Fixed points of (ψ,ϕ) contractions on rectangular metric spaces

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R E S E A R C H

Open Access

Fixed points of

(

ψ

,

φ

)

contractions on

rectangular metric spaces

˙Inci M Erhan

1*

_{, Erdal Karapınar}

1

_{and Tanja Sekuli´c}

2

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Atilim University, Incek, Ankara 06836, Turkey

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

Abstract

Existence and uniqueness of ﬁxed points of a general class of (ψ,

φ) contractive

mappings on complete rectangular metric spaces are discussed. One of the theorems is a generalization of a ﬁxed point theorem recently introduced by Lakzian and Samet. Fixed points of (ψ,

φ) contractions under conditions involving rational expressions are

also investigated. Several particular cases and applications as well as an illustrative example are given.

MSC: 46T99; 47H10; 54H25

Keywords: rectangular metric spaces; ﬁxed point theorem; Hausdorﬀ spaces

1 Introduction and preliminaries

Fixed point theory has been one of the most rapidly developing fields in analysis during the last few decades. Wide application potential of this theory has accelerated the research activities which resulted in an enormous increase in publications [–]. In a large class of studies the classical concept of a metric space has been generalized in different directions by partly changing the conditions of the metric. Among these generalizations, one can mention the partial metric spaces introduced by Matthews [, ] (see also [, , –]), and rectangular metric spaces defined by Branciari [].

Branciari deﬁned a rectangular metric space (RMS) by replacing the sum at the right-hand side of the triangle inequality by a three-term expression. He also proved an analog of the Banach Contraction Principle. The intriguing nature of these spaces has attracted attention, and ﬁxed points theorems for various contractions on rectangular metric spaces have been established (see,e.g., [–]).

In  Boyd and Wong [] deﬁned a class of contractive mappings calledφ contrac-tions. In , Alber and Guerre-Delabriere [] generalized this concept by introducing weakφcontraction. A self-mappingT on a metric space (X,d) is said to be weakφ con-tractive if there exists a mapφ: [, +∞) –→[, +∞) withφ() =  andφ(t) >  for allt>  such that

d(Tx,Ty)≤d(x,y) –φd(x,y), (.)

for allx,y∈X. Contractions of this type have been studied by many authors (see,e.g., [, ]). The larger class of (ψ,φ) weakly contractive mappings has also been a subject of interest (see,e.g., [, –]).

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In a recent paper, Lakzian and Samet [] stated and proved a ﬁxed point theorem for the (ψ,φ) weakly contractive mappings on complete rectangular metric spaces. They also provided interesting examples as particular cases of such mappings. In this paper, we gen-eralize the result of Lakzian and Samet [] and, in addition, investigate maps satisfying rational type contractive conditions. We also give applications and an example.

We state some basic deﬁnitions and notations to be used throughout this paper. Rect-angular metric spaces are deﬁned as follows.

Deﬁnition ([]) LetXbe a nonempty set and letd:X×X–→[,∞] satisfy the fol-lowing conditions for allx,y∈Xand all distinctu,v∈Xeach of which is diﬀerent fromx

andy.

(RM) d(x,y) =  if and only ifx=y,

(RM) d(x,y) =d(y,x), (.)

(RM) d(x,y)≤d(x,u) +d(u,v) +d(v,y).

Then the mapdis called a rectangular metric and the pair (X,d) is called a rectangular metric space.

The concepts of convergence, Cauchy sequence and completeness in a RMS are deﬁned below.

Deﬁnition 

() A sequence{xn}in a RMS(X,d)is RMS convergent to a limitxif and only if

d(xn,x)→asn→ ∞.

() A sequence{xn}in a RMS(X,d)is RMS Cauchy if and only if for everyε> there exists a positive integerN(ε)such thatd(xn,xm) <εfor alln>m>N(ε).

() A RMS(X,d)is called complete if every RMS Cauchy sequence inXis RMS convergent.

We also use the following modiﬁed notations of Lakzian and Samet [].

Letdenote the set of all continuous functionsψ: [, +∞)→[, +∞) for whichψ(t) =  if and only ift= . Nondecreasing functions which belong to the classare also known as altering distance functions (see []).

