Fixed point theorems for a class of generalized weak cyclic compatible contractions

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R E S E A R C H

Open Access

Fixed point theorems for a class of

generalized weak cyclic compatible

contractions

P.S. Kumari

1

_{, J. Nantadilok}

2*

_{and M. Sarwar}

3

*_{Correspondence:}

[email protected]; [email protected]

2_{Department of Mathematics,}

Faculty of Science, Lampang Rajabhat University, Lampang, Thailand

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

Abstract

In this manuscript, we establish a coincidence point and a unique common ﬁxed point theorem for (ψ,ϕ)-weak cyclic compatible contractions. We also present a ﬁxed point theorem for a class of

-weak cyclic compatible contractions

via altering

distance functions. Our results extend and improve some well-known results in the literature. We provide examples to analyze and illustrate our main results.

MSC: 54H25; 47H10

Keywords: Cyclic maps;b-quasi-metric family;b-dislocated metric family; (ψ-ϕ)-weak cyclic compatible contraction;

-cyclic compatible contraction

1 Introduction and preliminaries

The Banach contraction principle [1] is one of the most powerful and useful tools in mod-ern analysis. Over time, this principle has been extended and improved in many ways and a variety of fixed point theorems have been obtained. In 2003, one of the more notable generalizations of the Banach contraction principle was introduced via cyclic contraction by Kirk et al. [2]. Following the publication of [2], many fixed point theorems for cyclic contractive mappings have been obtained. For more results on cyclic maps, we refer the reader to [3–9] and the references therein. Here, we present some essential definitions.

Deﬁnition 1.1(see [2]) LetA,Bbe non-empty subsets of a setXand letU:A∪B→A∪B. Uis called a cyclic map, ifU(A)⊆BandU(B)⊆A.

Throughout this manuscript, we assume that R+_{= [0,}_∞_), _N_{= the set of all positive}

integers.

Deﬁnition 1.2(see [10]) Let (X,d) be a complete metric space and letU,V:X→X

be self-mappings. ThenUandV are said to be weakly compatible ifUx=Vximplies UVx=VUx.

Deﬁnition 1.3(see [11]) A functionη:R+→R+is called an altering distance function if the following conditions are satisﬁed:

1. η(0) = 0;

(2)

2. ηis monotonically nondecreasing; 3. ηis continuous.

We will denote the set of all altering distance functions by.

The concepts of (ψ,ϕ)-weak contractions, weakly compatible maps and altering dis-tance functions is interesting, though brief. These concepts were widely used in the con-struction of existence theorems and many results, a number of applications have been obtained; see [12–17] for examples.

Below, we provide necessary deﬁnitions.

Deﬁnition 1.4 (see [6]) LetX be a non-empty set and{dα:α∈(0, 1]} a family of the mappingdαofX×XintoR+. Then (X,dα) is called a generating space of ab-quasi-metric family (abbreviated asGbq-family), if it satisﬁes the following conditions, for anyx,y,z∈X

ands≥1:

(a) dα(x,y) = 0if and only ifx=y. (b) dα(x,y) =dα(y,x).

(c) For anyα∈(0, 1]there existsβ∈(0,α]such thatdα(x,z)≤s[dβ(x,y) +dβ(y,z)]. (d) For anyx,y∈X,dα(x,y)is non-increasing and left continuous inα.

Deﬁnition 1.5 (see [6]) LetX be a non-empty set and{dα:α∈(0, 1]} a family of the mappingdαofX×XintoR+. Then (X,dα) is called a generating space ofb-dislocated metric family (abbreviated asGbd-family), if it satisﬁes the following conditions, for any

x,y,z∈Xands≥1:

(a) dα(x,y) = 0impliesx=y. (b) dα(x,y) =dα(y,x).

(c) For anyα∈(0, 1]there existsβ∈(0,α]such thatdα(x,z)≤s[dβ(x,y) +dβ(y,z)]. (d) For anyx,y∈X,dα(x,y)is non-increasing and left continuous inα.

The construction of topological concepts of the above spaces can be found in [6, 18]. Recently, Kumari and Panthi [7] introducedcyclic compatible contractionsand established ﬁxed point theorems in the generating space of a b-quasi-metric family (X,dα).

