of the properties ofψ, for alln∈Nandt> , we get

M(xn,xn+,t)≥ψ

>M(xn–,xn,t)M(xn,xn+,t) >M(xn,xn+,t),

a contradiction. Hence equation () is valid. Moreover, in view of the monotonicity ofψ one has

M(xn,xn+,t)≥ψ

M(xn–,xn,t)

>M(xn–,xn,t)

for alln∈Nandt> . Repeating this procedure, we have

M(xn,xn+,t)≥ψ

M(xn–,xn,t)

≥ · · · ≥ψnM(x,x,t)

for alln∈Nandt> . Now, for allm>n,m,n∈Nandt> , we can write

M(xn,xm,t)≥M(xn,xn+,tn+)∗M(xn+,xn+,tn+)∗ · · · ∗M(xm–,xm,tm)

≥ m–

i=n

M(xi,xi+,ti+)≥ m–

i=n

ψiM(x,x,ti+)

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where ti>  (i=n+ ,n+ , . . . ,m) and m_i₌_n_+ti=t. In the light of Lemma ., we can assume thatψi(M(x,x,ti)) >  – /i wheni is large enough. Note that the series

∞

i=/iis convergent, and the inﬁnite product

∞

i=( – (/i)) is convergent, too. Hence, lim_n_→∞∞_i₌_n( – (/i_{)) = . This implies that}_{_x_n}_{is an}_M_{-Cauchy sequence.}

In view of the completeness of (X,M,∗), there existsy∈Xsuch thatxn→yasn→ ∞. By means of (ii), we haveα(xn,y,t)≤η(xn,y,t) for alln∈Nandt> . From the contractive condition it follows that

HM(Txn,Ty,t)≥ψN(xn,y,t).

We observe thatxn∈/Txn, soM(xn,Txn,t) <  by Lemma .. IfM(y,Ty,t) < , then

N(xn,y,t) =min

M(xn,y,t),

M(xn,Txn,t)M(y,Ty,t)

< .

Therefore,HM(Txn,Ty,t)≥ψ(N(xn,y,t)) >N(xn,y,t) forn∈Nandt> . Moreover, by

equation () we obtain

M(xn+,Ty,t)≥HM(Txn,Ty,t) >N(xn,y,t).

Noting that M(xn,y,t) → , M(xn,Txn,t) =M(xn,xn+,t) →, we have N(xn,y,t) →

M(y,Ty,t) asn→ ∞. By taking the limit asn→ ∞in the above inequality, we obtain

M(y,Ty,t)≥M(y,Ty,t).

This is a contradiction. Therefore,M(y,Ty,t) = . From Lemma . it follows thaty∈Ty,

i.e. yis a ﬁxed point ofT.

We present the following interesting corollaries.

Corollary . Let (X,M,∗)be a complete fuzzy metric space and let T be an α∗-η∗ -admissible set-valued mapping withη= .Assume that the following assertions hold:

(i) forψ∈,x,y∈Xandt> ,

α(x,y,t)≤ ⇒ HM(Tx,Ty,t)≥ψN(x,y,t);

(ii) there existx∈Xandx∈PTx(x)such thatα(x,x,t)≤for eacht> ;

(iii) for any sequence{xn} ⊂Xconverging tox∈Xandα(xn,xn+,t)≤,for alln∈N andt> ,we haveα(xn,x,t)≤for alln∈Nandt> .

Remark . Letα(x,y,t) =M(x,y,t). Then the assumptions (i) and (ii) of Corollary . hold andα(x,y,t)≤ for allx,y∈X andt> . ThusHM(Tx,Ty,t)≥ψ(N(x,y,t)) for all

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Corollary . Let all hypotheses of Corollary.hold except(i)changed into one of the following conditions:

(I) forψ∈,x,y∈Xandt> ,

α(x,y,t)HM(Tx,Ty,t)≥ψ

N(x,y,t);

(II) forψ∈,x,y∈Xandt> ,

α(x,y,t) +λHM(Tx,Ty,t)≥( +λ)ψ(N(x,y,t)), λ> ;

(III) forψ∈,x,y∈Xandt> ,

HM(Tx,Ty,t) +λα(x,y,t)≥ψN(x,y,t)+λ, λ> .

Corollary . Let (X,M,∗)be a complete fuzzy metric space and let T be an α∗-η_∗ -admissible set-valued mapping withα= .Assume that the following assertions hold:

(i) forψ∈,x,y∈Xandt> ,

η(x,y,t)≥ ⇒ HM(Tx,Ty,t)≥ψ

N(x,y,t);

(ii) there existx∈Xandx∈PTx(x)such thatη(x,x,t)≥for eacht> ;

(iii) for any sequence{xn} ⊂Xconverging tox∈Xandη(xn,xn+,t)≥,for alln∈N andt> ,we haveη(xn,x,t)≥for alln∈N.

4 Examples

In this section, we conclude the paper with several examples to illustrate the usability of the obtained results.

Example . LetX= [,∞),a∗b=abfor anya,b∈[, ] andM(x,y,t) = _maxmin{_{x_x,_,y_y}_} for

x,y∈Xandt> . Then, for givenλ> , the set-valued mappingT:X→X, where

Tx=

{x+λ}, x+λ< , [√x,√x+λ], x+λ≥,

has a ﬁxed point inX.