In their paper, Lakzian and Samet [] stated the following ﬁxed point theorem.

Theorem  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

ψd(Tx,Ty)≤ψd(x,y)–φd(x,y)

for all x,y∈X andψ,φ∈, whereψ is nondecreasing. Then T has a unique ﬁxed point in X.

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2 Main results and applications

We present our main results in this section. First, we state the following ﬁxed point theo-rem.

Theorem  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

ψd(Tx,Ty)≤ψM(x,y)–φM(x,y)+Lm(x,y) (.)

for all x,y∈X andψ,φ∈, where L> , the functionψis nondecreasing and M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty),

m(x,y) =mind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

(.)

Then T has a unique ﬁxed point in X.

Proof First, we prove the existence part. Letx∈Xbe an arbitrary point. Deﬁne the

se-quence{xn} ⊂Xas

xn=Txn–, n= , , , . . . .

Assume thatxn=xn+=Txnfor alln≥. Substitutex=xn–andy=xnin (.) and note that

m(xn–,xn) =min

d(xn–,xn),d(xn,xn+),d(xn–,xn+),d(xn,xn)

= .

Then we obtain,

ψd(Txn–,Txn)

=ψd(xn,xn+)

≤ψM(xn–,xn)

–φM(xn–,xn)

+Lm(xn–,xn)

=ψM(xn–,xn)

–φM(xn–,xn)

, (.)

where

M(xn–,xn) =max

d(xn–,xn),d(xn–,xn),d(xn,xn+)

=maxd(xn–,xn),d(xn,xn+)

.

IfM(xn–,xn) =d(xn,xn+), then we have

ψd(xn,xn+)

≤ψd(xn,xn+)

–φd(xn,xn+)

,

which impliesφ(d(xn,xn+)) = , and henced(xn,xn+) = . Thenxn=xn+=Txn, which contradicts the initial assumption. Therefore, we must haveM(xn–,xn) =d(xn–,xn), that is,

ψd(xn,xn+)

≤ψd(xn–,xn)

–φd(xn–,xn)

≤ψd(xn–,xn)

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Sinceψis nondecreasing, thend(xn,xn+)≤d(xn–,xn) for alln≥, that is, the sequence

{d(xn,xn+)}is decreasing and positive. Hence, it converges to a positive number, says> .

Taking limit asn→ ∞in (.), we obtain

ψ(s)≤ψ(s) –φ(s),

which leads toφ(s) =  and hence tos= . Thus, lim

n→∞d(xn,xn+) = . (.)

Next, we will show thatT has a periodic point, that is, there exist a positive integerp

and a pointz∈Xsuch thatz=Tpz. Assume the contrary, that is,Thas no periodic point. Then, all elements of the sequence{xn}are distinct,i.e.,xn=xmfor alln=m. Suppose also that{xn}is not a RMS Cauchy sequence. Therefore, there existsε>  for which one can ﬁnd subsequences{xn(i)}and{xm(i)}of{xn}withn(i) >m(i) >isuch that

d(xm(i),xn(i))≥ε, (.)

wheren(i) is the smallest integer satisfying (.), that is,

d(xm(i),xn(i)–) <ε. (.)

We apply the rectangular inequality (RM) and use (.) and (.) to obtain

ε≤d(xm(i),xn(i))

≤d(xm(i),xn(i)–) +d(xn(i)–,xn(i)–) +d(xn(i)–,xn(i))

≤ε+d(xn(i)–,xn(i)–) +d(xn(i)–,xn(i)). (.)

Taking limit asi→ ∞in (.) and using (.), we get

lim

i→∞d(xn(i),xm(i)) =ε. (.)