Deﬁnition 1.6(see [7]) LetA,Bbe non-empty subsets of aGbq-family (X,dα) and let U,V:A∪B→A∪Bbe cyclic mappings such thatU(X)⊂V(X). ThenU,Vare said to becyclic compatible contraction, if for somex∈A, there exists aγ ∈(0, 1) such that

dα

U2nx,Uy≤γdα

U2n–1x,Vy,

for alln∈Nandy∈B.

In this paper, motivated and inspired by the above deﬁnitions, we investigate the weak

cyclic compatible contractionsvia (ψ,ϕ)-weak contractions and altering distance func-tions.

2 Main results

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Deﬁnition 2.1 LetAandBbe non-empty subsets of aGbd-family (X,dα). SupposeU,V:

A∪B→A∪Bare cyclic mappings withU(X)⊂V(X). ThenU,Vare called (ψ,ϕ)-weak cyclic compatible contractions, if for somex∈A

ψdα

U2nx,Uy≤ψdα

U2n–1x,Vy–ϕdα

U2n–1x,Vy, (1)

whereψ,ϕ: [0,∞)→[0,∞) are both continuous and monotone nondecreasing functions withψ(t) =ϕ(t) = 0 if and only ift= 0 andn∈N,y∈B.

We state and prove our main results.

Theorem 2.2 Let A and B be non-empty closed subsets of a complete Gbd-family(X,dα)

and letU,V:A∪B→A∪B be cyclic mappings withU(X)⊂V(X)andV(X)closed in X.

SupposeU,Vare(ψ,ϕ)-weak cyclic compatible contractions,thenUandVhave a point of coincidence and a unique common ﬁxed point in A∩B.

Proof Letx0∈Xbe ﬁxed. AsU(X)⊂V(X), we may choosex1∈Xsuch that

Ux0=Vx1.

Hence we can deﬁne the sequence{xn}inXbyUnx0=Uxn=Vxn+1=xnforn∈N∪ {0}.

Now consider

ψdα(x2n,x2n+1)

=ψdα

U2nx0,Ux2n+1

≤ψdα

U2n–1x0,Vx2n+1

–ϕdα

U2n–1x0,Vx2n+1

=ψdα(x2n–1,x2n)

–ϕdα(x2n–1,x2n)

≤ψdα(x2n–1,x2n)

. (2)

This implies that

dα(x2n,x2n+1)≤dα(x2n–1,x2n).

Similarly, we have

dα(x2n+1,x2n+2)≤dα(x2n,x2n+1).

Inductively, we have

dα(xn,xn+1)≤dα(xn–1,xn), ∀n∈N.

Thus the sequence{dα(xn,xn+1)}is non-increasing and hence it is convergent. So, there

existsκ≥0 such thatlimn→∞dα(xn,xn+1) =κ.

From (2), we have

ψdα(x2n,x2n+1)

≤ψdα(x2n–1,x2n)

–ϕdα(x2n–1,x2n)

(4)

Now taking the limit asn→ ∞, we get

ψ(κ)≤ψ(κ) –ϕ(κ).

This is a contradiction, unlessκ= 0. Thus

lim

n→∞dα(xn,xn+1) = 0. (4)

Forn,m∈N,m>n, consider

dα

Un_x

0,Umx0

=dα(xn,xm)

≤sdβ(xn,xn+1) +dβ(xn+1,xm)

≤sdβ(xn,xn+1) +s2dβ(xn+1,xn+2) +s3dβ(xn+2,xn+3) +· · ·. (5)

Lettingn,m→ ∞, we get

lim

n,m→∞dα

Unx0,Umx0

= 0.

Therefore {Un_x

0} is a Cauchy sequence. Since (X,dα) is a complete Gbd-family, there

exist sequences {U2n_x

0} in A and {U2n–1x0} in B such that limn→∞U2nx0 →u and

limn→∞U2n–1x0→u. SinceAandBare closed inX,u∈A∩B. SinceV(X) is closed inX,

there existszinXsuch thatVz=u. Now we shall prove thatUz=u. Consider

ψdα

U2n_x

0,Uz

≤ψdα

U2n–1_x

0,Vz

–ϕdα

U2n–1_x

0,Vz

≤ψdα

U2n–1x0,Vz

. (6)

