Comments on some recent generalization of the Banach contraction principle

R E S E A R C H

Open Access

Comments on some recent generalization

of the Banach contraction principle

Tomonari Suzuki

*_{Correspondence:}

[email protected] Department of Basic Sciences, Faculty of Engineering, Kyushu Institute of Technology, Tobata, Kitakyushu 804-8550, Japan

Abstract

We study Browder and CJM contractions of integral type. As a result, we give an alternative proof of some recent generalization of the Banach contraction principle by Jleli and Samet.

MSC: Primary 47H09; secondary 54H25

Keywords: the Banach contraction principle; Browder contraction; CJM contraction; contraction of integral type; fixed point

1 Introduction

We have introduced many types of contractions. We give the definitions of three of them. DefineΦas follows:ϕ∈Φiffϕis a nondecreasing, right continuous function from [,∞) into itself satisfyingϕ(t) <tfor anyt> . It is obvious thatϕ() =  holds.

Definition  LetT be a mapping on a metric space (X,d).

• Tis said to be a (usual)contraction(C, for short) [, ] if there existsr∈[, )such thatd(Tx,Ty)≤rd(x,y)for anyx,y∈X.

• Tis said to be aBrowder contraction(BroC, for short) [] if there existsϕ∈Φsuch thatd(Tx,Ty)≤ϕ◦d(x,y)for anyx,y∈X.

• Tis said to be aCJM contraction(CJMC, for short) [–] if the following hold: (j) For everyε> , there existsδ> such thatd(x,y) <ε+δimpliesd(Tx,Ty)≤ε. (jj) x=yimpliesd(Tx,Ty) <d(x,y).

We know the following implications:

C⇒BroC⇒CJMC.

There are some conditions equivalent to BroC; see []. Lemma  in [] gives seven equiv-alent conditions connected with BroC. See [] and the references therein for further in-formation as regards contractions.

Branciari [] introduced contractions of integral type as follows: A mappingT on a metric space (X,d) is aBranciari contractionif there existr∈[, ) and a locally integrable

functionf from [,∞) into itself such that

s



f(t)dt>  and

d(Tx,Ty)



f(t)dt≤r d(x,y)



f(t)dt

for alls>  andx,y∈X.

We have studied contractions of integral type in [, ]. Very recently, Jleli and Samet [] introduced the following new type of contractions.

Theorem (Jleli and Samet []) Let(X,d)be a complete generalized metric space and letθ be a function from(,∞)into(,∞)satisfying the following:

(θ) θ is nondecreasing.

(θ) For any sequence{tn}in(,∞),limnθ(tn) = ifflimntn= .

(θ) There existr∈(, )and∈(,∞]such thatlim_t→+(θ(t) – )/tr_=.

Let T be a mapping on X.Assume that there exists k∈(, )such that

Tx=Ty implies θ◦d(Tx,Ty)≤θ◦d(x,y)k

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

Remark

• Considering the domain and range ofθand (θ), it is obvious that (θ) is equivalent to inf{θ(t) :t∈(,∞)}= .

• The underlying space of Theorem  is a generalized metric space. This interesting concept was introduced by Branciari []. See also [–] and others. However, we omit the statement of the definition of a generalized metric space because it is not essential in this paper.

In this paper, motivated by Theorem , we deepen the study of contractions of integral type. We also give an alternative proof of Theorem .

2 Preliminaries

Throughout this paper we denote byNthe set of all positive integers and byRthe set of all real numbers. For a functionf, we denote byDom(f) the domain off.

Letfbe a function from a subset ofRintoR. Thenf is said to satisfy (U)fif the following

holds:

(U)f For anyt∈Dom(f), there existδ> andε> such thatf(s)≤t–εholds for any

s∈(t–δ,t+δ)∩Dom(f).

We list some further notation in order to simplify the statement of the results of this paper:

(A) LetYbe an arbitrary set and lethbe a function fromYinto[,∞). LetSbe a mapping onYsatisfying thath(x) = impliesh(Sx) = for anyx∈Y.

(A) Letθ be a function from(,∞)intoR. Put=θ((,∞))and

The purpose of this study is to obtain the mathematical structure of contractions men-tioned in Section . So, we simplify our setting as follows:

Definition  Assume (A).

