Edelstein type fixed point theorems

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R E S E A R C H

Open Access

Edelstein type ﬁxed point theorems

Erdal Karapınar

*

*_{Correspondence:}

[email protected]; [email protected] Department of Mathematics, Atılım University, ˙Incek, Ankara 06836, Turkey

Abstract

In this manuscript, we investigate certain conditions that imply the existence of ﬁxed points for almost contraction mappings deﬁned on compact metric spaces.

Furthermore we introduce a criteria establishing the uniqueness of ﬁxed points for the mentioned operators. As a result we obtain generalized results by unifying some recent related ﬁxed point theorems on the topic.

1 Introduction and Preliminaries

In nonlinear functional analysis, ﬁxed point theory is being investigated increasingly by reason of the fact that it has a wide range of applications in ﬁelds such as economics (seee.g.

[, ]), computer science (seee.g.[–]), and many others. One of the pioneering theorems in this direction is the Banach contraction mapping principle [] which states that each contraction defined on a complete metric spaceXhas a unique fixed point. Banach’s result is the origin and antecedents results by the fact that he not only proved the existence and uniqueness of a fixed point of a contraction, but also showed how to evaluate this point. After this celebrated result[], a number of authors have observed various other types of contraction mappings and proved related fixed point theorems (seee.g.such as Kannan [], Reich [], Hardy and Rogers [], Ćirić [–], Zamfirescu [], Arshadet al.[]). By following this trend Suzuki recently proved the following fixed point theorems:

Theorem (Suzuki []) Let(X,d)be a compact metric space and let T be a mapping

on X.Assume that _d(x,Tx) <d(x,y)implies d(Tx,Ty) <d(x,y)for all x,y∈X.Then T has a unique ﬁxed point.

Theorem (Suzuki []) Deﬁne a non-increasing functionθfrom[,)onto(/,]by

θ(r) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 if≤r≤(√ – )/,

( –r)r– _if₍√_{ – )/}_≤_r_≤√_/,

( +r)– _if√_/_≤_r_{< .}

Then for a metric space(X,d),the followings are equivalent:

. Xis complete;

. Every mappingTonXsatisfying the following has a ﬁxed point:There existsr∈[, )

such thatθ(r)d(x,Tx)≤d(x,y)impliesd(Tx,Ty)≤rd(x,y)for allx,y∈X.

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In the literature Theorem  and Theorem  attracted considerable attention from many authors (seee.g.[–]). Notice that these theorems are inspired by Edelstein’s Theorem []:

Theorem  Let(X,d)be a compact metric space and let T be a mapping on X.Assume

d(Tx,Ty) <d(x,y)for all x,y∈X with x=y.Then T has a unique ﬁxed point.

Motivated by these developments in this area, in this manuscript, we combine well-known results of Suzuki [], Edelstein [] and Berinde [] to complement a multitude of related results from the literature. For the sake of completeness we include the results of Berinde as well:

Theorem (See []) Let(X,d)be a complete metric space and T:X→X be an almost

contraction,that is,a mapping for which there exist a constant k∈[, )and some L≥

such that

d(Tx,Ty)≤kd(x,y) +Ld(y,Tx), ()

for all x,y∈X.ThenFix(T) ={x∈X:Tx=x} =∅.

Theorem (See []) Let(X,d)be a complete metric space and T:X→X be an almost

contraction,that is,a mapping for which there exist a constant k∈(, )and some L≥

such that

d(Tx,Ty)≤kd(x,y) +Ld(x,Tx), ()

for all x,y∈X.Then has a unique ﬁxed point.

Main Theorems

We start this section by proving the following theorem:

Theorem  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y)

⇒ d(Tx,Ty) < 

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

for all x,y∈X with x=y and L> .Then,T has a ﬁxed point z∈X,that is,Tz=z.