By taking the limit asn→ ∞, we getψ(dα(u,Uz)) = 0. Thusdα(u,Uz) = 0. ThenUz=u. HenceUz=Vz=u. Souis a coincidence point ofUandV. From weak compatibility, we get

Uu=Vu. (7)

Now we prove thatVu=u=Uu. We assumeu=Uu, then

ψdα(u,Uu, )

= lim

n→∞ψ

dα

U2nx0,Uu

≤ lim

n→∞

ψdα

U2n–1x0,Vu

–ϕdα

U2n–1x0,Vu

=ψdα(u,Vu) –ϕ

dα(u,Vu)

≤ψdα(u,Uu)

. (8)

That is a contradiction. Therefore

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From (7) and (9), we haveUu=Vu=u. Thereforeuis a common ﬁxed point ofUandV. To prove uniqueness, supposevis another ﬁxed point ofUandVsuch thatu=v, then

ψdα(u,v)

≤ lim

n→∞

ψdα

U2nx0,Uv

≤ lim

n→∞

ψdα

U2n–1x0,Vv

–ϕdα

U2n–1x0,Vv

=ψdα(u,v)

–ϕdα(u,v)

<ψdα(u,v)

. (10)

This is a contradiction. Henceu=v. This completes our proof.

Remark2.3 We will obtain special cases of Theorem 2.2, if we

1. replace aGbq-family(X,dα)by aGq-family, according to Deﬁnition 1.4 above, by

puttings= 1;

2. replace aGbq-family(X,dα)by abd-metric space, and takingdinstead ofdα; 3. replace aGbq-family(X,dα)by a complete dislocated metric space, by takingd

instead ofdαand lettings= 1.

For more details ofGq-family,bd-metric space and a complete dislocated metric space, we

refer the reader to [6].

Example2.4 LetA=B=X= [0, 1]. Deﬁnebd:X×X→R+bybd(x,y) = (x+y)2. Then

(X,bd) is abd-metric space withs= 2, but not a dislocated metric space. Deﬁne

Ux= 0, if 0≤x≤1

and

Vx= ⎧ ⎨ ⎩

0, if 0≤x<1 2, 1

3, if 1

2≤x≤1.

ClearlyU(X)⊂V(X). Deﬁneψ(t) = 2tandϕ(t) =t. Fix anyx∈[0, 1], we have

Ux=U2x=· · ·=Unx= 0, ∀n.

For anyy∈[0, 1],

Vy= ⎧ ⎨ ⎩

0, if 0≤y<1₂,

1 3, if

1

2≤y≤1.

Consider

Case(i). If 0≤y<1₂,Vy= 0, we get

ψbd

U2n–1x,Vy–ϕbd

U2n–1x,Vy=ψbd(0, 0)

–ϕbd(0, 0)

= 0 =ψbd

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Case(ii). If 1₂≤y≤1,Uy= 0,Vy=1₃, we get

ψbd

U2nx,Uy=ψbd(0, 0)

= 0

and

ψbd

U2n–1x,Vy=ψ bd 0,

1 3

–ϕ bd 0,

1 3

=ψ 1 9

–ϕ 1 9

=2 9–

1 9

=1 9.

From both cases, we obtain

ψbd

U2nx,Uy≤ψbd

U2n–1x,Vy.

ThusU,Vare (ψ,ϕ)-weak cyclic compatible contractions. All the conditions of Theorem 2.2 hold true, andUandVhave a unique common ﬁxed point. Hereu= 0 is the unique common ﬁxed point ofUandV.

Example2.5 LetA=B=X= [0, 20]. Deﬁned:X×X→R+byd(x,y) =x+y. Then (X,d) is a complete dislocated metric space.

Deﬁne

Ux= ⎧ ⎨ ⎩

0, ifx∈ {0} ∪(3, 20],

2, if 0 <x≤3,

and

Vx= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0, ifx= 0,

x+ 10, if 0 <x≤3,

x– 2, if 3 <x≤20.

Clearly,U(X)⊂V(X). Deﬁneψ(t) = 2tandϕ(t) =t. Fix anyx∈(3, 20], we have

Ux=U2_x₌_{· · ·}₌_Un_x_{= 0,} _∀_n_.

For anyy∈[0, 20].