• Sis said to be aBrowder contractionif there existsϕ∈Φsuch that

h(Sx)≤ϕ◦h(x)

for anyx∈Y.

• Sis said to be aCJM contractionif the following hold:

(j) For anyε> , there existsδ> such thath(x) <ε+δimpliesh(Sx)≤ε. (jj) h(x) > impliesh(Sx) <h(x).

Remark Let (X,d) andT be as in Definition . Put Y=X×X and definehandSby h((x,y)) =d(x,y) andS(x,y) = (Tx,Ty). Then the concept of Browder contraction in Defi-nition  becomes that in DefiDefi-nition  and the concept of CJM contraction in DefiDefi-nition  becomes that in Definition .

We give some lemmas concerning (U)f.

Lemma  Let f be a function from a subset ofRintoR.Then f satisfies(U)fiff

lim supf(u) :u→t,u∈Dom(f) <t

holds for any t∈Dom(f).

Remark We definelim sup[f(u) :u→t,u∈Dom(f)] =γ iff the following hold:

• There exists a sequence{un}inDom(f)such that{un}converges totand{f(un)} converges toγ.

• lim sup_nf(un)≤γ holds for any sequence{un}inDom(f)converging tot.

Note that we do not exclude the case ofun=t.

Proof of Lemma Obvious.

Lemma  Let f be an upper semicontinuous function from a subset ofRintoRsuch that f(t) <t for any t∈Dom(f).Then f satisfies(U)f.

Proof Obvious.

Lemma  Letψbe a function from a subset D ofRintoRsatisfying(U)ψ.Let M be a real

number with M> .Define a functionϕfrom D intoRby

ϕ(t) = t +

 max

supψ(u) :u∈D,u≤t,

supψ(u) +M(t–u) :u∈D,t≤u

(i) ϕsatisfies(U)ϕ.

(ii) ϕis strictly increasing and Lipschitz continuous. (iii) ψ(t) <ϕ(t)holds for anyt∈D.

Proof In this proof, we always assumes,t,u∈D. Define a functionωfromDintoRby

ω(t) =maxsupψ(u) :u≤t,supψ(u) +M(t–u) :t≤u.

For eachu, we define a functionψufromDintoRby

ψu(t) =

⎧ ⎨ ⎩

ψ(u) +M(t–u) ift≤u,

ψ(u) ift≥u.

Then from the definition ofω, we note

ω(t) =supψu(t) :u

.

So it is obvious that ψ(t)≤ω(t). We can easily check that ψu is nondecreasing and |ψu(s) –ψu(t)| ≤M|s–t|holds. Henceωis nondecreasing and

ω(s) –ω(t)≤M|s–t|

holds. Henceωis continuous. Fixt; and chooseδ andεas in the definition of (U)f. We

have

ω(t) = max

sup u≤t–δ

ψu(t), sup t–δ<u≤t

ψu(t), sup t≤u<t+δ

ψu(t), sup t+δ≤u

ψu(t)

=maxsup u≤t–δ

ψ(u), sup t–δ<u≤t

ψ(u),

sup t≤u<t+δ

ψ(u) +M(t–u), sup t+δ≤u

ψ(u) +M(t–u)

≤max

sup u≤t–δ

u, sup t–δ<u<t+δ

ψ(u), sup t+δ≤u

u+M(t–u)

≤max{t–δ,t–ε,t+δ–Mδ}

<t.

Hence by Lemma ,ωsatisfies (U)ω. It is obvious that

ϕ(s) –ϕ(t)≤M+ 

 |s–t| and ψ(t)≤ω(t) <ϕ(t) <t

hold. We can easily prove the remainders.

3 Browder contraction

In this section, we discuss Browder contractions.

Lemma  Assume(A).Letψ be a function from(,∞)into[,∞)such that(U)ψ holds

and

h(Sx)≤ψ◦h(x)

for any x∈Y with h(x) > and h(Sx) > .Then S is a Browder contraction.

Proof By Lemma , there exists a nondecreasing continuous functionηfrom (,∞) into [,∞) such that (U)ηis satisfied andψ(t)≤η(t) holds for anyt∈(,∞). Define a function

ϕfrom [,∞) into itself by

ϕ(t) = ⎧ ⎨ ⎩

η(t) ift> ,

 ift= .

