Fixed point theorems for complex non-self mappings satisfying an implicit relation in Kaleva-Seikkala’s type fuzzy metric spaces

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R E S E A R C H

Open Access

Fixed point theorems for complex non-self

mappings satisfying an implicit relation in

Kaleva-Seikkala’s type fuzzy metric spaces

Ming-Liang Song

*

_{and Zhong-qian Wang}

*_{Correspondence:}

[email protected] Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing, 210013, P.R. China

Abstract

In this paper, by usingAϕ-type real functions, some ﬁxed point theorems for complex non-self mappings satisfying an implicit relation in two fuzzy metric spaces in the sense of Kaleva and Seikkala are established. As applications, we obtain the corresponding ﬁxed point theorems for complex non-self mappings satisfying an implicit relation in two Menger probabilistic metric spaces. Also, an example, which shows the validity of the hypotheses of our main results, is given.

MSC: 47H10; 46S40; 54E70

Keywords: fuzzy metric space;Aϕ-type real functions; implicit relation; ﬁxed point; Menger probabilistic metric space

1 Introduction

In , Kaleva and Seikkala [] introduced the concept of a fuzzy metric space by setting the distance between two points to be a nonnegative fuzzy real number. From then on, some important results for mappings in fuzzy metric spaces, such as variational principles, coincidence theorems and various ﬁxed point theorems,etc., were stated in subsequent work (see [–],etc.). It is well known that the Kaleva-Seikkala’s type fuzzy metric space is an important generalization of a Menger probabilistic metric space and a metric space as well (see [, , ]) and possesses rich structure with suitable choices of binary operations. Recently, Huang and Wu [] investigated the completion of the Kaleva-Seikkala’ type fuzzy metric space. Previous work on this question is based on thet-normminand thet-conorm

max. Their meaningful work does away with this restriction. Shortly afterwards, Xiaoet al.[] proved some ﬁxed point theorems of self-mapping for nonlinear contraction in a complete fuzzy metric space. Their main results do away with the restriction of the t-conormmaxand are based on a generic class of binary operations.

On the other hand, fixed point theorems and their applications for complex non-self mappings in two metric spaces were considered by Fisher [] in . Telci [] general-ized the fixed point theorems by introducing the notion of the real functionsF. Recently, Aliouche and Fisher [] proved two new fixed point theorems for complex non-self map-pings by introducing the notion of the real functionsF satisfying an implicit relation in two metric spaces, which is a generalization of the Telci fixed point theorem in [].

Inspired by the work of [, , ], in this paper we discuss the unique existence of ﬁxed points for complex non-self mappings in two fuzzy metric spaces in the sense of Kaleva

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and Seikkala. In Sections -, we first introduce the new real function classAϕsatisfying an implicit relation. Then, by usingAϕ-type real functions, some fixed point theorems for complex non-self mappings satisfying an implicit relation in fuzzy metric spaces are estab-lished. Our main results do away with the restriction of thet-conormmaxand are based on a generic class of binary operations. As some immediate consequences of this theo-rems, we obtain Theorem . and Theorem .. It is a generalization of the main results in []. In Section , as applications, we obtain the corresponding fixed point theorems in Menger probabilistic metric spaces. Also, an example, which shows the validity of the hypotheses of our main results, is given.

2 Basic concepts and lemmas

Throughout this paper, let N be the set of all positive integers, R= (–∞, +∞) and

R+_{= [, +}_∞_{). For the details of a fuzzy real number, we refer the reader to Kaleva and}

Seikkala [], Dubois and Prade [], Bag and Samanta [].

Deﬁnition .(Dubois and Prade []) A mappingu:R→[, ] is called a fuzzy real number or a fuzzy interval, whoseα-level set is denoted by [u]α={t∈R:u(t)≥α}, if it satisﬁes two axioms:

() There existst∈Rsuch thatu(t) = ;

() [u]α= [λα,ρα]is a closed interval ofRfor eachα∈(, ], where –∞<λα≤ρα< +∞.

Let us denote the set of all such fuzzy real numbers byG. Ifu∈Gandu(t) =  whenever t< , thenuis called a nonnegative fuzzy real number, and byG+ _{we mean the set of}

all nonnegative fuzzy real numbers. If λα= –∞andρα= +∞are admissible, then, for the sake of clarity,uis called a generalized fuzzy real number. The sets of all generalized fuzzy real numbers or all generalized nonnegative fuzzy real numbers are denoted byG∞

andG+

∞, respectively. In that case, ifλα= –∞, for instance, then [λα,ρα] means the interval (–∞,ρα]. Since eachr∈Rcan be considered as a fuzzy real numberrdeﬁned by

r(t) =

⎧ ⎨ ⎩

, ift=r,

, ift=r,

Rcan be embedded inG, and ∈G+.

Lemma .(Xiao and Zhu []) Let u∈G,α∈(, ],and[u]α= [λα,ρα].Then () limt→–∞u(t) =  =limt→+∞u(t);

() u(t)is a left-continuous and non-increasing function fort∈(λ, +∞);

() ραis a left-continuous and non-increasing function forα∈(, ].

Deﬁnition .(Kaleva and Seikkala []) LetXbe a non-empty set,dbe a mapping from X×XintoG+_{, and let the mappings}_L,_R_{: [, ]}_×_{[, ]}_→_{[, ] be symmetric,}

nonde-creasing in both arguments and satisfyL(, ) =  andR(, ) = . Forα∈(, ] andx,y∈X, deﬁne the mapping

d(x,y)_α=λα(x,y),ρα(x,y)

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The quadruple (X,d,L,R) is called a fuzzy metric space (brieﬂy, FMS), anddis called a fuzzy metric onXif

(FM-) d(x,y) =¯if and only ifx=y; (FM-) d(x,y) =d(y,x)for allx,y∈X; (FM-) for allx,y,z∈X:

(FM-L) d(x,y)(s+t)≥L(d(x,z)(s),d(z,y)(t)), whenevers≤λ(x,z),t≤λ(z,y)

ands+t≤λ(x,y);

(FM-R) d(x,y)(s+t)≤R(d(x,z)(s),d(z,y)(t)), whenevers≥λ(x,z),t≥λ(z,y)

ands+t≥λ(x,y).

If d is a mapping from X×X into G+

∞ and (X,d,L,R) satisﬁes (FM-)-(FM-), then

(X,d,L,R) is called a generalized fuzzy metric space (brieﬂy, GFMS).

