Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces

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R E S E A R C H

Open Access

Common coupled fixed point theorems for

weakly compatible mappings in fuzzy metric

spaces

Xin-Qi Hu

1*

, Ming-Xue Zheng

1

, Boško Damjanovi´c

2

and Xiao-Feng Shao

3

*Correspondence:

xqhu.math@whu.edu.cn 1School of Mathematics and

Statistics, Wuhan University, Wuhan, 430072, P.R. China

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

Abstract

In this paper, we prove a common fixed point theorem for weakly compatible mappings under

φ

-contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of Hu (Fixed Point Theory Appl. 2011:363716, 2011,

doi:10.1155/2011/363716). We also indicate a minor mistake in Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716).

1 Introduction

In , Zadeh [] introduced the concept of fuzzy sets. Then many authors gave the im-portant contribution to development of the theory of fuzzy sets and applications. George and Veeramani [, ] gave the concept of a fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space, which have very important applications in quantum particle physics, particularly, in connection with both string and E-infinity theory.

Bhaskar and Lakshmikantham [], Lakshmikantham and Ćirić [] discussed the mixed monotone mappings and gave some coupled fixed point theorems, which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghiet al.[] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Jin-xuan Fang [] gave some common fixed point theorems for compatible and weakly compatibleφ-contractions mappings in Menger probabilistic metric spaces. Xin-Qi Hu [] proved a common fixed point theorem for mappings underϕ-contractive conditions in fuzzy metric spaces. Many authors [–] proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.

In this paper, we give a new coupled fixed point theorem under weaker conditions than in [] and give an example, which shows that the result is a genuine generalization of the corresponding result in [].

2 Preliminaries

First, we give some definitions.

Definition . (see []) A binary operation∗: [, ]× [, ]→[, ] is a continuous

t-norm if∗satisfies the following conditions: () ∗is commutative and associative,

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() ∗is continuous,

() a∗ =afor alla∈[, ],

() abcdwheneveracandbdfor alla,b,c,d∈[, ].

Definition .(see []) Letsup<t<(t,t) = . At-normis said to be of H-type if the family of functions{m(t)}∞m=is equicontinuous att= , where

(t) =tt, m+(t) =tm(t), m= , , . . . ,t∈[, ]. (.)

The t-normM =minis an example of t-norm of H-type, but there are some other

t-normsof H-type [].

Obviously,is at-norm of H-type if and only if for anyλ∈(, ), there existsδ(λ)∈(, ) such thatm(t) >  –λfor allm∈N, whent>  –δ.

Definition .(see []) A -tuple (X,M,∗) is said to be a fuzzy metric space ifXis an

arbitrary nonempty set,∗is a continuoust-norm andMis a fuzzy set onX×(, +∞) satisfying the following conditions for eachx,y,zXandt,s> ,

(FM-) M(x,y,t) > ,

(FM-) M(x,y,t) = if and only ifx=y, (FM-) M(x,y,t) =M(y,x,t),

(FM-) M(x,y,t)∗M(y,z,s)≤M(x,z,t+s), (FM-) M(x,y,·) : (,∞)→[, ]is continuous.

We shall consider a fuzzy metric space (X,M,∗), which satisfies the following condition:

lim

t→+∞M(x,y,t) = , ∀x,yX. (.)

Let (X,M,∗) be a fuzzy metric space. Fort> , the open ballB(x,r,t) with a centerxX

and a radius  <r<  is defined by

B(x,r,t) =yX:M(x,y,t) >  –r. (.)

A subsetAXis called open if for eachxA, there existt>  and  <r<  such that

B(x,r,t)⊂A. Letτdenote the family of all open subsets ofX. Thenτis called the topology onX, induced by the fuzzy metricM. This topology is Hausdorff and first countable.

Example . Let (X,d) be a metric space. Definet-normab=aborab=min{a,b}

and for allx,yXandt> ,M(x,y,t) =t+dt(x,y). Then (X,M,∗) is a fuzzy metric space.