Vy= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0, ify= 0,

y+ 10, if 0 <y≤3,

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Consider

Case(i). Ify= 0,Uy= 0,Vy= 0, we have

ψdU2nx,Uy=ψd(0, 0)= 0

and

ψdU2n–1x,Vy–ϕdU2n–1x,Vy=ψd(0, 0)–ϕd(0, 0)

= 0.

Hence

ψdU2nx,Uy=ψdU2n–1x,Vy–ϕdU2n–1x,Vy.

Case(ii). If 0 <y≤3,Uy= 2 andVy=y+ 10. Then we have

ψdU2nx,Uy=ψd(2, 2)=ψ(4) = 8

and

ψdU2n–1x,Vy–ϕdU2n–1x,Vy=ψd(2,y+ 10)–ϕd(2,y+ 10)

=ψ(y+ 12) –ϕ(y+ 12)

= 2(y+ 12) – (y+ 12) =y+ 12.

Therefore, for 0 <y≤3,

ψdU2nx,Uy<ψdU2n–1x,Vy–ϕdU2n–1x,Vy.

Case(iii). If 3 <y≤20,Uy= 0 andVy=y– 1. Then we have

ψdU2nx,Uy=ψd(0, 0)= 0

and

ψdU2n–1x,Vy–ϕdU2n–1x,Vy=ψd(0,y– 2)–ϕd(0,y– 2)

=ψ(y– 2) –ϕ(y– 2)

= 2(y– 2) – (y– 2) =y– 2.

Hence

ψdU2nx,Uy<ψdU2n–1x,Vy–ϕdU2n–1x,Vy.

Therefore, from all cases, we have

(8)

Thus U,V are (ψ,ϕ)-weak cyclic compatible contractions. All the conditions of Theo-rem 2.2 hold true, andU andV have a unique common ﬁxed point. Hereu= 0 is the unique common ﬁxed point ofUandV.

If we takeV=UandV=Iin Definition 1.6 and Definition 2.1, we obtain the following definition.

Deﬁnition 2.6 LetUbe a cyclic mapping; then

(1) Uis called acyclic idle contraction⇔dα(U2nx,Uy)≤γdα(U2n–1x,Uy); (2) Uis called a(ψ,ϕ)-weak cyclic idle contraction

⇔ψ(dα(U2nx,Uy))≤ψ(dα(U2n–1x,Uy)) –ϕ(dα(U2n–1x,Uy)); (3) Uis called a(ψ,ϕ)-weak cyclic orbital contraction

⇔ψ(dα(U2nx,Uy))≤ψ(dα(U2n–1x,y)) –ϕ(dα(U2n–1x,y)),

whereψ,ϕ: [0,∞)→[0,∞) are both continuous and monotone nondecreasing functions withψ(t) =ϕ(t) = 0 if and only ift= 0.

and letU:A∪B→A∪B be a(ψ-ϕ)-weak cyclic idle contraction.Assume thatU(X)is closed in X.ThenUhas a unique ﬁxed point in A∩B.

Proof TakeU=Vin Theorem 2.2.

and letU:A∪B→A∪B be a(ψ-ϕ)-weak cyclic orbital contraction.Assume thatU(X)

is closed in X.ThenUhas a unique ﬁxed point in A∩B.

Proof TakeV=Iin Theorem 2.2.

Next we will state and prove a ﬁxed point theorem via altering distance functions. We introduce the following deﬁnition.

Deﬁnition 2.9 LetAandBbe non-empty closed subsets of the generating space of a b-quasi-metric family (X,dα). SupposeU,V:A∪B→A∪Bare cyclic mappings such thatU(X)⊂V(X). ThenU,Vare said to be-cyclic compatible contractions,if for some

x∈A, there existsγ ∈(0, 1) such that

ηdα

U2nx,Uy≤γ ηdα

U2n–1x,Vy, (11)

for alln∈N,y∈Bandη∈.

Theorem 2.10 Let A and B be non-empty closed subsets of the generating space of a com-plete b-quasi-metric family(X,dα).LetU,V:A∪B→A∪B be cyclic mappings such that U(X)⊂V(X).Suppose

1. U,Vare-cyclic compatible contractions, 2. U,Vare weakly compatible,

3. V(X)is a closed subset ofX.

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Proof Takex=x0∈A. SinceU(X)⊂V(X), we may choosex1∈Xsuch thatUx0=Vx1.