It is obvious thatϕ is nondecreasing continuous function such thatϕ(t) <tfor anyt∈ (,∞). Thus,ϕ∈Φ. Fixx∈Y. We consider the following two cases:

• h(Sx) = , • h(Sx) > .

In the first case,h(Sx)≤ϕ◦h(x) obviously holds. In the second case, we note thath(Sx) >  impliesh(x) > . Soh(x) >  holds. We have

h(Sx)≤ψ◦h(x)≤η◦h(x) =ϕ◦h(x).

ThereforeSis a Browder contraction.

Proposition  Assume(A), (A)and the following:

(i) θ is nondecreasing and continuous.

(ii) There exists a functionψ fromintoRsatisfying(U)ψand

θ◦h(Sx)≤ψ◦θ◦h(x) ()

for anyx∈Ywithh(x) > andh(Sx) > .

Then S is a Browder contraction.

Proof Define a functionθ–

+ fromRinto [,∞] by

θ₊–(τ) = ⎧ ⎨ ⎩

sup{s∈(,∞) :θ(s)≤τ} if{s∈(,∞) :θ(s)≤τ} =∅,

 otherwise.

Putϕ=θ–

+ ◦ψ◦θ. We note

Sinceψ◦θ(t) <θ(t)≤θ(s) for anyt,s∈(,∞) witht≤s, we haveϕ(t)≤tfor anyt∈ (,∞). Thus,ϕis a function from (,∞) into [,∞). Arguing by contradiction, we assume that (U)ϕdoes not hold. Then there existt∈(,∞) and a sequence{tn}in (,∞) such that

{tn}converges totand

ϕ(tn) > ( – /n)t

holds forn∈N. Sinceϕ(tn) > ,

sups∈(,∞) :θ(s)≤ψ◦θ(tn)

> ( – /n)t

holds. Hence there exists a sequence{un}in (,∞) satisfying

θ(un)≤ψ◦θ(tn) <θ(tn) and un> ( – /n)t

forn∈N. Sinceθ is nondecreasing,un<tnholds for anyn∈N. Thus{un}also converges

tot. Hence

θ(t)≤lim sup n→∞

ψ◦θ(tn)≤lim sup

ψ(τ) :τ→θ(t),τ∈,

becauseθ is continuous. This contradicts (U)ψ. Therefore (U)ϕholds. For anyx∈Y with

h(x) >  andh(Sx) > , we have by ()

h(Sx)≤sups∈(,∞) :θ(s)≤ψ◦θ◦h(x)=ϕ◦h(x).

Therefore by Lemma ,Sis a Browder contraction.

By Lemma , we obtain the following.

Corollary  Assume(A), (A), (i)in Propositionand the following:

(ii) There exists an upper semicontinuous functionψ fromintoRsatisfyingψ(t) <t

for anyt∈and()for anyx∈Y withh(x) > andh(Sx) > .

Then S is a Browder contraction.

The following examples tell that the continuity ofθ in Proposition  is needed. In Ex-ample ,θis right continuous, however, it is not left continuous. On the other hand, in Example ,θis left continuous, however, it is not right continuous.

Example (Example . in []) Define a complete metric space (X,d) by

X= [, ]∪[,∞) and d(x,y) = ⎧ ⎨ ⎩

min{x+y, } ifx=y,

 ifx=y.

Define a mappingTonXand functionsθandψfrom (,∞) into itself by

Tx= ⎧ ⎨ ⎩

 ifx≤,

 – /x ifx≥,

θ(t) = ⎧ ⎨ ⎩

 ift< ,

andψ(t) =t/. PutY=X×Xand definehandSbyh((x,y)) =d(x,y) andS(x,y) = (Tx,Ty) for (x,y)∈Y. Then all the assumptions of Proposition  except the left continuity ofθare satisfied. However,Tis not a Browder contraction.

Example (Example . in []) Define a complete metric space (X,d) byX= [,∞) and d(x,y) =x+yforx,y∈Xwithx=y. Define a mappingTonXand functionsθandψfrom (,∞) into itself by

Tx= ⎧ ⎨ ⎩

 ifx≤,

 ifx> ,

θ(t) = ⎧ ⎨ ⎩

t ift≤,

 +t ift> ,

andψ(t) =t/. LetY,h, andSbe as in Example . Then all the assumptions of Proposi-tion  except the right continuity ofθare satisfied. However,Tis not a Browder contrac-tion.