Definition .(see []) Let (X,M,∗) be a fuzzy metric space. Then

() a sequence{xn}inXis said to be convergent tox(denoted bylimn→∞xn=x) if

lim

n→∞M(xn,x,t) = 

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() A sequence{xn}inXis said to be a Cauchy sequence if for anyε> , there exists

n∈N, such that

M(xn,xm,t) >  –ε

for allt> andn,mn.

() A fuzzy metric space(X,M,∗)is said to be complete if and only if every Cauchy sequence inXis convergent.

Remark .(see []) Let (X,M,∗) be a fuzzy metric space. Then () for allx,yX,M(x,y,·)is non-decreasing;

() ifxnx,yny,tnt, then

lim

n→∞M(xn,yn,tn) =M(x,y,t);

() ifM(x,y,t) >  –rforx,yinX,t> , <r< , then we can find at, <t<tsuch thatM(x,y,t) >  –r;

() for anyr>r, we can find arsuch thatr∗r≥r, and for anyr, we can find ar such thatr∗r≥r(r,r,r,r,r∈(, )).

Define ={φ:R+R+}, whereR+= [, +) and eachφ satisfies the following conditions:

(φ-) φis non-decreasing,

(φ-) φis upper semi-continuous from the right,

(φ-) ∞n=φn(t) < +for allt> , whereφn+(t) =φ(φn(t)),nN.

It is easy to prove that ifφ∈ , thenφ(t) <tfor allt> .

Lemma .(see []) Let(X,M,∗)be a fuzzy metric space,whereis a continuous t-norm

of H-type.If there existsφsuch that

Mx,y,φ(t)≥M(x,y,t) (.)

for all t> ,then x=y.

Definition .(see []) An element (x,y)∈X×Xis called a coupled fixed point of the

mappingF:X×XXif

F(x,y) =x, F(y,x) =y. (.)

Definition .(see []) An element (x,y)∈X×Xis called a coupled coincidence point

of the mappingsF:X×XXandg:XXif

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Definition .(see []) An element (x,y)∈X×Xis called a common coupled fixed point of the mappingsF:X×XXandg:XXif

x=F(x,y) =g(x), y=F(y,x) =g(y). (.)

Definition .(see []) An elementxXis called a common fixed point of the mappings

F:X×XXandg:XXif

x=g(x) =F(x,x). (.)

Definition .(see []) The mappingsF:X×XXandg :XX are said to be compatible if

lim

n→∞M

gF(xn,yn),F

g(xn),g(yn)

,t=  (.)

and

lim

n→∞M

gF(yn,xn),F

g(yn),g(xn)

,t=  (.)

for allt>  whenever{xn}and{yn}are sequences inX, such that

lim

n→∞F(xn,yn) =nlim→∞g(xn) =x, nlim→∞F(yn,xn) =nlim→∞g(yn) =y, (.)

for allx,yXare satisfied.

Definition . (see []) The mappings F :X×XX and g :XX are called weakly compatible mappings ifF(x,y) =g(x),F(y,x) =g(y) implies thatgF(x,y) =F(gx,gy),

gF(y,x) =F(gy,gx) for allx,yX.

Remark . It is easy to prove that ifFandgare compatible, then they are weakly com-patible, but the converse need not be true. See the example in the next section.

3 Main results

For simplicity, denote

M(x,y,t)n=M(x,y,t)∗M(x,y,t)∗ · · · ∗M(x,y,t)

n

for alln∈N.

Xin-Qi Hu [] proved the following result.

Theorem .(see []) Let(X,M,∗)be a complete FM-space,whereis a continuous t-norm of H-type satisfying(.).Let F:X×XX and g:XX be two mappings,and there existsφsuch that

MF(x,y),F(u,v),φ(t)≥Mg(x),g(u),tMg(y),g(v),t (.)

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Suppose that F(X×X)⊆g(X),g is continuous,F and g are compatible.Then there exist x,yX such that x=g(x) =F(x,x);that is,F and g have a unique common fixed point in X.