Hence we can deﬁne the sequence{xn}inXbyUnx0=Uxn=Vxn+1=xn+1forn∈N∪

{0}. Then{x2n} ∈Aand{x2n–1} ∈B. Thus we have

ηdα(x2n,x2n+1)

=ηdα(Vx2n,Vx2n+1)

=ηdα(Ux2n–1,Ux2n)

=ηdα

U2nx0,Ux2n–1

≤γ ηdα

U2n–1x0,Vx2n–1

=γ ηdα(x2n,x2n–1)

=γ ηdα(x2n–1,x2n)

. (12)

Similarly,

ηdα(x2n+1,x2n+2)

=ηdα(Vx2n+1,Vx2n+2)

=ηdα(Ux2n,Ux2n+1)

=ηdα

U2n+1x0,Ux2n

≤γ ηdα

U2nx0,Vx2n

=γ ηdα(Ux2n,x2n)

=γ ηdα(x2n+1,x2n)

=γ ηdα(x2n,x2n+1)

. (13)

In general, we have

ηdα(xn,xn+1)

≤γ ηdα(xn–1,xn)

.

Inductively, for eachn∈N, we get

ηdα(xn,xn+1)

≤γnηdα(x0,x1)

.

Since 0≤γ < 1, lettingn→ ∞, we getlimn→∞η(dα(xn,xn+1)) = 0. From the deﬁnition of

altering distance functions, we get

lim

n→∞

dα(xn,xn+1)

= 0. (14)

Therefore

lim

n→∞

dα(Vxn,Vxn+1)

= 0,

or

lim

n→∞

dα(Uxn–1,Uxn)

(10)

Now we claim that{xn}is a Cauchy sequence. According to the deﬁnition of generating

space of a b-quasi-metric family (X,dα), and forn,m∈N,m>n, we have

dα(xn,xm)≤s

dβ(xn,xn+1) +dβ(xn+1,xm)

=sdβ(xn,xn+1) +sdβ(xn+1,xm)

≤sdβ(xn,xn+1) +s2dβ(xn+1,xn+2) +s3dβ(xn+2,xn+3) +· · ·. (15)

Lettingn,m→ ∞, and from (14), we havelimn,m→∞dα(xn,xm) = 0 for allα∈(0, 1].

There-fore{xn}is a Cauchy sequence inX. SinceXis complete, there exist sequences{V2nx0}in Aand{V2n–1_x

0}inBsuch that both sequences converge to someωinX. SinceAandB

are closed inX,ω∈A∩B.

SinceV(X) is closed inX, there existszinXsuch that

Vz=ω. (16)

Consider

ηdα

U2nx0,Uz

≤γ ηdα

U2n–1x0,Vz

.

Lettingn→ ∞, we getlimn→∞η(dα(U2nx0,Uz)) = 0. From the deﬁnition of altering

dis-tance functions, we get

lim

n→∞dα

U2nx0,Uz

= 0.

This implies

dα(ω,Uz) = 0.

Therefore

ω=Uz. (17)

From (16) and (17), it follows thatUz=Vz=ω. Thusωis a point of coincidence ofU andV. By weak compatibility, we getVω=Uω. Now we proveVω=ω. SupposeVω=ω, then consider

ηdα(Vω,ω)

= lim

n→∞η

dα

Uω,U2n–1x0

= lim

n→∞η

dα

U2n–1x0,Uω

≤γ lim

n→∞η

dα

U2n–2x0,Vω

=γ ηdα(ω,Vω)

≤ηdα(ω,Vω)

(11)

which is a contradiction. HenceVω=ω. ThereforeUω=Vω=ω. To prove uniqueness, suppose thatω1andω2are two common ﬁxed points ofUandV. Assume thatω1=ω2.

ThenUω1=Vω1=ω1andUω2=Vω2=ω2.

Consider

ηdα(ω1,ω2)

= lim

n→∞η

dα

U2n–1x0,Uω2

≤γ lim

n→∞η

dα

U2n–2x0,Vω2

=γ ηdα(ω1,Vω2)

≤ηdα(ω1,ω2)

, (19)

which is a contradiction. Henceω1=ω2and our proof is ﬁnished.