4 CJM contraction

In this section, we discuss CJM contractions.

The following is a modification of Proposition . in [].

Proposition  Assume(A), (A),and the following:

(i) θis nondecreasing.

(ii) For anyε∈_≤,there existsδ> such thath(x) > ,h(Sx) > ,andθ◦h(x) <ε+δ

implyθ◦h(Sx)≤εfor allx∈Y.

(iii) h(x) > andh(Sx) > implyθ◦h(Sx) <θ◦h(x).

Then S is a CJM contraction.

Proof In order to show (jj), we fixx∈Ywithh(x) > . Arguing by contradiction, we assume h(Sx)≥h(x). Thenh(Sx) >  obviously holds. We have from (iii),

θ◦h(Sx) <θ◦h(x),

which contradicts (i). Therefore we obtain h(Sx) <h(x). We have shown (jj). We shall prove (j). Fixε>  and putβ=lim[θ(t) :t→ε+ ]. We consider the following two cases:

• β<θ(ε+γ)holds for anyγ> .

• There existsδ> such thatβ=θ(ε+δ).

In the first case, sinceθ(ε)≤β,β∈≤holds. From (ii), there existsα>  such that

h(x) > , h(Sx) >  and θ◦h(x) <β+α imply θ◦h(Sx)≤β.

We can chooseδ>  satisfyingθ(ε+δ) <β+α. Fixx∈Ywithh(x) <ε+δ. Arguing by

contradiction, we assumeh(Sx) >ε. Then we haveh(Sx) >  and henceh(x) > . We have

θ◦h(x)≤θ(ε+δ) <β+α

and hence

which is a contradiction. Therefore we obtainh(Sx)≤ε. In the second case, we also fix x∈Y withh(x) <ε+δ. Arguing by contradiction, we assumeh(Sx) >ε. Then we have

h(Sx) >  and henceh(x) > . By (iii), we have

β≤θ◦h(Sx) <θ◦h(x)≤θ(ε+δ) =β,

which is a contradiction. Therefore we obtainh(Sx)≤ε. We have proven (j).

The proof of the following is obvious, however, we give a proof in order to show how differentlyand_≤work.

Corollary  Assume(A), (A), (i)in Propositionand the following:

(ii) There exists a functionψ from≤intoRsatisfying(U)ψ and()for anyx∈Ywith

h(x) > andh(Sx) > .

Then S is a CJM contraction.

Proof We first note that from (U)ψ,ψ(t) <t for anyt∈Dom(ψ) =≤. Forx∈Y with

h(x) >  andh(Sx) > , we have

θ◦h(Sx)≤ψ◦θ◦h(x) <θ◦h(x).

We have shown (iii) in Proposition . Let us prove (ii) in Proposition . Fixε∈_≤. Then from (U)ψ, there existsδ>  such thatψ(s) <εholds for anys∈[ε,ε+δ). Fixx∈Y with

h(x) > ,h(Sx) > , andθ◦h(x) <ε+δ. Then ifθ◦h(x) <ε, then we have

θ◦h(Sx) <θ◦h(x) <ε.

Ifθ◦h(x)≥ε, then we have

θ◦h(Sx)≤ψ◦θ◦h(x) <ε.

Therefore by Proposition ,Sis a CJM contraction.

By Lemma , we obtain the following.

Corollary  Assume(A), (A), (i)in Propositionand the following:

(ii) There exists an upper semicontinuous functionψ from≤intoRsatisfyingψ(t) <t

for anyt∈≤and()for anyx∈Ywithh(x) > andh(Sx) > .

Then S is a CJM contraction.

There is a counterexample if we replace≤within Corollary .

Example  Define a complete metric space (X,d) by

X= [,∞) and d(x,y) = ⎧ ⎨ ⎩

 + /x+ /y ifx=y,

Define a mappingTonXbyTx= xforx∈X. Define functionsθ from (,∞) intoRand ψ from (, ]∪(,∞) intoRby

θ(t) = ⎧ ⎨ ⎩

t ift≤,

 +t ift> , and ψ(t) = ⎧ ⎨ ⎩

t/ ift≤,

 +t/ ift> .

