Common fixed point results in function weighted metric spaces

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R E S E A R C H

Open Access

Common ﬁxed point results in function

weighted metric spaces

Obaid Alqahtani

1

_{, Erdal Karapınar}

2*

_{and Priya Shahi}

3

*_{Correspondence:}

[email protected] 2_{Department of Medical Research,}

China Medical University, Taichung, Taiwan

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

Abstract

In this paper, we consider a common ﬁxed point result in the context of a very recently deﬁned abstract space: “function weighted metric space”. We present also some examples to illustrate the validity of the given results.

MSC: 54E50; 54A20; 47H10

Keywords: Function weighted metric space; Topological properties; Banach contraction principle

1 Introduction

One of the very natural trends of mathematical research is to refine the framework of the known theorems and results. For instance, Banach observed the first metric fixed point results in the setting of complete normed spaces. Immediate extension of this theorem was given by Caccioppoli who observed the characterization of Banach fixed point theo-rem in the context of complete metric spaces. After then, for the various abstract spaces, several analogs of the Banach contraction principle have been reported. Among them we can underline some of the interesting abstract structures such as modular metric space, symmetric space, semi-metric space, quasi-metric space, partial metric space,b-metric space, dislocated (metric-like) space, fuzzy metric space, probabilistic metric space, 2-metric space,δ-metric space,G-metric space,S-metric space, function weighted metric space, and so on.

In this paper, we shall restrict ourselves to the recently introduced generalization of a metric space, namely, function weighted metric space [1]. Our aim is to obtain a ﬁxed point result for two mappings. More precisely, we shall consider coincidence points and common ﬁxed points for certain operators in the setting of the function weighted metric space.

For the sake of the self-contained text, we recall the definition of the newly introduced metric space. For this purpose, we first recall two basic notions for functions that we need: A functionf : (0, +∞)→Ris called logarithmic-like if each sequence{tn} ⊂(0, +∞) sat-isfies

lim

n→+∞tn= 0 ⇐⇒ n→lim+∞f(tn) = –∞.

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A functionf : (0, +∞)→Ris called non-decreasing if 0 <s<t ⇒ f(s)≤f(t).

The letterFdenotes the set of all functions that are non-decreasing (in symbols, (1))

and logarithmic-like (in symbols, (2)).

By using the auxiliary functions of F, Jleli–Samet [1] introduced a new metric space, more precisely, a function weighted metric space. Indeed, in this new metric space deﬁ-nition, Jleli–Samet [1] proposed a new condition instead of triangle inequality by using a function from the setF. Henceforth, we presume thatXis a nonempty set and avoid to repeat this in all statements. For the sake of the self-contained text, we put the deﬁnition here:

Deﬁnition 1.1 Letδ:X×X→[0, +∞) be a given mapping. Suppose that there exist an f ∈Fand a constantC∈[0, +∞) such that

(1) (Self-distance axiom)δ(x,y) = 0⇐⇒x=y, forx,y∈X; (2) (Symmetry axiom) For allx,y∈X, we haveδ(x,y) =δ(y,x);

(3) (Generalized functionf-weighted triangle inequality axiom) For any pair(x,y)∈ X×Xand for anyN∈NwithN≥2, we have

δ(x,y) > 0 ⇒ fδ(x,y)≤f

_N_–1

i=1

δ(ui,ui+1)

+C,

for every(ui)Ni=1⊂Xwith(u1,uN) = (x,y).

Then, the functionδis called a “function weighted metric” or “F-metric” onX, and the pair (X,δ) is named as a “function weighted metric space” or “F-metric space”.

Throughout the text, we prefer to use the name “function weighted metric space” instead of “F-metric space”.

As it seen clearly, the only diﬀerence between a “standard metric space” and a “function weighted metric space” is the last axiom: In a “function weighted metric space” instead of “the triangle inequality”, another axiom has been used, namely “generalizedf-weighted triangle inequality axiom.” Based on this observation, we also easily conclude that any met-ric onXis anF-metric onXby lettingf(t) =lntfor the axiom (3). Indeed, on account

of the triangle inequality, for all distinctx,y∈Xand for eachN∈NwithN≥2, and for any (ui)Ni=1⊂Xwith (u1,uN) = (x,y), we ﬁnd

d(x,y) > 0 ⇒ lnd(x,y)≤ln

_N_–1

i=1

d(ui,ui+1)

,

sinced(x,y)≤N_i₌₁–1d(ui,ui+1), andf(t) =lntis non-decreasing. Here, we takeC= 0.