Proof Setθ=inf{d(x,Tx) :x∈X}and choose a sequence{xn}inXsuch thatlimn→∞d(xn, Txn) =θ. Regarding thatXis compact, without loss of generality, assume that{xn}and

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We claim thatθis equal to zero. To show this, assume to the contrary thatθ> . Observe that we have

lim

n→∞d(xn,w) =d(z,w) =nlim→∞d(xn,Txn) =θ. ()

We can choosek∈Nin such a way that



θ<d(xn,w) and d(xn,Txn) < 

θ, ()

for eachn≥k. As a consequence, we have_d(xn,Txn) <d(xn,w) for eachn≥k. Due to (), we getd(Txn,Tw) <_[d(Txn,xn) +d(Tw,w)] for eachn≥k. Accordingly, we obtain

d(w,Tw)

= lim

n→∞d(Txn,Tw)

≤ lim

n→∞

 

d(Txn,xn) +d(Tw,w)+Lmind(w,Txn),d(xn,Tw),d(xn,w) ()

which implies that

d(w,Tw)≤d(w,z) =θ. ()

By taking the deﬁnition ofθ into account, we conclude thatd(w,Tw) =θ. Notice that the inequality_d(w,Tw) <d(w,Tw) always holds. By applying the condition () again, we ﬁnd

dTw,Tw<  

d(Tw,w) +dTw,Tw

+Lmind(Tw,Tw),dw,Tw,d(w,Tw) ()

which is equivalent to the inequalityd(Tw,T_w_{) <}_d₍_Tw_,_w_{) =}_θ_{. This contradicts with the}

deﬁnition ofθ. Hence, we conclude thatθ= .

We next assert that T has a ﬁxed point. We use the method of Reductio ad absur-dum to show this assertion. Suppose that T has no ﬁxed points. Since the inequality  <_d(Txn,xn) <d(Txn,xn) holds for eachn, we derive, for everyn∈N, that

dTxn,Txn<  

d(Txn,xn) +dTxn,Txn

+Lmind(Txn,Txn),dxn,Txn,d(xn,Txn)

<  

d(Txn,xn) +dTxn,Txn.

Hence, we ﬁnd that

dTxn,Txn

<d(Txn,xn) ()

for eachn∈Nand

lim

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Thus, we getz=w. In other words,{xn}and{Txn}converge to the same point. Due to the triangular inequality and (), we obtain that

lim

n→∞d

z,Txn≤ lim

n→∞

d(z,Txn) +dTxn,Txn

≤ lim

n→∞

d(z,Txn) +d(xn,Txn)= d(z,z) = . ()

Hence,{T_xn_}_{too converges to}_z_.

Assume that

d(xn,z)≤ 

d(xn,Txn), and d(Txn,z)≤  d

Txn,Txn. ()

We use (), () and the triangular inequality, we ﬁnd that

d(xn,Txn)≤d(xn,z) +d(Txn,z)≤



d(xn,Txn) +  d

Txn,Txn

< 

d(xn,Txn) + 

d(xn,Txn) =d(xn,Txn). ()

This is a contradiction. Thus, either

d(xn,z) >



d(xn,Txn), or d(Txn,z) >  d

Txn,Txn

holds for eachn∈N. Then regarding (), one of the below holds:

d(Txn,Tz) < 

d(Txn,xn) +d(Tz,z)+Lmind(z,Txn),d(xn,Tz),d(xn,z) , ()

dTxn,Tz

< 



dTxn,Txn

+d(Tz,z)

+Lmindz,Txn,d(Txn,Tz),d(Txn,z) . ()

This is equivalent to stating that either

(i) there is an inﬁnite subsetIofNso that the inequality () holds for alln∈I, or, (ii) there is an inﬁnite subsetJofNso that the inequality () holds for alln∈J.

We ﬁrst consider the case (). The inequality

d(z,Tz) = lim

n∈I,n→∞d(Txn,Tz)≤n∈I,nlim→∞  

d(Txn,xn) +d(Tz,z)=

d(z,Tz), ()

yields that

d(z,Tz) = .

Thus, we conclude thatTz=z. For the other case in () we get

d(z,Tz) = lim

n∈J,n→∞d

Txn,Tz≤ lim

n∈J,n→∞

 

dTxn,Txn+d(z,Tz)

= 

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which implies that



d(z,Tz) < 

d(z,z) = .

Thus, we reach the conclusionTz=zagain. This contradicts with the assumption thatT

has no ﬁxed point. Hence,Thas a ﬁxed point.

Corollary  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y)

⇒ d(Tx,Ty) < 

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

for all x,y∈X with x=y and L> .Then,T has a unique ﬁxed point z∈X,that is,Tz=z.