Now we give our main result.

Theorem . Let(X,M,∗)be a FM-space,whereis a continuous t-norm of H-type

sat-isfying(.).Let F:X×XX and g:XX be two weakly compatible mappings,and there existsφsatisfying(.).

Suppose that F(X×X)⊆g(X)and F(X×X)or g(X)is complete.Then F and g have a unique common fixed point in X.

Proof Letx,y∈Xbe two arbitrary points inX. SinceF(X×X)⊆g(X), we can choose

x,y∈Xsuch thatg(x) =F(x,y) andg(y) =F(y,x). Continuing this process, we can construct two sequences{xn}and{yn}inXsuch that

g(xn+) =F(xn,yn), g(yn+) =F(yn,xn), for alln≥. (.)

The proof is divided into  steps.

Step : We shall prove that{gxn}and{gyn}are Cauchy sequences.

Since∗is at-norm of H-type, for anyλ> , there exists anμ>  such that

( –μ)∗( –μ)∗ · · · ∗( –μ)

k

≥ –λ

for allk∈N.

SinceM(x,y,·) is continuous andlimt→+∞M(x,y,t) =  for allx,yX, there existst>  such that

M(gx,gx,t)≥ –μ, M(gy,gy,t)≥ –μ. (.)

On the other hand, sinceφ∈ , by condition (φ-), we haven=φn(t) <. Then for

anyt> , there existsn∈Nsuch that

t> ∞

k=n

φk(t). (.)

From condition (.), we have

Mgx,gx,φ(t)

=MF(x,y),F(x,y),φ(t)

M(gx,gx,t)∗M(gy,gy,t),

Mgy,gy,φ(t)=MF(y,x),F(y,x),φ(t)

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Similarly, we have

Mgx,gx,φ(t)=MF(x,y),F(x,y),φ(t)

Mgx,gx,φ(t)∗Mgy,gy,φ(t)

M(gx,gx,t)∗M(gy,gy,t),

Mgy,gy,φ(t)=MF(y,x),F(y,x),φ(t)

M(gy,gy,t)∗M(gx,gx,t) 

.

From the inequalities above and by induction, it is easy to prove that

Mgxn,gxn+,φn(t)

M(gx,gx,t)n–M(gy,gy,t)n–,

Mgyn,gyn+,φn(t)

M(gy,gy,t)n–M(gx,gx,t) n–

.

So, from (.) and (.), form>nn, we have

M(gxn,gxm,t)≥M

gxn,gxm,

k=n

φk(t)

M

gxn,gxm, m–

k=n

φk(t)

Mgxn,gxn+,φn(t)

Mgxn+,gxn+,φn+(t)

∗ · · · ∗Mgxm–,gxm,φm–(t)

M(gy,gy,t) n–

M(gx,gx,t)n–M(gy,gy,t)n

M(gx,gx,t)n∗ · · · ∗M(gy,gy,t) m–

M(gx,gx,t) m–

=M(gy,gy,t)

m––n–

M(gx,gx,t)m––n–

≥( –μ)∗( –μ)∗ · · · ∗( –μ) m–n

≥ –λ,

which implies that

M(gxn,gxm,t) >  –λ (.)

for allm,n∈Nwithm>nnandt> . So{g(xn)}is a Cauchy sequence.

Similarly, we can prove that{g(yn)}is also a Cauchy sequence.

Step : Now, we prove thatgandFhave a coupled coincidence point.

Without loss of generality, we can assume thatg(X) is complete, then there existx,yg(X), and exista,bXsuch that

lim

n→∞g(xn) =nlim→∞F(xn,yn) =g(a) =x,

lim

n→∞g(yn) =nlim→∞F(yn,xn) =g(b) =y.

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From (.), we get

MF(xn,yn),F(a,b),φ(t)

Mgxn,g(a),t

Mgyn,g(b),t

.

SinceMis continuous, taking limit asn→ ∞, we have

Mg(a),F(a,b),φ(t)= ,

which implies thatF(a,b) =g(a) =x.

