Coupled fixed point theorems for partially contractive mappings

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R E S E A R C H

Open Access

Coupled ﬁxed point theorems for partially

contractive mappings

Thabet Abdeljawad

*

*_{Correspondence:}

[email protected] Mathematics and Computer Science Department, Çankaya University, Eski¸sehir Yolu, Yenimahalle, Ankara, 06810, Turkey

Abstract

Recently, some authors have started to generalize fixed point theorems for contractive mappings in a class of generalized metric spaces in which the self-distance need not be zero. These spaces, partial metric spaces, were first introduced by Matthews in 1994. The proved fixed point theorems have been obtained for mappings satisfying contraction type conditions empty of the self-distance. In this article, we prove some coupled fixed point theorems for mappings satisfying contractive conditions allowing the appearance of self-distance terms. These partially contractive mappings do reflect the structure of the partial metric space, and hence their coupled fixed theorems generalize the previously obtained by (Aydi in Int. J. Math. Sci. 2011:Article ID 647091, 2011). Some examples are given to support our claims.

MSC: 47H10; 54H25

Keywords: partial metric space; coupled ﬁxed point; partially contractive map; strong partially contractive; 0-complete

1 Introduction and preliminaries

The Banach contraction mapping principle is considered to be the soul of many extended fixed point theorems. It has widespread applications in many branches of mathematics, engineering and computer science. During the last decades many authors were able to gen-eralize this principle [–]. After the appearance of partial metric spaces as a place for dis-tinct research work into flow analysis, non-symmetric topology and domain theory [, ], many authors started to generalize this principle to these spaces (see [–]. However, the contraction type conditions used in those generalizations do not reflect the structure of a partial metric space apparently. Later, the authors in [] proved a more reasonable con-traction principle in the partial metric space in which they used self-distance terms. On the other hand, the theory of coupled fixed point theorems has recently attracted some authors (see [–]). Also, Meir-Keeler type common and tripled fixed point theorems have been recently considered over partial metric spaces [, ]. In this article, we prove a coupled partial contraction principle generalizing the recently published coupled fixed point theorems in []. An example is presented to show that our coupled partial contrac-tion principle is worthy of investigacontrac-tion.

A partial metric space (PMS) (see,e.g., [, ]) is a pair (X,p:X×X→R+_{) (where}_R+

denotes the set of all nonnegative real numbers) such that (P) p(x,y) =p(y,x)(symmetry)

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(P) If≤p(x,x) =p(x,y) =p(y,y)thenx=y(equality) (P) p(x,x)≤p(x,y)(small self-distances)

(P) p(x,z) +p(y,y)≤p(x,y) +p(y,z)(triangularity) for allx,y,z∈X.

For a partial metricponX, the functiondp:X×X→R+given by

dp(x,y) = p(x,y) –p(x,x) –p(y,y) ()

is a (usual) metric onX. Each partial metricponXgenerates aTtopologyτponXwith a

base of the family of openp-balls{Bp(x,ε) :x∈X,ε> }, whereBp(x,ε) ={y∈X:p(x,y) <

p(x,x) +ε}for allx∈Xandε> .

Deﬁnition (see,e.g., [, , ])

(i) A sequence{xn}in a PMS(X,p)converges tox∈Xif and only if p(x,x) =lim_n→∞p(x,xn);

(ii) A sequence{xn}in a PMS(X,p)is called Cauchy if and only iflimn,m→∞p(xn,xm)

exists (and is ﬁnite);

(iii) A PMS(X,p)is said to be complete if every Cauchy sequence{xn}inXconverges, with respect toτp, to a pointx∈Xsuch thatp(x,x) =limn,m→∞p(xn,xm);

(iv) A mappingf :X→Xis said to be continuous atx∈X, if for everyε> , there

existsδ> such thatf(Bp(x,δ))⊂Bp(f(x),ε).