Proof The proof of Theorem  applies, mutatis mutandis, to show the existence of a ﬁxed point. Letz∈Xbe a ﬁxed point ofT.

We shall provezis the unique ﬁxed point ofT. Suppose, to the contrary that, there exists

y∈Xso thaty=zandTy=y. Then the inequalitiesd(y,z) >  and  =_d(z,Tz) <d(z,y) are satisﬁed. Due to (), we have

≤d(z,Ty) =d(Tz,Ty) <  

d(Tz,z) +d(Ty,y)+Lmind(z,Tz),d(y,Ty),d(z,y) = 

which is a contradiction. Hence,zis the unique ﬁxed point ofT.

Theorem  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y)

⇒ d(Tx,Ty) < 

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

Proof The proof of Theorem  applies, mutatis mutandis, to prove Theorem .

Corollary  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y)

⇒ d(Tx,Ty) < 

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

for all x,y∈X with x=y and L> .Then,T has a unique ﬁxed point z∈X,that is,Tz=z.

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Theorem  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y) ⇒ d(Tx,Ty) <  

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

+Lmind(y,Tx),d(x,Ty),d(x,y) , ()

Proof As in the proof of Theorem , we setθ=inf{d(x,Tx) :x∈X}and choose a sequence

{xn}inXsuch thatlimn→∞d(xn,Txn) =θ. SinceXis compact, without loss of generality, we assume that{xn}and{Txn}converge to the pointszandwinX, respectively.

We aim to show thatθis equal to zero. Let assume the contrary. Recall that

lim

n→∞d(xn,w) =d(z,w) =nlim→∞d(xn,Txn) =θ. ()

It is possible to choosek∈Nin such a way that



θ<d(xn,w) and d(xn,Txn) < 

θ, ()

for eachn≥k. Consequently, we see that 

d(xn,Txn) <d(xn,w) for eachn≥k. By (), we

derive that

d(Txn,Tw) <  

d(xn,w) +d(Txn,xn) +d(Tw,w)

+Lmind(w,Txn),d(xn,Tw),d(xn,w)

for eachn≥k. Then it follows that

d(w,Tw) = lim

n→∞d(Txn,Tw)≤nlim→∞ _

[d(xn,w) +d(Txn,xn) +d(Tw,w)]

+Lmin{d(w,Txn),d(xn,Tw),d(xn,w)}

()

which implies that

d(w,Tw) = lim

n→∞d(Txn,Tw)≤

 

d(z,w) +d(w,z) +d(Tw,w). ()

Hence, we ﬁnd thatd(w,Tw) <d(z,w) =θ. As a result, we conclude thatd(w,Tw) =θwhen we take the deﬁnition ofθ into account. Since we always have the inequality _d(w,Tw) <

d(w,Tw), we obtain

dTw,Tw<  

d(w,Tw) +d(Tw,w) +dTw,Tw

+Lmind(Tw,Tw),dw,Tw,d(w,Tw) ()

by applying (). But this is equivalent to stating thatd(Tw,T_w_{) <}_d₍_Tw_,_w_{) =}_θ_{. This}

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We are ready to show thatT has a ﬁxed point. We shall use the method of Reductio ad absurdum again. Suppose thatThas no ﬁxed point. Since the inequality  <_d(Txn,xn) <

d(Txn,xn) is true for eachn, the expression

dTxn,Txn<  

d(xn,Txn) +d(Txn,xn) +dTxn,Txn

+Lmind(Txn,Txn),dxn,Txn,d(xn,Txn)

holds for everyn∈N. In other words, we see that the inequality

dTxn,Txn

<d(Txn,xn) ()

is satisﬁed for eachn∈N. Therefore, we infer that

lim

n→∞d(z,Txn) =d(z,w) =nlim→∞d(Txn,xn) =θ= . ()

So, we also ﬁndz=w. Thus, the sequences{xn}and{Txn}converge to the same point. By the triangular inequality, together with the inequality (), we derive

lim

n→∞d

z,Txn≤ lim

n→∞

d(z,Txn) +dTxn,Txn

≤ lim

n→∞

d(z,Txn) +d(xn,Txn)= d(z,z) = . ()