Similarly, we can show thatF(b,a) =g(b) =y.

SinceF andgare weakly compatible, we get thatgF(a,b) =F(g(a),g(b)) andgF(b,a) =

F(g(b),g(a)), which implies thatg(x) =F(x,y) andg(y) =F(y,x). Step : We prove thatg(x) =yandg(y) =x.

Since∗is at-norm of H-type, for anyλ> , there exists anμ>  such that

( –μ)∗( –μ)∗ · · · ∗( –μ)

k

≥ –λ

for allk∈N.

SinceM(x,y,·) is continuous andlimt→+∞M(x,y,t) =  for allx,yX, there existst>  such thatM(gx,y,t)≥ –μandM(gy,x,t)≥ –μ.

On the other hand, sinceφ∈ , by condition (φ-), we have∞n=φn(t) <∞. Thus, for anyt> , there existsn∈Nsuch thatt>

k=nφ

k(t). Since

Mgx,gyn+,φ(t)

=MF(x,y),F(yn,xn),φ(t)

M(gx,gyn,t)∗M(gy,gxn,t),

lettingn→ ∞, we get

Mgx,y,φ(t)

M(gx,y,t)∗M(gy,x,t). (.)

Similarly, we can get

Mgy,x,φ(t)

M(gx,y,t)∗M(gy,x,t). (.)

From (.) and (.), we have

Mgx,y,φ(t)

Mgy,x,φ(t)

M(gx,y,t) 

M(gy,x,t) 

.

From this inequality, we can get

Mgx,y,φn(t)∗Mgy,x,φn(t)≥Mgx,y,φn–(t)∗Mgy,x,φn–(t)

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for alln∈N. Sincet>k=nφk(t), then, we have

M(gx,y,t)∗M(gy,x,t)≥M

gx,y, ∞

k=n

φk(t)

M

gy,x, ∞

k=n

φk(t)

Mgx,y,φn(t)Mgy,x,φn(t)M(gx,y,t)n∗M(gy,x,t)n

≥( –μ)∗( –μ)∗ · · · ∗( –μ) n

≥ –λ.

Therefore, for anyλ> , we have

M(gx,y,t)∗M(gy,x,t)≥ –λ (.)

for allt> . Hence conclude thatgx=yandgy=x. Step : Now, we prove thatx=y.

Since∗is at-norm of H-type, for anyλ> , there exists anμ>  such that ( –μ)∗( –μ)∗ · · · ∗( –μ)

k

≥ –λ

for allk∈N.

Since M(x,y,·) is continuous andlimt+M(x,y,t) = , there exists t>  such that

M(x,y,t)≥ –μ.

On the other hand, sinceφ∈ , by condition (φ-), we have∞n=φn(t) <∞. Then, for anyt> , there existsn∈Nsuch thatt>

k=nφ

k(t

). From (.), we have

Mgxn+,gyn+,φ(t)

=MF(xn,yn),F(yn,xn),φ(t)

M(gxn,gyn,t)∗M(gyn,gxn,t).

Lettingn→ ∞yields

Mx,y,φ(t)

M(x,y,t)∗M(y,x,t).

Thus, we have

M(x,y,t)≥M

x,y, ∞

k=n

φk(t)

Mx,y,φn(t

)

M(x,y,t)n–∗M(y,x,t)n–

≥( –μ)∗( –μ)∗ · · · ∗( –μ) n–

≥ –λ,

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Thus, we proved thatFandghave a common fixed point inX.

The uniqueness of the fixed point can be easily proved in the same way as above. This

completes the proof of Theorem ..

Takingg=I(the identity mapping) in Theorem ., we get the following consequence.

Corollary . Let(X,M,∗)be a FM-space,whereis a continuous t-norm of H-type sat-isfying(.).Let F:X×XX,and there existsφsuch that

MF(x,y),F(u,v),φ(t)≥M(x,u,t)∗M(y,v,t) (.)

for all x,y,u,vX,t> .F(X)is complete.