From Lemma . and Deﬁnition ., we obtain the following consequences.

Lemma . Let(X,d,L,R)be aFMS, [d(x,y)]α= [λα(x,y),ρα(x,y)]forα∈(, ],where x,y∈X are any two ﬁxed elements.Then

() limt→–∞d(x,y)(t) =  =limt→+∞d(x,y)(t);

() d(x,y)(t)is a left-continuous and non-increasing function fort∈(λ(x,y), +∞);

() ρα(x,y)is a left-continuous and non-increasing function forα∈(, ].

Lemma .(Xiao and Zhu []) Let(X,d,L,R)be aFMS,and suppose that (R-) R≤max;

(R-) for eachα∈(, ],there existss∈(,α]such thatR(s,t) <αfor allt∈(,α); (R-) lima→+R(a,a) = .

Then(R-)⇒(R-)⇒(R-).

Remark .(Xiaoet al.[, ]) SinceR(t,s) = [min(t,s)]/ _{satisﬁes (R-) and does not}

satisfy (R-), (R-) does not imply (R-). SinceR(t,s) = [max(t,s)]/_{satisﬁes (R-) and does}

not satisfy (R-), (R-) does not imply (R-).

Lemma . Let(X,d,L,R)be aFMS.Then

() (R-)⇒for eachα∈(, ],ρα(x,y)≤ρα(x,z) +ρα(z,y)for allx,y,z∈X(cf.[]); () (R-)⇒for eachα∈(, ],there existsμ=μ(α)∈(,α]such that

ρα(x,y)≤ρμ(x,z) +ρα(z,y)for allx,y,z∈X(cf.[]);

() (R-)⇒for eachα∈(, ],there existsμ=μ(α)∈(,α]such that

ρα(x,y)≤ρμ(x,z) +ρμ(z,y)for allx,y,z∈X(cf.[]).

Lemma .(Kaleva and Seikkala []) Let(X,d,L,R)be aFMSwith(R-).Then the family {U(ε,α) :ε> ,α∈(, ]}of sets U(ε,α) ={(x,y)∈X×X:ρα(x,y) <ε}forms a basis for a Hausdorﬀ uniformity on X×X.Moreover,the sets

Nx(ε,α) =

y∈X:ρα(x,y) <ε

form a basis for a Hausdorﬀ topology on X and this topology is metrizable.

According to Lemma ., convergence in a FMS (X,d,L,R) can be deﬁned by sequences.

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() A sequence{xn}inXis said to be convergent tox(we writexn→xor

limn→∞xn=x) iflimn→∞d(xn,x) =,¯ i.e.,limn→∞ρα(xn,x) = for allα∈(, ]; () A sequence{xn}inXis said to be a Cauchy sequence inXiflimm,n→∞d(xm,xn) =,¯

equivalently, for any givenε> andα∈(, ], there existsN=N(ε,α)∈Z+_such

thatρα(xm,xn) <ε, wheneverm,n≥N;

() (X,d,L,R)is said to be complete if each Cauchy sequence inXis convergent to some point inX.

Lemma .(Xiaoet al.[]) Let(X,d,L,R)be aFMSwith(R-).Then,for eachα∈(, ],

ρα(x,y)is continuous at(x,y)∈X×X.

Deﬁnition . Let ϕ:R+_→_R+ _{be a function and let} _ϕn_{(t) denote the} _{nth iteration}

ofϕ(t).

() ϕis said to satisfy condition()if it is nondecreasing, right-continuous andϕ(t) <t for allt> .

() ϕis said to satisfy condition()if it is nondecreasing, right-continuous and ∞

n=ϕn(t) < +∞for allt> .

Remark . Obviously, ifϕ(t) satisﬁes condition (), thenϕ(t) <tfor allt> ,i.e., ()⊂

(). Sinceϕ(t) = _+t_t satisﬁes condition () and does not satisfy condition (), () does

not imply (),i.e., ()⊂(). Also, ifϕ(t) satisﬁes condition (), thenϕ() = .

Deﬁnition . A functionF:R+

→Ris called a real function satisfying an implicit

rela-tion if the following condirela-tions are satisﬁed: (A-) Fis right-lower semi-continuous;

(A-) There existsϕ∈()or()such thatF(u,v, ,u)≤orF(u,v,u, )≤for

u,v∈R+_implies_u_≤_ϕ_(v).

We denote byAϕ the collection of all real functions F:R+_→Rsatisfying an implicit relation.

The following examples show that theAϕis a largish class of real functions.

Example  Letϕ∈() or (). We deﬁne the functionsF,F:R+→Ras follows:

F(t,t,t,t) =t–ϕ

max{t,t,t}

,

F(t,t,t,t) =t–max

ϕ(t),ϕ(t),ϕ(t)

,

respectively. ThenF,F∈Aϕ.

In fact, since ϕ∈() or (), by the right-continuity ofϕ, we know thatF satisﬁes

condition (A-). IfF(u,v, ,u)≤ orF(u,v,u, )≤, then we haveu≤ϕ(max{u,v}) < max{u,v}foru=v. This impliesu<v, and sou≤ϕ(max{u,v}) =ϕ(v). In addition, ifu=v,

thenu= ≤ϕ(v) =ϕ() = . This shows thatFsatisﬁes condition (A-). HenceF∈Aϕ.

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Example  Letϕ∈() or (). We deﬁne the functionsF,F:R+→R, respectively, as

follows:

F(t,t,t,t) =t–ϕ(λt+λt+λt),

F(t,t,t,t) =t–

λϕ(t) +λϕ(t) +λϕ(t)

,

whereλ,λ,λ≥ andλ+λ+λ= . ThenF,F∈Aϕ. Obviously,Fsatisﬁes condition (A-).

Now suppose that F(u,v, ,u)≤ orF(u,v,u, )≤. Ifu= , then u≤ϕ(v) holds

evidently. Ifu> , then we haveϕ(u) <u≤ϕ(λv+λu) orϕ(u) <u≤ϕ(λv+λu), which implies thatu<λv+λuoru<λv+λu, and sou<v. Hence, we haveu≤ϕ(λv+λu)≤ ϕ(v) oru≤ϕ(λv+λu)≤ϕ(v),i.e., (A-) holds. ThereforeF∈Aϕ.

Similarly, it is easy to prove thatF∈Aϕ.