The main goal of this paper is to obtain some common ﬁxed point result in the context of function weighted metric spaces.

2 Main results

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Theorem 2.1 Let T,g:X→X be self-mappings on a function weighted complete metric space(X,δ)such that T(X)⊆g(X).Suppose that there exists k∈(0, 1)such that

δT(x),T(y)≤kδ(gx,gy), (x,y)∈X×X.

Also suppose g(X)is closed.Then,T and g have a unique coincidence point.

Proof SinceT(X)⊆g(X), we can choose a pointx1 ∈X such thatTx0=gx1. We shall

construct a sequencexninXsuch that

yn=Txn=gxn+1, n= 0, 1, 2, . . . . (2.1)

First, observe thatT andgpossess a unique coincidence point. Indeed, suppose on the contrary that (u,v)∈X×Xare two distinct coincidence points ofTandg. Thus,δ(u,v) > 0, g(u) =T(u) andg(v) =T(v). Then from (ii), we have

δ(u,v) =δ(Tu,Tv)≤kδ(gu,gv) =kδ(u,v) <δ(u,v),

a contradiction.

Suppose (f,C)∈F×[0, +∞) so that (3) is fulﬁlled. For a given> 0 and on account of

(2), there existsγ > 0 such that

0 <t<γ ⇒ f(t) <f() –C. (2.2)

Consider the sequence{yn} ⊂Xdeﬁned in (2.1). Now, without loss of generality, we as-sume thatδ(Tx0,Tx1) > 0. Otherwise,x1 will be a coincidence point ofT andg. By the

contraction condition,

δ(Txn,Txn+1)≤kδ(gxn,gxn+1)

=kδ(Txn–1,Txn)

≤k2δ(gxn–1,gxn),

thereby implying that

δ(Txn,Txn+1)≤knδ(Tx0,Tx1).

So we have

m–1

i=n

δ(Txi,Txi+1)≤

kn

1 –kδ(Tx0,Tx1), m>n.

Since

lim

n→+∞

kn

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there exists someN∈Nsuch that 0 < k

n

1 –kδ(Tx0,Tx1) <γ, n≥N. (2.3) Hence, by (2.2) and (1), we have

f

_m_–1

i=n

δ(Txi,Txi+1)

≤f

kn

1 –kδ(Tx0,Tx1)

<f() –C, m>n≥N. (2.4)

Employing (3) together with (2.4), we ﬁnd

δ(yn,ym) > 0, m>n≥N ⇒ f

δ(yn,ym)

≤f

_m_–1

i=n

δ(yi,yi+1)

+C<f(),

which implies by (1) that

δ(yn,ym) <, m,n≥N.

This proves that{Txn}is Cauchy. Since{Txn}={gxn+1} ⊆g(X) andg(X) is closed, there

existsz∈Xsuch that

lim

n→∞δ(gxn,gz) = 0. (2.5)

As a next step, we shall indicate thatzis a coincidence point ofTandg. On the contrary, assume thatδ(Tz,gz) > 0. We have

fδ(Tz,gz)≤fδ(Tz,Txn) +δ(Txn,gz)

+C, n∈N

≤fkδ(gz,gxn) +δ(gxn+1,gz)

+C.

Asn→ ∞in the inequality above, and due to (2.5), we get

lim

n→+∞f

kδ(gz,gxn) +δ(gxn+1,gz)

+C= –∞,

which is a contradiction. Therefore, we conclude thatδ(Tz,gz) = 0, and hencezis a unique

coincidence point ofT andg.

Example2.1 ConsiderX=R+₀. Deﬁneδ:X×X→[0,∞) as δ(x,y) =|x–y|, x,y∈X.

Soδis a function weighted metric withf(t) =ln(t) anda= 0. Consider,

T(x) =

⎧ ⎨ ⎩

x–3

2 ifx> –1,

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and

g(x) =

⎧ ⎨ ⎩

5–x

2 ifx> 1,

0 otherwise.

Clearly,x= 4 is a coincidence point ofTandg. We shall show that there exists ak∈(0, 1) such that

δ(Tx,Ty) =x– 3 2 –

y– 3 2

=1

2|x–y| ≤ 1

2|y–x|=k|gx–gy|=kδ(gx,gy).

Therefore, all the hypothesis of Theorem2.1are satisﬁed. Furthermore,x= 4 is a unique coincidence point ofT andg.