Then there exist xX such that x=F(x,x);that is,F admits a unique fixed point in X.

Remark . Comparing Theorem . with Theorem . in [], we can see that Theo-rem . is a genuine generalization of TheoTheo-rem ..

() We only need the completeness ofg(X)orF(X×X). () The continuity ofgis relaxed.

() The concept of compatible has been replaced by weakly compatible.

Remark . The Example  in [] is wrong, since thet-normab=abis not thet-norm of H-type.

Next, we give an example to support Theorem ..

Example . LetX={, ,,, . . . ,n, . . .},∗=min,M(x,y,t) =|xty|+t, for allx,yX,t> . Then (X,M,∗) is a fuzzy metric space.

Letφ(t) =t. Letg:XXandF:X×XXbe defined as

g(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

, x= ,

, x=n+, 

n+, x=  n,

F(x,y) = ⎧ ⎨ ⎩

(n+), (x,y) = (n,n),

, others.

Letxn=yn=n. We havegxn=n+→,F(xn,yn) =(n+)→, but

MF(gxn,gyn),gF(xn,yn),t

=M(, ,t),

sogandFare not compatible. FromF(x,y) =g(x),F(y,x) =g(y), we can get (x,y) = (, ), and we havegF(, ) =F(g,g), which implies thatFandgare weakly compatible.

The following result is easy to verify

t

X+t≥min

t Y+t,

t Z+t

X≤max{Y,Z}, ∀X,Y,Z≥,t> .

By the definition ofMandφand the result above, we can get that inequality (.)

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is equivalent to the following

F(x,y) –F(u,v)≤maxg(x) –g(u),g(y) –g(v). (.)

Now, we verify inequality (.). LetA={n,n∈N},B=XA. By the symmetry and without loss of generality, (x,y), (u,v) have  possibilities.

Case : (x,y)∈B×B, (u,v)∈B×B. It is obvious that (.) holds. Case : (x,y)∈B×B, (u,v)∈B×A. It is obvious that (.) holds.

Case : (x,y)∈B×B, (u,v)∈A×A. Ifu=v, (.) holds. Ifu=v, letu=v=n, then

F(x,y) –F(u,v)= 

(n+ ), maxg(x) –g(u),g(y) –g(v)= n

n+ ,

which implies that (.) holds.

Case : (x,y)∈B×A, (u,v)∈B×A. It is obvious that (.) holds.

Case : (x,y)∈B×A, (u,v)∈A×A. Ifu=v, (.) holds. If u=v, letxB,y= j,

u=v=n, then

F(x,y) –F(u,v)=  (n+ ),

maxg(x) –g(u),g(y) –g(v)=max

 n+ ,

j+ –  n+ 

,

or

maxg(x) –g(u),g(y) –g(v)=max

n

n+ ,

j+ –  n+ 

,

(.) holds.

Case : (x,y)∈A×A, (u,v)∈A×A. Ifx=y,u=v, (.) holds.

Ifx=y,u=v, letx=  i,y=

j,i=j,u=v=

 n. Then

F(x,y) –F(u,v)=  (n+ ),

maxg(x) –g(u),g(y) –g(v)=max 

i+ –  n+ 

, 

j+ –  n+ 

,

(.) holds.

Ifx=y,u=v, letx=y=i,u=v=n. Then

F(x,y) –F(u,v)=   (i+ )–

 (n+ )

,

maxg(x) –g(u),g(y) –g(v)=  i+ –

 n+ 

,

(.) holds.

(11)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China.2Faculty of Agriculture, University of

Belgrade, Nemanjina 6, Belgrade, Serbia.3The General Course Department of Huanggang Polytechnic College, Huanggang, Hubei 438002, P.R. China.

Acknowledgements

This work of Xin-qi Hu was supported by the National Natural Science Foundation of China (under grant No. 71171150). The research of B. Damjanovi´c was supported by Grant No. 174025 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.

Received: 1 March 2013 Accepted: 1 August 2013 Published: 19 August 2013 References

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