Example  Letϕ∈() or (). The functionF:R+→Ris deﬁned by

F(t,t,t,t) =t–ϕ

tt+tt

t+t+t+ 

,

thenF∈Aϕ.

In fact, it is easy to see thatFsatisﬁes condition (A-). Now suppose thatF(u,v, ,u)≤

 orF(u,v,u, )≤. Ifu= , thenu≤ϕ(v) holds evidently. Ifu> , then we haveu≤ ϕ(_u₊uv_v_+), which implies thatu≤ϕ(v),i.e., (A-) holds. ThereforeF∈Aϕ.

Example  Letϕ∈() or (). The functionF:R+→Ris deﬁned by

F(t,t,t,t) = ( +t)t–ϕ

max{tt,tt}

–ϕmax{t,t,t}

,

thenF∈Aϕ.

In fact, it is easy to see thatFsatisﬁes condition (A-). Now suppose thatF(u,v, ,u)≤

 orF(u,v,u, )≤. Ifu= , thenu≤ϕ(v) holds evidently. Ifu> , then we have (+v)u≤ ϕ(uv) +ϕ(max{u,v}) <uv+max{u,v}, which implies thatu<max{u,v}, and sou<v. Hence, we have ( +v)u≤ϕ(uv) +ϕ(v) <uv+ϕ(v)⇒u<ϕ(v),i.e., (A-) holds. ThereforeF∈Aϕ.

Deﬁnition . (Aliouche and Fisher []) Letf :R+

→Rbe a function satisfying the

following conditions:

(F-) f is lower semi-continuous;

(F-) There exists <c< such thatf(u,v, ,u)≤orf(u,v,u, )≤foru,v∈R+ impliesu<cv.

We denote byF the collection of all real functionsf :R+_→Rsatisfying the conditions of Deﬁnition ..

Remark . Takingϕ(t) =ct,c∈(, ),t≥, we haveϕ∈()⊂(). Then it is easy to see

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3 Main results

Theorem . Let(X,d,L,R)and(Y,d,L,R)be two completeFMSs with R andR satisfying (R-).Let T:X→Y and S:Y→X be two non-self mappings,andϕ,ϕ∈().If there

exist F∈Aϕand G∈Aϕsuch that

Fρα(Tx,TSy),ρα(x,Sy),ρα(y,Tx),ρα(y,TSy)

≤, (.)

Gρα(Sy,STx),ρα(y,Tx),ρα(x,Sy),ρα(x,STx)

≤ (.)

for all x∈X,y∈Y andα∈(, ],then ST has a unique ﬁxed point x∗∈X,and TS has a

unique ﬁxed point y∗∈Y with Tx∗=y∗,Sy∗=x∗.

Proof For any givenx∈X, we construct a sequence{xn}n∞=inXand a sequence{yn}∞n=

inY, respectively, as follows:

xn= (ST)nx, yn=T(ST)n–x for alln= , , , . . . .

Obviously, we haveyn=Txn–,Syn=xn,TSyn=Txn=yn+for alln= , , , . . . .

Foryn,xn–, applying (.), we obtain for eachα∈(, ]

Fρα(Txn–,TSyn),ρα(xn–,Syn),ρα(yn,Txn–),ρα(yn,TSyn)

=Fρα(yn,yn+),ρα(xn–,xn), ,ρα(yn,yn+)

≤.

Note thatF∈Aϕ, it is not diﬃcult to see that

ρα(yn,yn+)≤ϕ

ρα(xn–,xn)

for eachα∈(, ]. (.)

Again, foryn,xn, applying (.), we obtain for eachα∈(, ]

Gρα(Syn,STxn),ρα(yn,Txn),ρα(xn,Syn),ρα(xn,STxn)

=Gρα(xn,xn+),ρα(yn,yn+), ,ρα(xn,xn+)

≤.

Note thatG∈Aϕ, we have

ρα(xn,xn+)≤ϕ

ρα(yn,yn+)

for eachα∈(, ]. (.)

Sinceϕ,ϕ∈(), by Remark ., (.) and (.), we can obtain

ρα(xn,xn+)≤ϕ

ϕ

ρα(xn–,xn)

≤ϕ

ρα(xn–,xn)

for eachα∈(, ],

and

ρα(yn,yn+)≤ϕ

ϕ

ρα(yn–,yn)

≤ϕ

ρα(yn–,yn)

for eachα∈(, ].

Using the inductive method, forn= , , , . . . , we have

ρα(xn,xn+)≤ϕn

ρα(x,x)

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and

ρα(yn,yn+)≤ϕn–

ρα(y,y)

for eachα∈(, ]. (.)

In the next step, we show that{xn}is a Cauchy sequence inX. Since (X,d,L,R) is with (R-), it follows from Lemma .() that for eachα∈(, ], there existsμ=μ(α)∈(,α] such that

ρα(x,y)≤ρμ(x,z) +ρα(z,y) for allx,y,z∈X. (.)

Form,n∈Nandn<m, by (.), (.) and Lemma .(), we have

ρα(xn,xm)≤ρμ(xn,xn+) +ρα(xn+,xm)

≤ρμ(xn,xn+) +ρμ(xn+,xn+) +· · ·+ρμ(xm–,xm–) +ρα(xm–,xm)

≤ρμ(xn,xn+) +ρμ(xn+,xn+) +· · ·+ρμ(xm–,xm–) +ρμ(xm–,xm)

≤

m–

i=n

ϕ_iρμ(x,x)

≤

∞

i=n

ϕ_iρμ(x,x)

. (.)

Since ϕ ∈(), i.e., ∞n=ϕn(t) < +∞ for all t> , it follows from (.) that{xn} is a Cauchy sequence inX. Hence, by the completeness of (X,d,L,R), there existsx∗∈Xsuch

that limn→∞xn=x∗. By the similar reasoning process, from (.), (.),ϕ∈() and

Lemma .(), we can prove that{yn}is a Cauchy sequence inY. Hence, by the complete-ness of (Y,d,L,R), there existsy∗∈Ysuch thatlimn→∞yn=y∗.

Now we prove thatx∗ is a ﬁxed point ofST andy∗ is a ﬁxed point ofTS. Forx∗,yn–,

applying (.), we have for eachα∈(, ]

Fρα(Tx∗,TSyn–),ρα(x∗,Syn–),ρα(yn–,Tx∗),ρα(yn–,TSyn–)

=Fρα(Tx∗,yn),ρα(x∗,xn–),ρα(yn–,Tx∗),ρα(yn–,yn)

≤.