Employing the rectangular inequality (RM) once again, we write the following inequali-ties:

d(xn(i),xm(i))≤d(xn(i),xn(i)–) +d(xn(i)–,xm(i)–) +d(xm(i)–,xm(i)),

d(xn(i)–,xm(i)–)≤d(xn(i)–,xn(i)) +d(xn(i),xm(i)) +d(xm(i),xm(i)–),

(.)

from which we obtain

ε≤ lim

i→∞d(xn(i)–,xm(i)–)≤ε, (.)

using (.) and (.); and therefore,

lim

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Now we substitutex=xn(i)–andy=xm(i)–in (.), which yields

ψd(Txn(i)–,Txm(i)–)

=ψd(xn(i),xm(i))

≤ψM(xn(i)–,xm(i)–)

–φM(xn(i)–,xm(i)–)

+Lm(xn(i)–,xm(i)–), (.)

where

M(xn(i)–,xm(i)–) =max

d(xn(i)–,xm(i)–),d(xn(i)–,xn(i)),d(xm(i)–,xm(i)

,

m(xn(i)–,xm(i)–) =min

d(xn(i)–,xn(i)),d(xm(i)–,xm(i)),d(xn(i)–,xm(i)),d(xm(i)–,xn(i)

.

Clearly, asi→ ∞we haveM(xn(i)–,xm(i)–)→max{ε, , }=εandm(xn(i)–,xm(i)–)→

due to (.) and (.). Then lettingi→ ∞in (.), we get

≤ψ(ε)≤ψ(ε) –φ(ε) + , (.)

which impliesφ(ε) = , and henceε= . This contradicts the assumption that{xn}is not RMS Cauchy, thus,{xn}must be RMS Cauchy. Since (X,d) is complete, then{xn}converges to a limit, sayu∈X. Letx=xnandy=uin (.). This gives

ψd(Txn,Tu)

≤ψM(xn,u)

–φM(xn,u)

+Lm(xn,u), (.)

with

M(xn,u) =max

d(xn,u),d(xn,Txn),d(u,Tu)

,

m(xn,u) =min

d(xn,Txn),d(u,Tu),d(xn,Tu),d(u,Txn)

.

Note thatm(xn,u)→ asn→ ∞. IfM(xn,u) =d(xn,u) orM(xn,u) =d(xn,xn+), then we

haveM(xn,u)→ asn→ ∞, due to (.) and the fact that the sequence{xn}converges tou. Regarding the continuity ofψandφ, we have

≤ψ

lim

n→∞d(Txn,Tu)

≤ lim n→∞

ψM(xn,u)

–φM(xn,u)

+Lm(xn,u) = .

Hence,limn→∞d(Txn,Tu) = , that is, in either case, we end up withxn+=Txn→Tu. SinceXis Hausdorﬀ, we deduce thatu=Tu. If, on the other hand,M(xn,u) =d(u,Tu) passing to limit asn→ ∞in (.), we getφ(M(u,Tu)) = , henced(u,Tu) = , that is

u=Tu. This result contradicts the assumption thatT has no periodic points. Therefore,

T has a periodic point, that is,z=Tpzfor somez∈Xand a positive integerp.

Ifp= , thenz=Tz, sozis a ﬁxed point ofT. Letp> . We claim that the ﬁxed point of

T isTp–_z_{. Suppose the contrary, that is,}_Tp–_z₌_T₍_Tp–_z_{). Then}_d₍_Tp–_z_,_Tp_x_{) >  and so} isφ(d(Tp–_z_,_Tp_x_{)). Letting}_x₌_Tp–_z_and_y₌_Tp_z_{in (.), we have}

ψd(z,Tz)=ψdTpz,Tp+z

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where

MTp–z,Tpz=maxdTp–z,Tpz,dTp–z,Tpz,dTpz,Tp+z,

mTp–z,Tpz=mindTp–z,Tpz,dTpz,Tp+z,dTp–z,Tp+z,dTpz,Tpz= .

ForM(Tp–z,Tpz) =d(Tpz,Tp+z), (.) becomes

ψd(z,Tz)=ψdTpz,Tp+z≤ψdTpz,Tp+z–φdTpz,Tp+z.

Thus, we getφ(d(Tp_z_,_Tp+_z_{)) =  and hence}_d₍_Tp_z_,_Tp+_z_{) =}_d₍_z_,_Tz_{) = , which is not}

pos-sible sincep> . If, on the other hand,M(Tp–_z_,_Tp_z_{) =}_d₍_Tp–_z_,_Tp_z_{), then (.) turns into}

ψd(z,Tz)=ψdTpz,Tp+z≤ψdTp–z,Tpz–φdTp–z,Tpz

<ψdTp–z,Tpz, (.)

and taking into account the fact thatψis nondecreasing, we deduce

d(z,Tz) <dTp–z,Tpz.