Next, we present the notion of generalizedθ–ψ contractive pair of mappings in the setting of function weighted metric spaces as follows:

Deﬁnition 2.1 LetT,gbe self-mappings on a function weighted metric space (X,δ). We say that the pair (T,g) is a generalizedθ–ψ contractive pair of mappings if there exist θ:X×X→[0, +∞) andψ∈Ψ such that

θ(gx,gy)δ(Tx,Ty)≤ψM(gx,gy), (2.6)

for allx,y∈X, where

M(gx,gy) =max

δ(gx,gy),δ(gx,Tx) +δ(gy,Ty)

2 ,

δ(gx,Ty) +δ(gy,Tx) 2

.

Theorem 2.2 Let(X,δ)be a complete function weighted metric space and T,g:X→X be such that T(X)⊆g(X)where g(X)is closed.Assume that the pair(T,g)is a generalized θ–ψcontractive pair of mappings and the following conditions hold:

(i) Tisθ-admissible with respect tog;

(ii) There existsx0∈Xsuch thatθ(gx0,Tx0)≥1;

(iii) If{gxn}is a sequence inXsuch thatθ(gxn,gxn+1)≥1for allnandgxn→gz∈g(X)

asn→ ∞,then there exists a subsequence{gxn(k)}of{gxn}such thatθ(gxn(k),gz)≥1 for allk.

Then,T and g have a coincidence point.

Proof In view of condition (ii), letx0∈X be such thatθ(gx0,Tx0)≥1. On account of

T(X)⊆g(X), we choose a pointx1∈Xin a way that Tx0=gx1. Iteratively, we build a

sequence{xn}inXso that

yn:=Txn=gxn+1, n= 0, 1, 2, . . . . (2.7)

SinceTisθ-admissible with respect tog, we have

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Using mathematical induction, we get

θ(gxn,gxn+1)≥1, ∀n= 0, 1, 2, . . . . (2.8)

IfTxn+1=Txnfor somen, then by (2.7),

Txn+1=gxn+1, n= 0, 1, 2, . . . ,

and thenx=xn+1forms a coincidence point ofT andg, which completes the proof.

Ac-cordingly, we suppose thatδ(Txn,Txn+1) > 0 for alln. Applying inequality (2.6) and using

(2.8), we obtain

δ(Txn,Txn+1)≤θ(gxn,gxn+1)δ(Txn,Txn+1)

≤ψM(gxn,gxn+1)

. (2.9)

On the other hand, we have

M(gxn,gxn+1) =max

δ(gxn,gxn+1),

δ(gxn,Txn) +δ(gxn+1,Txn+1)

2 ,

δ(gxn,Txn+1) +δ(gxn+1,Txn) 2

≤maxδ(Txn–1,Txn),δ(Txn,Txn+1)

.

Due to the monotonicity of functionψ and using inequalities (2.9), we have for alln≥1 δ(Txn,Txn+1)≤ψ

maxδ(Txn–1,Txn),δ(Txn,Txn+1)

. (2.10)

If, the inequalityδ(Txn–1,Txn)≤δ(Txn,Txn+1) is satisﬁed for somen∈N, from (2.10), we

obtain that

δ(Txn,Txn+1)≤ψ

δ(Txn,Txn+1)

<δ(Txn,Txn+1),

a contradiction. Thus, for alln≥1, we have δ(Txn–1,Txn) =max

δ(Txn–1,Txn),δ(Txn,Txn+1)

. (2.11)

Notice that in view of (2.10) and (2.11), we get for alln≥1 that δ(Txn,Txn+1)≤ψ

δ(Txn–1,Txn)

. (2.12)

Continuing this process inductively, we obtain

δ(Txn,Txn+1)≤ψn

δ(Tx0,Tx1)

, ∀n≥1. (2.13)

So,

m–1

i=n

δ(Txi,Txi+1)≤

m–1

p=n

ψpδ(Tx0,Tx1)

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Since

lim

p→+∞ψ

p_δ(Tx

0,Tx1)

= 0,

there exists someN∈Nsuch that

0 <ψpδ(Tx0,Tx1)

<γ, p≥N. (2.14)

Next, let (f,C)∈F×[0, +∞) be such that (3) is satisﬁed. Let> 0 be ﬁxed. By (2),

∃γ > 0 such that

0 <t<γ ⇒ f(t) <f() –C. (2.15)

Hence, by (2.15) and (1), we have

f

_m_–1

i=n

δ(Txi,Txi+1)

≤f

_m_–1

p=i

ψpδ(Tx0,Tx1)

<f() –C, m>n≥N. (2.16)

Using (3) and (2.16), we have

δ(yn,ym) > 0, m>n≥N ⇒ f

δ(yn,ym)

≤f

_m_–1

i=n

δ(yi,yi+1)

+C<f(),

which implies by (1) that

δ(yn,ym) <, m,n≥N.