Letn→ ∞, by the lower semi-continuity ofFand Lemma ., we have for eachα∈(, ]

Fρα(Tx∗,y∗), ,ρα(y∗,Tx∗), 

≤lim inf

n→∞ F

ρα(Tx∗,yn),ρα(x∗,xn–),ρα(yn–,Tx∗),ρα(yn–,yn)

≤.

Note thatF∈Aϕ and (A-) of Deﬁnition ., we can obtainρα(Tx∗,y∗)≤ϕ() =  for

eachα∈(, ],i.e.,Tx∗=y∗. Similarly, we can prove thatSy∗=x∗. Hence,STx∗=Sy∗=x∗

andTSy∗=Tx∗=y∗, which imply thatx∗is a ﬁxed point ofST andy∗ is a ﬁxed point of

TS.

Finally, we show the uniqueness of a ﬁxed point. Ify∗is another ﬁxed point ofTS, then by (.) we have for eachα∈(, ]

Fρα

Tx∗,TSy∗,ρα

x∗,Sy∗,ρα

y∗,Tx∗,ρα

y∗,TSy∗

=Fρα

y∗,y∗,ρα

Sy∗,Sy∗,ρα

y∗,y∗, ≤.

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α∈(, ]. We claim thatSy∗=Sy∗. In fact, ifSy∗=Sy∗, there existsα∈(, ] such that ρα(Sy∗,Sy∗) > . From (.), it follows that

Gρα

Sy∗,STx∗,ρα

y∗,Tx∗,ρα

x∗,Sy∗,ρα(x∗,STx∗)

=Gρα

Sy∗,Sy∗,ρα

y∗,y∗,ρα

Sy∗,Sy∗, ≤.

Note that G∈Aϕ, we haveρα(Sy∗,Sy∗)≤ϕ(ρα(y∗,y∗)). Since ϕ,ϕ∈(), by

Re-mark ., it is not diﬃcult to obtain that

ρα

y∗,y∗≤ϕ

ρα

Sy∗,Sy∗<ρα

Sy∗,Sy∗≤ϕ

ρα

y∗,y∗≤ρα

y∗,y∗,

which is a contradiction. Hence,Sy∗=Sy∗,i.e.,ρα(y∗,y∗)≤ϕ(ρα(Sy∗,Sy∗)) =ϕ() =  for

eachα∈(, ]. This shows thaty∗=y∗,i.e., the uniqueness of a ﬁxed point forTSis true. Similarly, we can prove the uniqueness of a ﬁxed point forST. So, the proof of Theorem .

is ﬁnished.

Theorem . Let(X,d,L,R)and(Y,d,L,R)be two completeFMSs with R andR satis-fying(R-).Let T:X→Y and S:Y→X be two non-self mappings,andϕ,ϕ∈().If there exist F∈Aϕand G∈Aϕsuch that u–F(u,v,w,s)≥λ(Tx,TSy),u–G(u,v,w,s)≥ λ(Sy,STx)and

d(Tx,TSy)u–F(u,v,w,s)

≤maxd(Tx,TSy)(u),d(x,Sy)(v),d(y,Tx)(w),d(y,TSy)(s), (.)

d(Sy,STx)u–G(u,v,w,s)

≤maxd(Sy,STx)(u),d(y,Tx)(v),d(x,Sy)(w),d(x,STx)(s) (.)

for all x∈X,y∈Y and u,v,w,s≥,then ST has a unique ﬁxed point x∗∈X,and TS has

a unique ﬁxed point y∗∈Y with Tx∗=y∗,Sy∗=x∗.

Proof Now, we use inequality (.) to prove that inequality (.) holds. In fact, for each x∈X, y∈Y andα∈(, ], if we setρα(Tx,TSy) =u, ρα(x,Sy) =v, ρα(y,Tx) =w,

ρα(y,TSy) =s, then for anyε> , it is obvious thatd(Tx,TSy)(u+ε) <α,d(x,Sy)(v+ε) <α,

d(y,Tx)(w+ε) <α,d(y,TSy)(s+ε) <α. By (.), we haved(Tx,TSy)(u+ε–F(u+ε,v+ε,w+

ε,s+ε)) <α, which implies that

ρα(Tx,TSy) =u<u+ε–F(u+ε,v+ε,w+ε,s+ε),

i.e.,

Fρα(Tx,TSy) +ε,ρα(x,Sy) +ε,ρα(y,Tx) +ε,ρα(y,TSy) +ε

<ε.

Then, by the arbitrariness ofεand the right-lower semi-continuity ofF, we have

≤lim inf

ε→+ F

ρα(Tx,TSy) +ε,ρα(x,Sy) +ε,ρα(y,Tx) +ε,ρα(y,TSy) +ε

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for eachx∈X,y∈Y andα ∈(, ],i.e., inequality (.) holds for all x,y∈Xandα ∈

(, ].

Similarly, by inequality (.), we can prove that inequality (.) also holds. Moreover, the other conditions in Theorem . are satisﬁed, thus by Theorem ., the theorem is

proved.

In Theorem ., taking (X,d,L,R) = (Y,d,L,R),S=T, we obtain the following corollary.

Corollary . Let(X,d,L,R)be a completeFMSwith R satisfying(R-).Let T:X→X be a self-mapping,andϕ∈().If there exists F∈Aϕsuch that u–F(u,v,w,s)≥λ(Tx,Ty)

and

dTx,Tyu–F(u,v,w,s)

≤maxdTx,Ty(u),d(x,Ty)(v),d(y,Tx)(w),dy,Ty(s) (.)

for all x,y∈X and u,v,w,s≥,then T has a unique ﬁxed point in X.

Corollary . Let(X,d,L,R)be a completeFMSwith R satisfying(R-),and let T:X→X be a self-mapping.If there existsϕ∈()such thatϕ(v)≥λ(Tx,Ty)and

d(Tx,Ty)ϕ(v)≤d(x,y)(v) (.)

for all x∈X,y∈T(X)and v≥,then T has a unique ﬁxed point in X.