Now we writex=Tp–_z_and_y₌_Tp–_z_{in (.) and get}

ψdTp–z,Tpz≤ψMTp–z,Tp–z–φMTp–z,Tp–z

+LmTp–z,Tp–z, (.)

where

MTp–z,Tp–z=maxdTp–z,Tp–z,dTp–z,Tp–z,dTp–z,Tpz,

mTp–z,Tp–z=mindTp–z,Tp–z,dTp–z,Tpz,

dTp–z,Tpz,dTp–z,Tp–z= .

ForM(Tp–_z_,_Tp–_z_{) =}_d₍_Tp–_z_,_Tp_z_{), we obtain}

ψdTp–z,Tpz≤ψdTp–z,Tpz–φdTp–z,Tpz,

which is possible only ifφ(d(Tp–_z_,_Tp_z_{)) =  and hence,}_d₍_Tp–_z_,_Tp_z_{) = . However, we} assumed thatd(Tp–_z_,_Tp_z_{) > . Thus, we must have}_M₍_Tp–_z_,_Tp–_z_{) =}_d₍_Tp–_z_,_Tp–_z_{), so}

that

ψdTp–z,Tpz≤ψdTp–z,Tp–z–φdTp–z,Tp–z

≤ψdTp–z,Tp–z, (.)

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We continue in this way and end up with the inequality

 <d(z,Tz) =dTp_z_,_Tp+_z_<_d_Tp–_z_,_Tp_z_≤_d_Tp–_z_,_Tp–_z_{≤ · · · ≤}_d₍_z_,_Tz_),

which yieldsd(z,Tz) <d(z,Tz). Therefore, the assumptiond(Tp–_z_,_Tp_x_{) >  is wrong, that} is,d(Tp–z,Tpx) =  andTp–zis the ﬁxed point ofT.

Finally, to prove the uniqueness, we assume thatT has two distinct ﬁxed points, sayz

andw. Then lettingx=zandy=win (.), we have

ψd(z,w)=ψd(Tz,Tw)≤ψM(z,w)–φM(z,w)+Lm(z,w), (.)

where

M(z,w) =maxd(z,w),d(z,Tz),d(w,Tw)=d(z,w),

m(z,w) =mind(z,Tz),d(w,Tw),d(w,Tz),d(z,Tw)= .

Thus, we have

ψd(z,w)≤ψd(z,w)–φd(z,w), (.)

implyingφ(d(z,w)) = , and henced(z,w) = , which completes the proof of the

unique-ness.

It is worth mentioning that the Theorem . given in [] is a particular case of Theo-rem . We next give some consequences of TheoTheo-rem .

Corollary  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

ψd(Tx,Ty)≤ψM(x,y)–φM(x,y) (.)

for all x,y∈X andψ,φ∈, where the functionψis nondecreasing and

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty). (.)

Proof Observe that

ψd(Tx,Ty)≤ψM(x,y)–φM(x,y)

≤ψM(x,y)–φM(x,y)

+Lmind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)

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Corollary  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)≤kmaxd(x,y),d(x,Tx),d(y,Ty) (.)

for all x,y∈X and some≤k< . Then T has a unique ﬁxed point in X.

Proof Letψ(t) =tandφ(t) = ( –k)t. Then by Corollary ,T has a unique ﬁxed point.

Corollary  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)≤kd(x,y) +d(x,Tx) +d(y,Ty)

+Lmind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx) (.)

for all x,y∈X and some≤k<_and L> . Then T has a unique ﬁxed point in X.

Proof Obviously,

kd(x,y) +d(x,Tx) +d(y,Ty) +Lmind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)

≤kmaxd(x,y),d(x,Tx),d(y,Ty)+Lmind(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx).

Letψ(t) =tandφ(t) = ( – k)t. Then by Theorem ,Thas a unique ﬁxed point. Our next corollary is concerned with weakφcontractions.

Corollary  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)≤M(x,y) –φM(x,y)+Lm(x,y) (.)

for all x,y∈X, where

M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty),

Proof Letψ(t) =t. Then by Theorem ,Thas a unique ﬁxed point. As the second result, we state the following existence and uniqueness theorem under conditions involving rational expressions.