Hence,{T_xn_}_{also converges to}_z_{. Assume that}

d(xn,z)≤ 

d(xn,Txn), and d(Txn,z)≤  d

Txn,Txn. ()

Regarding (), () and the triangular inequality, we ﬁnd

d(xn,Txn)≤d(xn,z) +d(Txn,z)≤



d(xn,Txn) +  d

Txn,Txn

< 

d(xn,Txn) + 

d(xn,Txn) =d(xn,Txn). ()

This is a contradiction. Thus, we have either

d(xn,z) > 

d(xn,Txn), or d(Txn,z) >  d

Txn,Txn

for eachn∈N. By (), we conclude that one of the inequalities below

d(Txn,Tz) < 

d(xn,z) +d(xn,Txn) +d(z,Tz)

+Lmind(z,Txn),d(xn,Tz),d(xn,z) , ()

dTxn,Tz<  

d(Txn,z) +dTxn,Txn+d(Tz,z)

+Lmindz,Txn

,d(Txn,Tz),d(Txn,z) ()

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(a) there is an inﬁnite subsetIofNso that

d(Txn,Tz) <  

+Lmind(z,Txn),d(xn,Tz),d(xn,z) , for alln∈Ior,

(b) there is an inﬁnite subsetJofNso that

dTxn,Tz<  

d(Txn,z) +dTxn,Txn+d(Tz,z)

+Lmindz,Txn,d(Txn,Tz),d(Txn,z) ,

for alln∈Jholds.

Considering the case (), we ﬁnd

d(z,Tz) = lim

n∈I,n→∞d(Txn,Tz) <n∈I,nlim→∞

 

+Lmind(z,Txn),d(xn,Tz),d(xn,z)

= 

d(z,z) +d(z,Tz), ()

which yields that



d(z,Tz)≤ 

d(z,z) = .

Thus, we conclude thatTz=z. For the other case (), we obtain

d(z,Tz) = lim

n∈J,n→∞d

Txn,Tz≤ lim

n∈J,n→∞  

d(Txn,z) +dTxn,Txn+d(z,Tz)

+Lmindz,Txn

,d(Txn,Tz),d(Txn,z)

= 

d(z,z) +d(z,Tz) ()

which implies



d(z,Tz) < 

d(z,z) = .

Thus, we reach the same conclusion, that is,Tz=z. This contradicts with assumption that

T has no ﬁxed point. Hence,Thas a ﬁxed point, sayz∈X.

Corollary  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y) ⇒ d(Tx,Ty) <  

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

+Lmind(y,Tx),d(x,Ty),d(x,y) , ()

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Proof The proof of Corollary  follows, mutatis mutandis, from the proofs of Corollary , Corollary  and Corollary . ThereforeThas a ﬁxed point, sayz∈X.

We need to provezis the unique ﬁxed point ofT. Suppose, to the contrary that, there existsy∈Xsuch thaty=zandTy=y. Then, we haved(y,z) >  and  =_d(z,Tz) <d(z,y). By (), we see that

d(z,Ty) =d(Tz,Ty) < 

d(z,y) +d(Tz,z) +d(Ty,y)+Lmind(y,Tz),d(z,Ty),d(z,y)

= 

d(z,y) +d(Ty,y)

≤



d(z,y) +d(Ty,z) +d(z,y)

which in turn implies thatd(z,Ty) <d(z,y). Soyis not a ﬁxed point ofT. Hence,zis the

unique.

Combining Theorem , Theorem  and Theorem  yields the following:

Theorem  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y) ⇒ d(Tx,Ty) <  

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

+Lmind(y,Tx),d(x,Ty),d(x,y) ()

for all x,y∈X.Then,T has a ﬁxed point z∈X,that is,Tz=z.

Combining Corollary , Corollary  and Corollary  yields the following:

Corollary  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y) ⇒ d(Tx,Ty) <  

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

+Lmind(y,Tx),d(x,Ty),d(x,y) ()

for all x,y∈X.Then,T has a unique ﬁxed point z∈X,that is,Tz=z.

The result below is a corollary of Theorem -Theorem :

Theorem  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y)

⇒ d(Tx,Ty) < 

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

+Lmind(y,Tx),d(x,Ty),d(x,y) ()

for all x,y∈X.Then,T has a ﬁxed point z∈X,that is,Tz=z.