Proof TakingF(t,t,t,t) =t–ϕ(λt+λt+λt),λ= ,λ=λ= , from Example ,

we obtainF=F∈Aϕ. Furthermore, for allx,y∈Xandu,v,w,s≥, byTy∈T(X) and (.), we have

dTx,Tyϕ(v)=dTx,Tyu–u–ϕ(v)≤d(x,Ty)(v)

≤maxdTx,Ty(u),d(x,Ty)(v),d(y,Tx)(w),dy,Ty(s),

which implies that (.) holds. Therefore, the conclusion follows from Corollary .

im-mediately.

Remark . Corollary . is a fuzzy version of the Boyd-Wong-type nonlinear contraction theorem (see []).

Theorem . Let(X,d,L,R)and(Y,d,L,R)be two completeFMSs with R andR satisfying (R-).Letϕ,ϕ∈().Suppose that T:X→Y and S:Y→X are two continuous non-self mappings satisfying the following conditions:

() There existsF∈Aϕsuch that

≤ (.)

for allx∈X,y∈Yandα∈(, ]withx=Sy; () There existsG∈Aϕ such that

Gρα(Sy,STx),ρα(y,Tx),ρα(x,Sy),ρα(x,STx)

≤ (.)

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() There existsx∈Xsuch that{(ST)nx}has an accumulation point inX.

Then ST has a unique ﬁxed point x∗∈X,and TS has a unique ﬁxed point y∗∈Y with

Tx∗=y∗,Sy∗=x∗.

Proof From condition (), we can construct a sequence{xn}∞n=inXand a sequence{yn}∞n=

inY, respectively, as follows:

xn= (ST)nx, yn=T(ST)n–x, for alln= , , , . . . .

Obviously, we haveyn=Txn–,Syn=xn,TSyn=Txn=yn+for alln= , , , . . . . Ifxn=xn+

for somen, thenx∗=xnis a ﬁxed point ofST. So, we can assume thatxn=xn+for all

n= , , , . . . . Then, for alln= , , , . . . , it is obvious thatyn=yn+.

Again by condition (), we can assume thatz∈Xis an accumulation point of{(ST)n_x

}=

{xn}. Then there exists a subsequence{xni}of{xn}such thatlimi→∞xni=z. LetTz=y∗.

Next we show thaty∗is a ﬁxed point ofTS.

IfTSy∗=y∗, then byTz=y∗, we haveTSTz=Tz, which implies thatSTz=z,i.e.,z=Sy∗.

It follows from (.) that for allα∈(, ],

Fρα(Tz,TSy∗),ρα(z,Sy∗),ρα(y∗,Tz),ρα(y∗,TSy∗)

=Fρα(Tz,TSy∗),ρα(z,Sy∗), ,ρα(Tz,TSy∗)

≤.

Note thatF∈Aϕ, it is not diﬃcult to see that for allα∈(, ],

ρα(Tz,TSy∗)≤ϕ

ρα(z,Sy∗)

⇔ ρα(Tz,TSTz)≤ϕ

ρα(z,STz)

. (.)

Similarly, byTSy∗=y∗and (.), we obtain for allα∈(, ]

Gρα(Sy∗,STSy∗),ρα(y∗,TSy∗),ρα(Sy∗,Sy∗),ρα(Sy∗,STSy∗)

=Gρα(Sy∗,STSy∗),ρα(y∗,TSy∗), ,ρα(Sy∗,STSy∗)

≤.

Note thatG∈Aϕ, we have for allα∈(, ]

ρα(Sy∗,STSy∗)≤ϕ

ρα(y∗,TSy∗)

⇔ ρα(STz,STSTz)≤ϕ

ρα(Tz,TSTz)

.

(.)

Since STz=z, we know that there existsα ∈(, ] such that ρα(STz,z) > . From ϕ,ϕ∈(), (.) and (.), we can obtain

ρα

STz, (ST)_z_≤_ϕ 

ϕ

ρα(z,STz)

≤ϕ

ρα(z,STz)

<ρα(z,STz). (.)

On the other hand, by (.) and (.), we can prove that{ρα(xn,xn+)}and{ρα(yn,yn+)}

are non-increasing. In fact, applying (.), we obtain

Gρα(Syn,STxn),ρα(yn,Txn),ρα(xn,Syn),ρα(xn,STxn)

=Gρα

S(Txn–),STxn

,ρα(Txn–,Txn),ρα(xn,xn),ρα(xn,STxn)

=Gρα(xn,xn+),ρα(yn,yn+), ,ρα(xn,xn+)

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Note thatG∈Aϕ, we have

ρα(xn,xn+)≤ϕ

ρα(yn,yn+)

≤ρα(yn,yn+). (.)

Similarly, by (.) andF∈Aϕ, we can obtain that

ρα(yn,yn+)≤ϕ

ρα(xn,xn–)

≤ρα(xn,xn–). (.)

Then from (.) and (.), we have ρα(xn,xn+)≤ρα(xn,xn–) and ρα(yn,yn+)≤

ρα(yn–,yn),i.e.,{ρα(xn,xn+)}and{ρα(yn,yn+)}are two non-increasing sequences, and

so there exist ξ,η≥ such that limn→∞ρα(xn,xn+) =ξ andlimn→∞ρα(yn,yn+) =η.

SinceTandSare continuous, by Lemma ., we can obtain that

ρα(z,STz) = lim

i→∞ρα(xni,STxni) = lim

i→∞ρα(xni,xni+) =ξ

and

ρα

STz, (ST)z= lim

i→∞ρα

STxni, (ST)xni

= lim

i→∞ρα(xni+,xni+) =ξ,

which imply thatρα(STz, (ST)z) =ρα(z,STz). This is a contradiction with (.). Hence,

y∗is a ﬁxed point ofTS.

Now we setSy∗=x∗, theny∗=TSy∗=Tx∗⇒STx∗=Sy∗=x∗,i.e.,x∗is a ﬁxed point of ST.

Finally, we show the uniqueness of a ﬁxed point. Assume thaty∗is another ﬁxed point ofTSwithy∗=y∗, thenSy∗=Sy∗. By (.), we have for eachα∈(, ]

Fρα

TSy∗,TSy∗,ρα

Sy∗,Sy∗,ρα

y∗,TSy∗,ρα

y∗,TSy∗

=Fρα

y∗,y∗,ρα

Sy∗,Sy∗,ρα

y∗,y∗, ≤.

Note that F∈Aϕ, it is not diﬃcult to obtain thatρα(y∗,y∗)≤ϕ(ρα(Sy∗,Sy∗)) for each

α∈(, ]. SinceSy∗=Sy∗, there existsα∈(, ] such thatρα(Sy∗,Sy∗) > . Fromϕ∈(),

it follows that

ρα

y∗,y∗≤ϕρα

Sy∗,Sy∗<ρα

Sy∗,Sy∗. (.)