Lemma (see,e.g., [, , ])

(A) A sequence{xn}is Cauchy in a PMS(X,p)if and only if{xn}is Cauchy in a metric space(X,dp);

(B) A PMS(X,p)is complete if and only if the metric space(X,dp)is complete. Moreover,

lim

n→∞dp(x,xn) =  ⇔ p(x,x) =n→∞lim p(x,xn) =n,mlim→∞p(xn,xm). ()

A sequence{xn}is called -Cauchy [] iflimm,np(xn,xm) = . The partial metric space

(X,p) is called -complete [, ] if every -Cauchy sequence inxconverges to a point x∈X with respect topandp(x,x) = . Clearly, every complete partial metric space is -complete. The converse need not be true.

Example (see []) LetX=Q∩[,∞) with the partial metricp(x,y) =max{x,y}. Then (X,p) is a -complete partial metric space which is not complete.

Letρp=inf{p(x,y) :x,y∈X}and deﬁneXp={x∈X:p(x,x) =ρp}.

The following theorem was presented in [].

Theorem  Let(X,p)be a complete metric space,α∈[, )and T:X→X a given map-ping. Suppose that for each x,y∈X the following condition holds:

p(x,y)≤maxαp(x,y),p(x,x),p(y,y).

Then

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() there is a uniqueu∈Xpsuch thatTu=u;

() for eachx∈Xpthe sequence{Tnx}n≥converges touwith respect to the metricdp.

If (X,p) is a partial metric space, then clearly (p×p) :X×X→Rdeﬁned by (p× p)((x,y), (u,v)) =p(x,u) +p(y,v) is a partial metric, and hence (X×X, (p×p)) is a par-tial metric space. Clearly, if (X,p) is complete (-complete) then so is (X×X, (p×p)). The minimum self partial distance in (X×X, (p×p)) is deﬁned by

ρ_p×p:=inf(p×p)(x,y), (x,y)=p(x,x) +p(y,y) :x,y∈X.

The set of all points inX×Xwith self partial distanceρp×pis denoted by

(X×X)p×p=(x,y)∈X×X: (p×p)(x,y), (x,y)=ρp×p.

Example  LetX= [, ] and provideXwith the partial metricp(x,y) =|x–y|if bothx,y∈ [, ) and p(x,y) =max{x,y} otherwise. Then clearly (X,p) is a complete partial metric space,ρ_p×p=  and (X×X)p×p= [, )×[, ).

Deﬁnition  Let (X,p) be a partial metric space andF:X×X→Xbe a mapping. Then Fis called

(a) partially contractive if there exist constants≤k,l< such that for allx,y,u,v∈X,

pF(x,y),F(u,v)

≤

max

kp(x,u) +lp(y,v), (p×p)(x,y), (x,y), (p×p)(u,v), (u,v), ()

(b) strong partially contractive if there exist constants≤k,l< such that for all x,y,u,v∈X,

pF(x,y),F(u,v)

≤

max

kp(x,u) +lp(y,v),(p×p)((x,y), (x,y)) + (p×p)((u,v), (u,v)) 

. ()

Aydi [] proved the following coupled ﬁxed point theorems in partial metric spaces:

Theorem  Let(X,p)be a complete partial metric space. Suppose that the mapping F: X×X→X satisﬁes the following contractive condition for all x,y,u,v∈X:

pF(x,y),F(u,v)≤kp(x,u) +lp(y,v),

where≤k,l< with k+l< . Then F has a unique coupled ﬁxed point. That is, there exists unique(x,y)∈X×X such that F(x,y) =x and F(y,x) =y.

Theorem  Let(X,p)be a complete partial metric space. Suppose that the mapping F: X×X→X satisﬁes the following contractive condition for all x,y,u,v∈X:

pF(x,y),F(u,v)≤kpF(x,y),x+lpF(u,v),u,

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2 Main results

Theorem  Let(X,p)be a complete partial metric space and F:X×X→X be a partially contractive mapping. Then

(a) (X×X)p×p=∅,

(b) there exists a unique(u,v)∈(X×X)p×psuch that

F(u,v) =u and F(v,u) =v; ()

(c) for each(x,y)∈(X×X)p×pthe sequence{(xn,yn)} ∈X×Xdeﬁned by

x=F(x,y), y=F(y,x), xn+=F(xn,yn), yn+=F(yn,xn), n≥,

converges to(u,v)with respect todp×dp. That is

lim

n→∞(dp×dp)

(xn,yn), (u,v)

= lim

n→∞

dp(xn,u) +dp(yn,v)

= .