Proof The proof of Theorem  follows from the proofs of the previous theorems

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Corollary  Let T be a self mapping on a compact metric space(X,d).Assume that



d(x,Tx) <d(x,y)

⇒ d(Tx,Ty) < 

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

+Lmind(x,Tx),d(y,Ty),d(x,y) ()

for all x,y∈X.Then,T has a unique ﬁxed point z∈X,that is,Tz=z.

Proof The proof of Corollary  follows from the proofs of the previous theorems

verba-tim.

Example (cf.[]) LetX=Zanddbe the discrete metric

d(x,y) = ⎧ ⎨ ⎩

 ifx=y,

 ifx=y.

Each self-mappingTonXsatisfying () has a unique fixed point. It is clear that (X,d) is complete, but it is not a compact metric space. LetT be a self-mapping onX. IfT has a fixed point, it is sufficient to prove that it is unique.

To show thatz∈Xis the unique ﬁxed point ofT, we takey∈Xwherey=z. Thus the inequalitiesd(y,z) =  >  and  =_d(z,Tz) <d(z,y) are satisﬁed. Due to (), we have

d(z,Ty) =d(Tz,Ty) < 

d(Tz,z) +d(Ty,y)+Lmind(z,Tz),d(y,Ty),d(z,y)

=

d(Ty,y)≤  

d(Ty,z) +d(z,y)

which implies thatd(z,Ty) <d(z,y). Soyis not a fixed point ofT. Hence,zis the unique fixed point ofT. SupposeThas no fixed point. Then, we have



d(x,Tx)≤ 

 <d(x,y) = , for allx,y∈X, withx=y.

Due to (), the inequality d(Tx,Ty) < 

[d(Tx,x) +d(Ty,y)] +Lmin{d(x,Tx),d(y,Ty), d(x,y)}=  holds. In other words, we getd(Tx,Ty) = . Thus the image ofTon the domain

Xconsists of only one point which is clearly a unique ﬁxed point. This is a contradiction.

Remark  Example  can be modiﬁed for Theorem -Theorem  just by replacing

the condition () with the relevant one. It is clear that proofs are obtained by apply the necessary manipulations in Example .

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Theorem  Let T be a self mapping on a metric space(X,d).Suppose that there exist k∈[, )and a T -invariant complete subset K of X such that

d(Tx,Ty)≤kd(x,y) and

d(Tz,Tw) <  

d(Tz,z) +d(Tw,w)+Lmind(w,Tw),d(z,Tz),d(z,w)

()

for all x,y∈K with x=y,and z,w∈X with z=w and L> .Then,T has a unique ﬁxed point u∈X,that is,Tu=u.

Proof Due to Banach [], there exists a unique ﬁxed pointu∈K. Consider

d(Tz,u) =d(Tz,Tu) <  

d(Tz,z) +d(Tu,u)

≤ 



d(Tz,u) +d(u,z)+Lmind(u,Tu),d(z,Tz),d(z,u) ,

for allz∈(X\K) which implies that

d(Tz,Tu) <d(u,z).

In other words, for allz∈(X\K) is not a ﬁxed point ofT. Hence,uis the unique ﬁxed

point ofTonX.

Remark  Theorem  holds also if we replace one of the conditions below instead of the condition ():

d(Tx,Ty)≤kd(x,y) and

d(Tz,Tw) <  

d(Tz,w) +d(Tw,z)+Lmind(w,Tw),d(z,Tz),d(z,w) , ()

d(Tz,Tw) <  

d(z,w) +d(Tz,z) +d(Tw,w)+Lmind(w,Tw),d(z,Tz),d(z,w) , ()

d(Tz,Tw) <  

d(z,w) +d(Tz,w) +d(Tw,z)+Lmind(w,Tw),d(z,Tz),d(z,w) , ()

d(Tz,Tw) < 

d(z,w) +d(Tz,z) +d(Tw,w) +d(Tz,w) +d(Tw,z)

+Lmind(w,Tw),d(z,Tz),d(z,w) . ()

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author express his gratitude to the referees for constructive and useful remarks and suggestions.

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