On the other hand, by y∗ =TSy∗, from (.) and G∈Aϕ, it is easy to obtain that ρα(Sy∗,Sy∗)≤ϕ(ρα(y∗,y∗))≤ρα(y∗,y∗). This is a contradiction with (.). Hence, the

uniqueness of a ﬁxed point forTSis true.

Assume thatx∗is also a ﬁxed point ofST. Lety=Tx∗, thenTSy=TSTx∗=Tx∗=y. By the uniqueness of a ﬁxed point forTS, we knowy=y∗. This shows thatx∗=Sy∗=Sy=

STx∗=x∗, i.e., the uniqueness of a ﬁxed point forST holds. This completes the proof.

Let (X,ρ) be a metric space and

d(x,y)(t) =

⎧ ⎨ ⎩

, t=ρ(x,y),

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Then (X,d,min,max) is a FMS (cf.[, ]). It is easy to see that (X,ρ) and (X,d,min,max) are homeomorphic andρ(x,y) =ρα(x,y) for allα∈(, ].

Theorem . Let(X,ρ)and(Y,ρ)be two complete metric spaces.Let T :X→Y and S:Y→X be two non-self mappings,andϕ,ϕ∈().If there exist F∈Aϕand G∈Aϕ

such that

Fρ(Tx,TSy),ρ(x,Sy),ρ(y,Tx),ρ(y,TSy)≤, (.)

Gρ(Sy,STx),ρ(y,Tx),ρ(x,Sy),ρ(x,STx)≤ (.)

for all x∈X,y∈Y,then ST has a unique ﬁxed point x∗∈X,and TS has a unique ﬁxed point y∗∈Y with Tx∗=y∗,Sy∗=x∗.

Proof Note that the topology and completeness of (X,ρ) and (Y,ρ), and the induced FMS (X,d,min,max) and FMS (Y,d,min,max) are coincident respectively, as well asρ(x,x) = ρα(x,x) for allx,x∈X andα∈(, ], andρ(y,y) =ρα(y,y) for ally,y∈Y and α∈(, ]. Then it is not diﬃcult to see that inequality (.) holds as a result of (.), and inequality (.) holds as a result of (.). Moreover, the other conditions of Theorem . are satisﬁed, thus by Theorem ., the theorem is proved.

Applying the same method, we can obtain the following theorem by virtue of Theo-rem ..

Theorem . Let(X,ρ)and(Y,ρ)be two complete metric spaces.Letϕ,ϕ∈().

Sup-pose that T :X→Y and S:Y→X are two continuous non-self mappings satisfying the following conditions:

() There existsF∈Aϕsuch that

Fρ(Tx,TSy),ρ(x,Sy),ρ(y,Tx),ρ(y,TSy)≤ (.)

for allx∈X,y∈Ywithx=Sy; () There existsG∈Aϕ such that

Gρ(Sy,STx),ρ(y,Tx),ρ(x,Sy),ρ(x,STx)≤ (.)

for allx∈X,y∈Ywithy=Tx;

() There existsx∈Xsuch that{(ST)nx}has an accumulation point inX.

Then ST has a unique ﬁxed point x∗in X,and TS has a unique ﬁxed point y∗ in Y with

Tx∗=y∗,Sy∗=x∗.

Remark . Takingϕ(t) =ct,c∈(, ),t≥ andF,G∈F in Theorems . and ., we can obtain Theorems  and  in [], respectively. This shows that our results improve and generalize Theorems - in [] , and so the main results in [, ].

4 Applications to Menger probabilistic metric spaces and example

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Deﬁnition . A functionF:R→[, ] is called a distribution function if it is nonde-creasing and left-continuous withinft∈RF(t) =  andsupt∈RF(t) = .

IfF is a distribution function which satisﬁesF() = , thenF is called a nonnegative distribution function. LetF+be the set of all nonnegative distribution functions. A special element ofF+is the Heaviside functionHdeﬁned by

H(t) =

⎧ ⎨ ⎩

, t> ,

, t≤.

Deﬁnition . (Had˘zić and Pap []) A function : [, ]×[, ]→[, ] is called a triangular norm (for short, a t-norm) if the following conditions are satisﬁed for any a,b,c,d∈[, ]:

(-) (a, ) =a; (-) (a,b) =(b,a);

(-) (a,b)≥ (c,d), fora≥c,b≥d; (-) ((a,b),c) =(a,(b,c)).

For each a∈ [, ], the sequence {n_(a)_}∞

n= is deﬁned by (a) =a and n(a) =

(n–_(a),_{a). A}_t-norm_{is said to be of}_{H-type if the sequence of functions}_{n_(a)_}∞ n=

is equicontinuous ata= .

Lemma .(Xiao et al. []) Let be a t-norm for each a,b∈[, ],R be deﬁned by R(a,b) =  –( –a,  –b),then

() Ris a symmetric and nondecreasing function such thatR(, ) = ; () Ifis ofH-type,thenRsatisﬁes(R-).

Remark .(Xiaoet al.[]) LetR: [, ]×[, ]→[, ] be a symmetric and nondecreas-ing function such thatR(, ) =  and(a,b) =  –R( –a,  –b) for alla,b∈[, ]. Then

satisﬁes (-) and (-). Butdoes not necessarily satisfy (-) and (-). Hence

is not necessarily at-norm. From Remark . we see thatR(s,t) = [min{s,t}]/satisﬁes

(R-); but(s,t) =  –R( –s,  –t) =  – [ –max{s,t}]/is not at-norm ofH-type.

Deﬁnition .(Hadžić and Pap []) A triplet (X,F,) is called a Menger probabilistic metric space ifXis a non-empty set,is at-norm andFis a mapping fromX×Xinto

F+_{satisfying the following conditions (F(x,}_{y) for}_x,_y_∈_X_{is denoted by}_F

x,y): (M-) Fx,y(t) =H(t)for allt∈Rif and only ifx=y;

(M-) Fx,y(t) =Fy,x(t)for allx,y∈Xandt∈R;

(M-) Fx,y(t+s)≥ (Fx,z(t),Fz,y(s))for allx,y,z∈Xandt,s∈R+.