Proof If (x,y)∈X×X, deﬁne the sequence{(xn,yn)} ∈X×Xby

x=F(x,y), y=F(y,x), xn+=F(xn,yn), yn+=F(yn,xn), n≥.

We divide the proof of the claim (a) into the following steps:

Step I: The sequence{(p×p)((xn,yn)(xn,yn))}={p(xn,xn) +p(yn,yn)}is non-increasing.

From the condition () we have

Tn=p

F(xn,yn),F(xn,yn)

+pF(yn,xn),F(yn,xn)

≤ 



p(xn,xn) +p(yn,yn) +

 

p(xn,xn) +p(yn,yn)

=p(xn,xn) +p(yn,yn) = (p×p)

(xn,yn), (xn,yn)

,

whereTn= (p×p)((xn+,yn+), (xn+,yn+)). As a result of Step I, there existsr(x,y)≥

such that

lim

n→∞(p×p)

(xn,yn), (xn,yn)

=inf

n(p×p)

(xn,yn), (xn,yn)

=r(x,y).

Step II: For eachn≥, we show that

(p×p)(xn,yn), (x,y)

≤M(x,y) :=



 –τ(p×p)

(x,y), (x,y)

+ (p×p)(x,y), (x,y)

, ()

whereτ=max{k,l}. We follow by induction. Forn= , , it is clear. Assume () is true for n≤r, and let us prove () forn=r+ ≥. Then by the help of (), the triangle inequality and Step I, we have

E = (p×p)(xr+,yr+), (x,y)

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=p(x,x) +p(y,y) +p

F(xr,yr),F(x,y)

+pF(yr,xr),F(y,x)

≤(p×p)(x,y), (x,y)

+ max

kp(xr,x) +lp(yr,y), (p×p)

(x,y), (x,y)

+ max

lp(xr,x) +kp(yr,y), (p×p)

(x,y), (x,y)

≤(p×p)(x,y), (x,y)

+maxτ(p×p)(xr,yr), (x,y)

, (p×p)(x,y), (x,y)

≤(p×p)(x,y), (x,y)

+maxτM(x,y), (p×p)

(x,y), (x,y)

≤M(x,y),

whereE=p(xr+,x) +p(yr+,y).

Step III: We show that

lim

m,n→∞(p×p)

(xn,yn), (xm,ym)

= lim

m,n→∞

p(xn.xm) +p(yn,ym) =r(x,y). ()

Let> . Choosensuch that (p×p)((xn,yn), (xn,yn)) <r(x,y)+and M(x,y)τ

n_<

r(x,y) +. Then form,n≥n, we have

S =pF(xn–,yn–),F(xm–,ym–)

+pF(yn–,xn–),F(ym–,xm–)

≤ 

max

kp(xn–,xm–) +lp(yn–,ym–), (p×p)

(xn–,yn–),

(xn–,yn–)

, (p×p)(xm–,ym–), (xm–,ym–)

+ max

lp(xn–,xm–) +kp(yn–,ym–), (p×p)

(xn–,yn–),

(xn–,yn–)

, (p×p)(xm–,ym–), (xm–,ym–)

≤maxτp(xn–,xm–) +p(yn–,ym–) , (p×p)

(xn–,yn–),

(xn–,yn–)

, (p×p)(xm–,ym–), (xm–,ym–)

≤ · · · ≤maxτn_p(x

n–n,xm–n) +p(yn–n,ym–n), (p×p)

(xn–n,yn–n), (xn–n,yn–n)

, (p×p)(xm–n,ym–n), (xm–n,ym–n)

≤maxr(x,y) +,r(x,y) +, M(x,y)τn

≤r(x,y) +, ()

whereS=p(xn,xm) +p(yn,ym). On the other hand, (P) impliesr(x,y) –≤p(xn,xn) +

p(yn,yn)≤p(xn,xm) +p(yn,ym) and so () is obtained.