Theorem  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

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for all x,y∈X andψ,φ∈, whereψis nondecreasing and

M(x,y) =max

d(x,y),d(y,Ty) +d(x,Tx)  +d(x,y)

. (.)

Proof Letx∈Xbe an arbitrary point. Deﬁne the sequence{xn} ⊂Xas

xn=Txn–, n= , , , . . . .

Assume thatxn=xn+=Txnfor alln≥. Substitutingx=xn–andy=xn in (.), we obtain

ψd(Txn–,Txn)

=ψd(xn,xn+)

≤ψM(xn–,xn)

–φM(xn–,xn)

, (.)

where

M(xn–,xn) =max

d(xn–,xn),d(xn,Txn)

 +d(xn–,Txn–)

 +d(xn–,xn)

=maxd(xn–,xn),d(xn,xn+)

.

For the rest of the proof, one can follow the same steps as in the proof of Theorem .

Settingψ(t) =tandφ(t) = ( –k)t, we obtain the following particular result.

Corollary  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)≤kmax

(.)

for all x,y∈X and some k∈[, ). Then T has a unique ﬁxed point in X.

We also generalize the applications of Theorem . in [] given by Lakzian and Samet. Letbe the set of functionsf: [, +∞)→[, +∞) such that

() f is Lebesgue integrable on each compact subset of[, +∞); () _εf(t)dt> for everyε> .

For this class of functions, we can state the following results.

Theorem  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)



f(t)dt≤

M(x,y)



f(t)dt– M(x,y)



g(t)dt+Lm(x,y) (.)

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Proof Letψ(t) =_tf(u)duandφ(t) =_tg(u)du. Thenψ andφ are functions in, and moreover, the functionψ is nondecreasing. By Theorem ,T has a unique ﬁxed point.

Corollary  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)



f(u)du≤k

M(x,y)



f(u)du+Lm(x,y) (.)

for all x,y∈X and f ∈and some≤k< , L> , where M(x,y) =maxd(x,y),d(x,Tx),d(y,Ty),

Proof Letg(t) = ( –k)f(t). Then by Theorem ,T has a unique ﬁxed point.

Theorem  Let(X,d)be a Hausdorﬀ and complete RMS and let T:X→X be a self-map satisfying

d(Tx,Ty)



f(t)dt≤

M(x,y)



f(t)dt– M(x,y)



g(t)dt (.)

for all x,y∈X and f,g∈, where

M(x,y) =max

.

Proof Letψ(t) =_tf(u)duandφ(t) =_tg(u)du. Thenψ andφ are functions in, and moreover, the functionψ is nondecreasing. By Theorem ,Thas a unique ﬁxed point. Finally, we give an illustrative example of (ψ,φ) contraction deﬁned on a generalized metric space.

Example  LetX=A∪B, whereA={_,_,_,_}andB= [, ]. Deﬁne the generalized metricdonXas follows:

d  ,   =d  ,  

= ., d

 ,   =d  ,   = ., d  ,   =d  ,  

= ., d

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and

d(x,y) =|x–y| ifx,y∈Borx∈A,y∈Borx∈B,y∈A.

It is clear thatddoes not satisfy the triangle inequality onA. Indeed,

. =d

 ,   ≥d  ,   +d  ,   = ..

Notice that (RM) holds, sodis a rectangular metric. LetT:X→Xbe deﬁned as

Tx= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 

 ifx∈[, ], 

 ifx∈ {  ,  ,  }, 

 ifx=  .

Define ψ(t) =tandφ(t) =_t. Then T satisfies the conditions of Theorem  and has a unique fixed point onX,i.e.,x=_.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Atilim University, Incek, Ankara 06836, Turkey.}2_{Higher Technical School of Professional} Studies in Zrenjanin, University of Novi Sad, Djordja Stratimirovica 23, Zrenjanin, Serbia.

Acknowledgements

Tanja Sekuli´c is thankful to the Ministry of Science and Technological Development of Republic Serbia.

Received: 14 March 2012 Accepted: 21 August 2012 Published: 4 September 2012 References

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