Lemma .(Kaleva and Seikkala []) Let(X,F,)be a Menger probabilistic metric space, x,y∈X andωxy=sup{t:Fx,y(t) = }.Let d:X×X→G+be a mapping deﬁned by

d(x,y)(t) =

⎧ ⎨ ⎩

, t<ωx,y,

 –Fx,y(t), t≥ωx,y.

(.)

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From Lemma . we see that each Menger probabilistic metric space can be consid-ered as a special Kaleva-Seikkala’s type fuzzy metric space. But Remark . shows that in general a Kaleva-Seikkala’s type fuzzy metric space cannot be considered as a Menger probabilistic metric space. Hence, as direct consequences of our results, we can obtain the corresponding ﬁxed point theorems in Menger probabilistic metric spaces. For example, from Theorem . and Lemmas .-. we can obtain the following consequence.

Theorem . Let (X,F,) and (Y,F,) be two complete Menger probabilistic metric

spaces such that andare two t-norms of H-type.Let T :X→Y and S:Y→X be

two non-self mappings,andϕ,ϕ∈().If there exist f ∈Aϕand g∈Aϕsuch that

FTx,TSy

u–f(u,v,w,s)≥minFTx,TSy(u),Fx,Sy(v),Fy,Tx(w),Fy,TSy(s)

, (.)

FSy,STx

u–g(u,v,w,s)≥minFSy,STx(u),Fy,Tx(v),Fx,Sy(w),Fx,STx(s)

(.)

for all x∈X,y∈Y and u,v,w,s≥,then ST has a unique ﬁxed point x∗∈X,and TS has

a unique ﬁxed point y∗∈Y with Tx∗=y∗,Sy∗=x∗.

Proof Let (X,d,L,R) and (Y,d,L,R) be deﬁned as in Lemma ., respectively. Then (X,d,L,R) and (Y,d,L,R) are two FMSs. Since andare ofH-type, by Lemma ., RandRsatisfy (R-). Now we check that (.) holds.

From (.) we see that

Fx,y(t) >  –α ⇔ d(x,y)(t) <α ⇔ ρα(x,y) <t,

for allx,y∈Xandα∈(, ]. (.)

For eachx∈X,y∈Yandα∈(, ], if we setρα(Tx,Tsy) =u,ρα(x,Sy) =v,ρα(y,Tx) =w,

ρα(y,TSy) =s, then for anyε> , it is obvious thatρα(Tx,Tsy) <u+ε,ρα(x,Sy) <v+ε,

ρα(y,Tx) <w+ε,ρα(y,TSy) <s+ε. By (.), we haveFTx,TSy(u+ε) >  –α,Fx,Sy(v+ε) >  –α,

Fy,Tx(w+ε) >  –α,Fy,TSy(s+ε) >  –α. Note (.), we can obtain that

FTx,TSy

u+ε–f(u+ε,v+ε,w+ε,s+ε)>  –α,

which implies that

ρα(Tx,TSy) =u<u+ε–f(u+ε,v+ε,w+ε,s+ε),

i.e.,

fρα(Tx,TSy) +ε,ρα(x,Sy) +ε,ρα(y,Tx) +ε,ρα(y,TSy) +ε

<ε.

Then by the arbitrariness ofεand the right-lower semi-continuity off, we have

fρα(Tx,TSy),ρα(x,Sy),ρα(y,Tx),ρα(y,TSy)

≤lim inf

ε→+ f

ρα(Tx,TSy) +ε,ρα(x,Sy) +ε,ρα(y,Tx) +ε,ρα(y,TSy) +ε

≤

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Similarly, by (.), we can prove that (.) also holds. Moreover, the other conditions in Theorem . are satisﬁed, thus by Theorem ., the theorem is proved.

In Theorem ., taking (X,F,) = (Y,F,),S=T, we obtain the following corollary.

Corollary . Let(X,F,)be a complete Menger probabilistic metric space such thatis a t-norm of H-type.Let T:X→X be a self-mapping,andϕ∈().If there exists f∈Aϕ such that

FTx,T_yu–f(u,v,w,s)≥minF_Tx_,_T_y(u),F_x_,_Ty(v),F_y_,_Tx(w),F_y_,_T_y(s) (.)

for all x,y∈X and u,v,w,s≥,then T has a unique ﬁxed point in X.

Corollary . Let(X,F,)be a complete Menger probabilistic metric space such that is a t-norm of H-type.Let T:X→X be a self-mapping.If there existsϕ∈()such that

FTx,Ty

ϕ(v)≥Fx,y(v) (.)

for all x∈X,y∈T(X)and v≥,then T has a unique ﬁxed point in X.

Proof Takingf(t,t,t,t) =t–ϕ(λt+λt+λt),λ= ,λ=λ= , from Example ,

we obtainf =F∈Aϕ. Furthermore, for allx,y∈Xandu,v,w,s≥, byTy∈T(X) and (.), we have

F_Tx_,_T_y

ϕ(v)=F_Tx_,_T_y

u–u–ϕ(v)≥Fx,Ty(v)

≥minF_Tx_,_T_y(u),F_x_,_Ty(v),F_y_,_Tx(w),F_y_,_T_y(s)

,

which implies that (.) holds. Therefore, the conclusion follows from Corollary .

im-mediately.

Remark . Recently, in [] Jachymski obtained the following result.

Theorem J Let(X,F,)be a complete Menger probabilistic metric space such thatis a continuous t-norm of H-type.Let a functionϕ:R+→R+be such that,for any r> ,

 <ϕ(r) <r and lim

n→∞ϕ

n_{(r) = .} _(.)

Let T:X→X be a mapping such that

FTx,Ty

ϕ(t)≥Fx,y(t) for all t>  and x,y∈X. (.)

Then there exists a unique x∗∈X such that Tx∗=x∗.

Remark . Comparing Corollary . with Theorem J, it is not diﬃcult to see the diﬀer-ences between them. Although Corollary . demands the functionϕ∈() to imply that

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Finally, we give an example to support the main results presented herein.

Example  Suppose thatX= [–, ]⊂R. Deﬁned:X×X→G+_by

d(x,y)(t) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

, ift< ,

, ift= ,

 –e–|x–ty|_, _if_t_{> ,}

for allx,y∈X. (.)

LetL,R: [, ]×[, ]→[, ] be deﬁned byL≡ andR(a,b) =max{a,b}. Then (X,d,L,R) is a complete FMS.