As a result of () and completeness of (X,p), there exists (x._,y._)∈X×Xsuch that r(x,y) = lim

m,n→∞(p×p)

(xn,yn), (xm,ym)

= lim

n→∞(p×p)

(xn,yn),

x._,y._= (p×p)x._,y._,x._,y._. () Step IV: We show that for eachn≥ the following holds:

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First, note that by (P) applied toporp×p, we have

(p×p)x._,y._,x._,y._≤(p×p)x._,y._,Fx._,y_.,Fy._,x._. () By the help of the triangle inequality applied toporp×p, for eachn≥, we have

T ≤(p×p)x._,y._, (xn+,yn+)

+ (p×p)(xn+,yn+),

Fx._,y._,Fy._,x._ – (p×p)(xn+,yn+), (xn+,yn+)

, ()

whereT= (p×p)((x.

,y.), (F(x.,y.),F(y.,x.))). By the condition () and deﬁnition ofp×p,

we have

L≤  max

kpxn,x.

+lpyn,y.

,

(p×p)(xn,yn), (xn,yn)

, (p×p)x._,y._,x._,y._ +

max

kpyn,y.

+lpxn,x.

,

(p×p)(yn,xn), (yn,xn)

, (p×p)y._,x._,y._,x._

≤maxτpxn,x.

+pyn,y. ,

(p×p)(xn,yn), (xn,yn)

, (p×p)x._,y._,x._,y._,

whereL= (p×p)((xn+,yn+), (F(x.,y.),F(y.,x.))). Without loss of generality (by passing

to subsequence if necessary), we may assume that for eachn≥, we have one of the fol-lowing:

(p×p)(xn+,yn+),

Fx._,y._,Fy._,x._≤τpxn,x.

+pyn,y. ,

or

(p×p)(xn+,yn+),

Fx._,y._,Fy._,x._≤(p×p)(xn,yn), (xn,yn)

,

or

(p×p)(xn+,yn+),

Fx._,y._,Fy._,x._≤(p×p)x._,y._,x._,y._.

Substituting each of these cases in (), lettingn→ ∞and using () together with (), we arrive at ().

Step V: We prove that (X×X)p×p=∅.

For eachk∈N, pick (xk,yk)∈X×Xwith

(p×p)(xk,yk), (xk,yk)

<ρp×p+

k. ()

We show that

lim

m,n→∞(p×p)

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Given>  putn= [(–τ)] + . Ifk≥nthen by () we have

ρp×p≤(p×p)Fx.k,y.k

,Fy._k,x._k,Fx_k.,y._k,Fy._k,x._k =pFx._k,y._k,Fx._k,y._k+pFy._k,x._k,Fy._k,x._k

≤px._k,x._k+py._k,y._k

= (p×p)x._k,y._k,x._k,y._k=r(xk,yk) ≤(p×p)(xk,yk), (xk,yk)

<ρp×p+

k

≤ρ_p×p+  n

<ρ_p×p+( –τ)

 . ()

Hence, we conclude that for allk≥n,

Uk:= (p×p)

x._k,y._k,x._k,y_k.– (p×p)Fx._k,y._k,Fy._k,x._k,Fx._k,y._k,Fy._k,x._k

≤ ( –τ)

 +

ρ_p×p– (p×p)Fx._k,y._k,Fy._k,x_k.,Fx._k,y_k.,Fy._k,x._k < ( –τ)

 ()

and

(p×p)x._k,y._k,x._k,y._k≤ρp×p+( –τ)

 . ()

Now, ifm,n≥n, then by the triangle inequality applied to (p×p), Step IV and (), we

have

(p×p)x._n,y._n,x._m,y._m =Un+Um+p

Fx._n,y._n,Fx._m,y._m+pFy._n,x._n,Fy._m,x._m

≤Un+Um+max

τ(p×p)x._n,y_n.,x._m,y._m ,

(p×p)x._n,y._n,x._n,y._n, (p×p)x._m,y._m,x._m,y._m. ()