Let Y, h, and Sbe as in Example . Then all the assumptions of Corollary  except

Dom(ψ) =_≤are satisfied. However,Tis not a CJM contraction.

Remark We note thatθ is left continuous, however,θ is not right continuous. Proof For anyx,y∈Xwithx=yandTx=Ty, we have

ψ◦θ◦d(x,y) =ψ◦θ  + x+  y =ψ  + x+  y =  + 

x+  y

and

θ◦d(Tx,Ty) =θ◦d(x, y) =θ

 +  x+

 y

=  + 

x+  y.

Thus, () holds. SinceThas no fixed point,T is not a CJM contraction.

We assume additionally thatθ is right continuous. Then we can prove the following.

Proposition  Assume(A), (A), (iii)in Propositionand the following:

(i) θ is nondecreasing and right continuous.

(ii) For anyε∈,there existsδ> such thath(x) > ,h(Sx) > ,andθ◦h(x) <ε+δ

implyθ◦h(Sx)≤εfor allx∈Y.

Then S is a CJM contraction.

Proof We will show (ii) in Proposition . Fixε∈≤\. Then there existsδ>  such that

[ε,ε+δ]⊂≤\.

If not so, then there exists a sequence{tn}in (,∞) such that{θ(tn)}is strictly

decreas-ing and converges toε. Sinceθ is nondecreasing,{tn}is strictly decreasing. Since{tn}is bounded from below,{tn}converges to somet∈[,∞). Sinceε∈≤,t>  holds. Since θ is right continuous, we obtainθ(t) =ε, which impliesε∈. This is a contradiction. Fix x∈Y withh(x) > ,h(Sx) > , andθ◦h(x) <ε+δ. Then we haveθ ◦h(x) <ε. Hence we obtain from (iii) (in Proposition )

θ◦h(Sx) <θ◦h(x) <ε.

So all the assumptions in Proposition  are satisfied. ThereforeSis a CJM contraction.

Proof We can prove (iii) in Proposition  (in Proposition ) as in the proof of Corol-lary . Also, we can prove (ii) in Proposition  as in the proof of CorolCorol-lary . Therefore

by Proposition ,Sis a CJM contraction.

By Lemma , we also obtain the following.

Corollary  Assume(A), (A), (i)in Propositionand(ii)in Corollary.Then S is a CJM contraction.

Finally, using Corollary , we give an alternative proof of Theorem .

Proof of Theorem LetY,h, andSbe as in Example . Define a functionψfrom (,∞) into (,∞) byψ(t) =tk_{. Sinceψ is continuous andψ(t) <tfor anyt∈(,∞), all the}

as-sumption of Corollary  are satisfied. So by Corollary ,Sis a CJM contraction. By The-orem  in [],Thas a unique fixed pointz. Moreover,limnd(Tnx,z) =  for anyx∈X.

Remark

(i) We do not need (θ) and (θ) in Theorem .

(ii) If we assume additionally thatθ is continuous, thenT is a Browder contraction.

5 Some comments

As stated in Section , Lemma  in Jachymski [] gives seven equivalent conditions con-nected with BroC. Finally, by Proposition  and Corollary  we can obtain the following, which is similar to Lemma  in [].

Lemma  Let D be a subset of(,∞).Then the following statements are equivalent:

(i) There existsϕ∈Φsuch that for any(t,u)∈D,u≤ϕ(t).

(ii) There exist a nondecreasing,continuous functionθfrom(,∞)intoRand a functionψfromθ((,∞))intoRsatisfying(U)ψ such that for any(t,u)∈D,

θ(u)≤ψ◦θ(t).

(iii) There exist a nondecreasing,continuous functionθfrom(,∞)intoRand an upper

semicontinuous functionψfromθ((,∞))intoRsatisfyingψ(t) <tsuch that for any(t,u)∈D,θ(u)≤ψ◦θ(t).

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author is grateful to the referees for many suggestions to improve the exposition of the paper. The author is supported in part by JSPS KAKENHI Grant Number 25400141 from Japan Society for the Promotion of Science.

Received: 2 December 2015 Accepted: 30 March 2016

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