It proves that{yn}={Txn}is Cauchy sequence. Due to the completeness of the considered space, there isy∈Xsuch that

lim

n→∞δ(yn,y) = 0 =nlim→∞δ(Txn,y) =nlim→∞δ(gxn+1,y).

Since{Txn}={gxn+1} ⊆g(X) andg(X) is closed, there existsz∈Xsuch thatgz=yand

lim

n→∞δ(gxn,gz) = 0. (2.17)

Next, we show that zis a coincidence point ofT andg. On the contrary, assume that δ(Tz,gz) > 0. We have from (iii) that

fδ(Tz,y)≤fδ(Tz,Txn(k)) +δ(Txn(k),y)

+C, n∈N

≤fθ(gxn(k),gz)δ(Tz,Txn(k)) +δ(Txn(k),y)

+C

≤fψM(gz,gxn(k))

+δ(Txn(k),y)

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where

M(gxn(k),gz) =max

δ(gxn(k),gz),

δ(gxn(k),Txn(k)) +δ(gz,Tz)

2 ,

δ(gxn(k),fz) +δ(gz,Txn(k))

2

≤max

δ(gxn(k),gz),

δ(gxn(k),Txn(k)) +δ(gz,Txn(k)) +δ(Txn(k),Tz)

2 ,

δ(gxn(k),fz) +δ(gz,Txn(k))

2

.

Keeping (2.17) in mind and lettingn→ ∞in (2.18), we conclude that the right-hand side tends to∞. This is a contradiction, and hencezis coincidence point of pointTandg.

We present the following example in support of our theorem:

Example2.2 ConsiderX=R. Deﬁneδ:X×X→[0,∞) as

δ(x,y) =

⎧ ⎨ ⎩

e|x–y| _if_x₌_y, 0 ifx=y.

Soδis a complete function weighted metric withf(t) = –1

t anda= 1. Consider the follow-ing self-mappfollow-ings:

T(x) =

⎧ ⎨ ⎩

3–x

2 ifx> –1,

0 otherwise,

and

g(x) =

⎧ ⎨ ⎩

x

2 ifx> 1,

0 otherwise.

Chooseψ(t) =√tfor allt≥0. So all the hypothesis of Theorem2.2are satisﬁed andx=3₂ is a unique coincidence point ofT andg.

Theorem 2.3 If we assume that T or g is continuous,in addition to the axioms of in The-orem2.2,then T and g possess a common point.

If we takeθ(gx,gy) = 1 in Theorem2.2, then we ﬁnd the following:

Theorem 2.4 Let(X,δ)be a complete function weighted metric space and T,g:X→X be such that T(X)⊆g(X)where g(X)is closed.Assume that the pair(T,g)satisﬁes

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for all x,y∈X withδ(Tx,Ty) > 0where

M(gx,gy) =max

δ(gx,gy),δ(gx,Tx) +δ(gy,Ty)

2 ,

δ(gx,Ty) +δ(gy,Tx) 2

.

Then,T and g possess a coincidence point.

If we takeg(x) =xfor allx∈Xin Theorem2.4, then we derive the following result. Theorem 2.5 Let(X,δ)be a complete function weighted metric space and assume T:X→ X satisﬁes

δ(Tx,Ty)≤ψM(x,y), (2.20)

for all x,y∈X withδ(Tx,Ty) > 0where

M(x,y) =max

δ(x,y),δ(x,Tx) +δ(y,Ty)

2 ,

δ(x,Ty) +δ(y,Tx) 2

.

Then,T possesses a ﬁxed point.

Acknowledgements

The authors thanks anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper. The authors would like to extend their sincere appreciation to the Deanship of Scientiﬁc Research at King Saud University for funding this group No. RG-1440-025.

Funding

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, King Saud University, Riyadh, Saudi Arabia.}2_{Department of Medical Research, China}

Medical University, Taichung, Taiwan.3_{St. Andrew’s College of Arts, Science & Commerce, Mumbai, India.}

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Received: 15 April 2019 Accepted: 31 May 2019 References

1. Jleli, M., Samet, B.: On a new generalization of metric spaces. J. Fixed Point Theory Appl.20, 128 (2018).