Hence, using () and (), we obtain

ρ_p×p≤(p×p)x._n,y._n,x._m,y._m

≤max



 ,

( –τ)

 + (p×p)

x._n,y._n,x._n,y._n, ( –τ)

 + (p×p)

x._m,y._m,x._m,y._m

≤max



 ,( –τ) +ρp×p

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This shows () and so{(x.

n,y.n)}is Cauchy in the complete partial metric space (X×X,p×

p). Therefore, there exists (x,y)∈X×Xsuch that

(p×p)(x,y), (x,y)

= lim

n→∞(p×p)

x._n,y._n,x._n,y._n= lim

n→∞(p×p)

x._n,y._n,x._m,y._m=ρ_p×p. ()

In particular (x,y)∈(X×X)p×pand so (X×X)p×p=∅.

Now, let (x,y)∈(X×X)p×pbe arbitrary. Then by (), (P) applied top×pand (),

we have

ρ_p×p≤(p×p)x._,y._,x._,y._

≤(p×p)x._,y._,Fx._,y._,Fy._,x._= (p×p)x._,y._,x._,y.=r(x,y)

≤(p×p)(x,y), (x,y)

=ρp×p. ()

Therefore, (P) applied top×pimplies that

(p×p)x._,y._,x._,y._= (p×p)x_.,y._,Fx._,y._,Fy._,x._

and so (x._,y._) is a coupled ﬁxed point ofF. Clearly, () implies that{(xn,yn)}converges

to the coupled ﬁxed point (x._,y._) with respect todp×dpwhich proves (c). To complete

the proof of (b), assume (x,y)∈(X×X)p×pand (u,v)∈(X×X)p×pare both coupled ﬁxed

points ofF. Then by () we have

(p×p)(x,y), (u,v)= (p×p)F(x,y),F(y,x),F(u,v),F(v,u)

≤maxτ(p×p)(x,y), (u,v), (p×p)(x,y), (x,y),

(p×p)(u,v), (u,v). ()

From which it follows either (p×p)((x,y), (u,v))≤τ(p× p)((x,y), (u,v)) and so (p× p)((x,y), (u,v)) =  implies that (x,y) = (u,v), or ρp×p ≤ (p × p)((x,y), (u,v))≤ (p× p)((x,y), (x,y)) = (p×p)((u,v), (u,v)) = ρ_p×p, and hence (P) applied to p× p implies

(x,y) = (u,v).

Remark  Although Theorem  does not imply the uniqueness of the fixed point, it is easy to see that, under the assumptions made, if (x,y) and (u,v) are both coupled fixed points forFsatisfying (p×p)((x,y), (x,y)) = (p×p)((u,v), (u,v)), then (x,y) = (u,v). If the partially contractive condition () is replaced by the somewhat stronger condition below, the uniqueness of the coupled fixed point is guaranteed.

Theorem  Let(X,p)be a complete partial metric space,≤k,l< and F:X×X→X be a strong partially contractive mapping. Then there exists a unique coupled ﬁxed point (x,y). Furthermore, (x,y)∈(X×X)p×p and for each(x,y)∈(X×X)p×p, the sequence {(xn,yn)} ∈X×X deﬁned by

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converges to(x,y)with respect to dp×dp. That is

lim

n→∞(dp×dp)

(xn,yn), (x,y)

= lim

n→∞

dp(xn,x) +dp(yn,y)

= .

Proof By Theorem , we only need to prove the uniqueness of the coupled ﬁxed point. If (x,y) and (u,v) are two coupled ﬁxed points, then

G=p(x,u) +p(y,v) =pF(x,y),F(u,v)+pF(y,x),F(v,u)

≤ 

max

+ max

≤max

τp(x,u) +p(y,v),(p×p)((x,y), (x,y)) + (p×p)((u,v), (u,v)) 

, ()

whereG= (p×p)((x,y), (u,v)) andτ =max{k,l}. From () we distinguish the following cases:

Case : If (p×p)((x,y), (u,v))≤τ(p(x,u) +p(y,v)) =τ(p×p)((x,y), (u,v)), then (p× p)((x,y), (u,v)) =  and so (x,y) = (u,v).