Proof Using similar arguments to the ones in [, Theorem ], one shows that (X,M,∗) is a complete GV-space. Letψ(t) =√tfort∈[, ]. Thenψ∈. Let

α(x,y,t) =

, xory∈[, ),



, otherwise,

x,y∈X,t> .

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Case . √x+λ <√y. In this case, we have, for any a∈ Tx, M(a,Ty,t) = √a

y and

inf_a_∈_TxM(a,Ty,t) = √

x

√

y. Similarly,infb∈TyM(Tx,b,t) =

√

x+λ √

y+λ. Consequently, we obtain

HM(Tx,Ty,t) =min

√ x √

y, √

x+λ

√ y+λ

=

√ x √

y=ψ

M(x,y,t).

Case.√x<√y≤√x+λ. We getinfa∈TxM(a,Ty,t) =

√

x

√

yandinfb∈TyM(Tx,b,t) =

√

x+λ √

y+λ. Consequently,

HM(Tx,Ty,t) =min

√ x √

y, √

x+λ

√ y+λ

=

√ x √

y=ψ

M(x,y,t).

Case . √y ≤ √x < √y+λ or √y+λ ≤ √x. We get infb∈TyM(Tx,b,t) =

√

y

√

x and

infa∈TxM(a,Ty,t) =

√

y+λ √

x+λ. Consequently,

HM(Tx,Ty,t) =min

√_y √

x, √

y+λ

√ x+λ

=

√_y √

x=ψ

M(x,y,t).

This shows that αandT satisfy Corollary .(i). Notice that √ +λ∈T√ andα(√,

√

 +λ,t) = _< ; that is, Corollary .(ii) is satisﬁed.

Now, if{xn}is a sequence inXsuch thatα(xn,xn+,t)≤, for allt>  andn∈N, and xn→xasn→ ∞, then {xn} ⊂[,∞), which implies thatx≥. This guarantees that α(xn,x,t)≤ for alln∈Nandt>  and hence Corollary .(iii) holds. Thus all condi-tions of Corollary . are satisﬁed. The conclusion is thatThas a ﬁxed point.

Remark . We observe thatT in Example . is not fuzzyψ-contractive. Hence there exists a mapping which is fuzzyα-ψ-contractive but not fuzzyψ-contractive, although every fuzzyψ-contractive mapping is obviously fuzzyα-ψ-contractive.

In fact, setλ= . and takex= .,y= . in Example .; we haveTx=x+λ= .,

Ty=y+λ= .. Note that

M(Tx,Ty,t) =M(., .,t) =. .=

 ,

M(x,y,t) =M(., .,t) = . .=

 ,

and we haveM(Tx,Ty,t) <ψ(M(x,y,t)) =



. This shows thatT is not fuzzyψ

-contrac-tive.

Example . LetX= [,∞) be endowed with the fuzzy metricM(x,y,t) =exp(–|x–y|/t) for allx,y∈Xandt> . Let the single-valued mappingsT:X→Xbe deﬁned by

Tx=

 x

_, _x_∈_{[, ],}

x+ , x∈(,∞).

Deﬁneα,η:X×X×[,∞)→Xandψ∈byα≡,

η(x,y,t) =

, xory∈[, ],



, x,y∈(,∞),

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andψ(s) =√sfors∈[, ], respectively. We prove that Corollary . can be applied toT. But the fuzzyψ-contraction cannot be applied toT.

Proof Clearly, (X,M,∗) is a complete GV-space. We show thatT is anα-η-admissible mapping. Letx,y∈Xandt> ; ifη(x,y,t)≥, thenx,y∈[, ]. On the other hand, for all

x∈[, ], we haveTx≤. It follows thatη(Tx,Ty,t)≥. Also,η(,T,t)≥.

Now, if{xn}is a sequence inXsuch thatη(xn,xn+,t)≥, for alln∈Nandxn→xas n→ ∞, then{xn} ⊂[, ] and hencex∈[, ]. This implies thatη(xn,x,t)≥ for alln∈N

andt> . Letη(x,y,t)≥. Thenx,y∈[, ]. We get

M(Tx,Ty,t) =exp

–|x

_–_y_|

t

≥exp

–|x–y| t

=ψM(x,y,t).

That is,

η(x,y,t)≥ ⇒ M(Tx,Ty,t)≥ψM(x,y,t).

Then all conditions of Corollary . hold. Hence,T has a ﬁxed point. Letx= ,y=  andt> . ThenTx= ,Ty=  and

M(Tx,Ty,t) =exp

–

t

<exp

–

t

<ψM(x,y,t).

This shows that theψ-contraction introduced in [] cannot be applied toT.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

This work was supported by Natural Science Foundation of Zhejiang Province (LY12A01002).

Received: 6 October 2013 Accepted: 17 December 2013 Published:10 Jan 2014

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10.1186/1687-1812-2014-12

Cite this article as:Hong:Fixed points for modiﬁed fuzzyψ-contractive set-valued mappings in fuzzy metric spaces.