In fact, (FM-), (FM-) and (FM-L) are easy to check. We only see (FM-R). Since d(x,y)() =  for allx,y∈X, and soλ(x,y) =  for allx,y∈X. To prove (FM-R), we assume

thats,t> ,x,y,z∈Xand

Rd(x,z)(t),d(z,y)(s)=max –e–|x–tz|_{,  –}_e–| z–y|

s _{=  –}_e–| x–z|

t _.

Then we haves|x–z| ≥t|z–y|, and sot+_ts|x–z|=|x–z|+s_t|x–z| ≥ |x–z|+|z–y| ≥ |x–y|. It follows that

d(x,y)(t+s) =  –e–|xt+–sy|≤_{ –}_e–| x–z|

t ₌_R_d(x,_z)(t),_d(z,_y)(s)_.

Hence (FM-R) holds. It is clear that (X,d,L,R) is complete.

In the same manner, if we takeY= [, ]⊂Randd:Y×Y→G+_{given by (.), then}

(Y,d,L,R) is a complete FMS. DeﬁneT:X→YandS:Y→Xby

Tx=

⎧ ⎨ ⎩ 

, x∈[–, ), 

, x∈[, ],

and Sy=

⎧ ⎨ ⎩ 

, y∈[,  ), 

, y∈[  , ].

Obviously, STx= _ for each x∈[–, ], TSy= _ for each y∈[, ], and ST(_) = _, TS(_) =_,T(_) =_,S(_) =_.

Next we check thatTandSsatisfy the conditions in Theorem .. Obviously, we have

ρα(x,y) =

⎧ ⎨ ⎩

, ifα= ,

–_ln|x_(––y_α|₎, ifα∈(, ), for allx,y∈X;

and

ρα(x,y) =

⎧ ⎨ ⎩

, ifα= ,

–_ln|x_(––y_α|₎, ifα∈(, ),

for allx,y∈Y.

We setF(u,v,w,s) =u–ϕ(max{v,w,s}),G(u,v,w,s) =u–ϕ(max{v,w,s}), whereϕ(t) =_t,

ϕ(t) =_t(∀t∈R). It is easily seen thatϕ,ϕ∈(),F∈Aϕ andG∈Aϕ. Ifα= , then

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Case . Ifx∈[–, ) andy∈[,_), then for eachα∈(, ) we have

ρα(Tx,TSy) =ρα

 ,   = – 

ln( –α), ρα(x,Sy) =ρα

x, 

= – |x–

 | ln( –α),

ρα(y,Tx) =ρα

y, 

= – |y–

 |

ln( –α), ρα(y,TSy) =ρα

y, 

= –|y–

 | ln( –α).

Sinceinf_x_∈_{[–,),}_y_∈_[,

){max{|x–  |,|y–

 |,|y–

 |}}=

 ,i.e.,

 –



max{|x–  |,|y–

 |,|y– 

|} ≤ for allx∈[–, ) andy∈[, 

), it follows that for eachα∈(, )

– 

ln( –α)–  max

– |x–

 | ln( –α), –

|y–_|

ln( –α), – |y–_|

ln( –α)

≤,

which implies that (.) holds. Similarly, we have

ρα(Sy,STx) =ρα

 ,   = – 

ln( –α), ρα(y,Tx) =ρα

y, 

= – |y–

 | ln( –α),

ρα(x,Sy) =ρα

x, 

= – |x–

 |

ln( –α), ρα(x,STx) =ρα

x, 

= – |x–

 | ln( –α).

Sinceinf_x_∈_{[–,),}_y_∈_[,

){max{|y–  |,|x–

 |,|x–

 |}}=

 ,i.e.,

 –



max{|y–  |,|x–

 |,|x– 

|} ≤ for allx∈[–, ) andy∈[, 

), it follows that for eachα∈(, )

– 

ln( –α)–  max

– |y–

 | ln( –α), –

|x–_|

ln( –α), – |x–_|

ln( –α)

≤,

which implies that (.) also holds.

Case . Ifx∈[–, ) andy∈[_, ], then for eachα∈(, ) we have

ρα(Tx,TSy) =ρα

 ,   = – 

ln( –α), ρα(x,Sy) =ρα

x, 

= – |x–

 | ln( –α),

ρα(y,Tx) =ρα

y, 

= – |y–

 |

ln( –α), ρα(y,TSy) =ρα

y, 

= –|y–

 | ln( –α).

Sinceinf_x_∈_{[–,),}_y_∈_[

,]{max{|x–  |,|y–

 |,|y–

 |}}=

 ,i.e.,

 –



max{|x–  |,|y–

 |,|y– 

|} ≤ for allx∈[–, ) andy∈[ 

, ], it follows that for eachα∈(, )

– 

ln( –α)–  max

– |x–

 | ln( –α), –

|y–_|

ln( –α), – |y–_|

ln( –α)

≤,

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Case . If x∈ [, ] and y∈ [,_), then for each α ∈ (, ) we have ρα(Tx,TSy) =

ρα(_,_) = , which shows that (.) holds. In addition, for eachα∈(, ) we have

ρα(Sy,STx) =ρα

 ,   = – 

ln( –α), ρα(y,Tx) =ρα

y, 

= – |y–

 | ln( –α),

ρα(x,Sy) =ρα

x, 

= – |x–

 |

ln( –α), ρα(x,STx) =ρα

x, 

= – |x–

 | ln( –α).

Sinceinf_x_∈_[,],_y_∈_[,

){max{|y– 

|,|x–  |,|x–

 |}}=

 ,i.e.,

 –



max{|y–  |,|x–

 |,|x– 

|} ≤ for allx∈[, ] andy∈[, 

), then we can obtain that for eachα∈(, )

– 

ln( –α)–  max

– |y–

 | ln( –α), –

|x–_|

ln( –α), – |x–_|

ln( –α)

≤,

which implies that (.) also holds.

Case . If x ∈[, ] and y∈[_, ], then for each α ∈(, ) we have ρα(Tx,TSy) =

˜

ρα(_,_) = ,ρα(Sy,STx) =ρα(_,_) = , which imply that (.) and (.) hold.

Thus, all the conditions of Theorem . are satisﬁed. This shows the validity of the hy-potheses of our main results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Acknowledgements

This work was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 13KJB110004) and Qing Lan Project of Jiangsu Province of China.

Received: 7 June 2013 Accepted: 28 August 2013 Published:07 Nov 2013 References

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