Case : If (p×p)((x,y), (u,v))≤(p×p)((x,y),(x,y))+(_p×p)((u,v),(u,v)), then (dp×dp)((x,y), (u,v)) = 

and so (x,y) = (u,v).

As a corollary, we obtain the already mentioned result [] stated in Theorem . As well, let us remark that the result of Aydi in Theorem  is valid also for -complete partial metric spaces.

Corollary  Let(X,p)be a-complete partial metric space. Suppose that the mapping F:X×X→X satisﬁes the following contractive condition for all x,y,u,v∈X:

pF(x,y),F(u,v)≤kp(x,u) +lp(y,v), ()

where≤k,l< with k+l< . Then F has a unique coupled ﬁxed point(u,v)such that (p×p)((u,v), (u,v)) = . Also for each(x,y)∈X×X the sequence{(xn+,yn+)}converges

to(u,v)with respect to the metric dp×dp.

Proof The condition () implies that (p× p)((xn+,yn+), (xn+,yn+)) ≤ (k + l)n(p×

p)((x,y), (x,y)), which, in turn, by () implies (p×p)((x.,y.)) = . But then by ()

we have (p×p)((x.,y.), (F(x.,y.),F(y.,x.))) =  so the dp ×dp-limit of the sequence {(xn+,yn+)}is actually the unique coupled ﬁxed point (x.,y.).

Example  Deﬁne p: [, ]×[, ]→[, ] byp(x,y) =max{x,y}. Then ([, ],p) is a complete partial metric space. LetF: [, ]×[, ]→[, ] be deﬁned by

F(x,y) =|x–y|  .

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(a) There are no ≤k,l<  withk+l<  such thatp(F(x,y),F(u,v))≤kp(x,u) +lp(y,v), for allx,y,u,v∈[, ]. That is,Fdoes not verify the assumptions of Theorem . Indeed, if we assume that there exist ≤k,l<  withk+l<  such that for allx,y,u,v∈X= [, ], we have

pF(x,y),F(u,v)≤kp(x,u) +lp(y,v),

then

pF(, ),F(, )=

≤kp(, ) +lp(, ) =k, ()

and

pF(, ),F(, )=

≤kp(, ) +lp(, ) =l, ()

then we havek+l≥, which is a contradiction.

(b) There are no ≤k,l<  withk+l<  such thatp(F(x,y),F(u,v))≤kp(F(x,y),x) + lp(F(u,v),u). That is,Fdoes not verify the assumptions of Theorem . Indeed, if we assume that there exist ≤k,l<  withk+l<  such that for allx,y,u,v∈X= [, ], we have

pF(x,y),F(u,v)≤kpF(x,y),x+lpF(u,v),u. ()

Then we conclude that

pF(, ),F(, )= ≤kp

 , 

+p(, ) =k <

 ,

which is a contradiction.

(c)Fis partially contractive. That isFverifies the assumptions of Theorem . Hence, it has the coupled fixed point (, ). The condition () is clearly satisfied since|x–y| ≤x+y for allx,y∈X= [, ].

(d) Note that (X×X)p×p={(, )}and hence the coupled Picard sequence{(xn,yn)}

con-verges for (x,y) = (, ) and for (x,y)∈(X×X)\{(, )}may converge or not depending

on the particular choice of (x,y).

(e) IfFis replaced byH(x,y) =|x–_y|, then it is strong partially contractive, and hence by Theorem , we guarantee the uniqueness of the coupled ﬁxed point (, ).

Competing interests

The author declares that they have no competing interests.

Received: 14 May 2012 Accepted: 24 August 2012 Published: 12 September 2012 References

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doi:10.1186/1687-1812-2012-148