Remark2.11 We will obtain special cases of Theorem 2.10 above, if we

1. replaceGbq-family(X,dα)byGq-family, according to Deﬁnition 1.4, by puttings= 1;

2. replaceGbq-family(X,dα)byb-metric space and takingdα=d;

3. replaceGbq-family(X,dα)by complete metric space and takingdα=dands= 1.

Example2.12 LetA=B=X= [0, 1]. Deﬁned:X×X→R+_by_d₍_x_,_y_{) = (}_x_–_y₎2_{. This is}

ab-metric space with s= 2, not the usual metric space, sinced(0, 1)d(0,1₂) +d(1₂, 1). Deﬁne

Ux= 0, if 0≤x≤1

and

Vx= ⎧ ⎨ ⎩

0, if 0≤x<1₂,

1 5, if

1

2≤x≤1.

ClearlyU(X)⊂V(X). Deﬁneη(t) =t2_{. For any}_x_∈_{[0, 1], we have}

Ux=U2x=· · ·Unx= 0, for alln.

For anyy∈[0, 1], we have

Vy= ⎧ ⎨ ⎩

0, if 0≤y<1₂,

1 5, if

1

2≤y≤1.

Case(i). If 0≤y<1₂,Uy= 0, we have

ηdU2nx,Uy=ηd(0, 0)

= 0 =γ ηdU2n–1x,Vy.

Case(ii). If 1

2≤y≤1,Uy= 0, we have

(12)

and

ηdU2n–1x,Vy=η d 0,1 5

=η 1 25

= 1

625.

So

ηdU2n_x_,_U_y_≤_{γ η}_d_U2n–1_x_,_V_y_, _{for 0}_≤_γ _{< 1.}

Therefore, from both cases, we get

ηdU2nx,Uy≤γ ηdU2n–1x,Vy, for 0≤γ < 1.

ThusU,Vare-cyclic compatible contractions.All the conditions of Theorem 2.10 hold true andU,Vhave a unique common ﬁxed point. Hereω= 0 is the unique common ﬁxed point ofUandV.

Example2.13 LetA=B=X= [0, 1]. Deﬁned:X×X→R+byd(x,y) =|x–y|. So (X,d) is a complete metric space. Deﬁne

Ux= 0, if 0≤x≤1

and

Vx= ⎧ ⎨ ⎩

0, if 0≤x<1₂,

1 9, if

1

2≤x≤1.

ClearlyU(X)⊂V(X). Deﬁneη(t) =t. For anyx∈[0, 1], we have

Ux=U2x=· · ·=Unx= 0, for alln.

For anyy∈[0, 1],

Vy= ⎧ ⎨ ⎩

0, if 0≤y<1₂,

1 9, if

1

2≤y≤1.

Consider

Case(i). If 0≤y<1₂,Uy= 0,Vy= 0. Then we have

ηdU2nx,Uy=ηd(0, 0)

= 0 =γ ηdU2n–1x,Vy.

Case(ii). If 1₂≤y≤1,Uy= 0,Vy=1₉. Then we have

(13)

and

ηdU2n–1x,Vy=η d 0,1 9

=1 9.

Hence, from both cases we conclude thatη(d(U2n_x_,_U_y₎₎_≤_{γ η(}_d_(U2n–1_x_,_V_y_{)), for all 0}_≤

γ < 1. ThusU,Vare-cyclic compatible contractions.All the conditions of Theorem 2.10 hold true. Hereω= 0 is the unique common ﬁxed point ofUandV.

In view of Deﬁnition 2.6, we introduce the following deﬁnition.

Deﬁnition 2.14 LetUbe a cyclic mapping, then

(1) Uis called-cyclic idle contraction.⇔η(dα(U2nx,Uy))≤γ η(dα(U2n–1x,Uy)). (2) Uis called-cyclic orbital contraction.⇔η(dα(U2nx,Uy))≤γ η(dα(U2n–1x,y)).

Theorem 2.15 Let A and B be non-empty closed subsets of the generating space of a com-plete b-quasi-metric family(X,dα).SupposeU:A∪B→A∪B is a-cyclic idle contrac-tion,thenUhas a unique ﬁxed point in A∩B.

Proof TakeU=Vin Theorem 2.10.

Theorem 2.16 Let A and B be non-empty closed subsets of the generating space of a com-plete b-quasi-metric family(X,dα).SupposeU:A∪B→A∪B is a-cyclic orbital con-traction,thenUhas a unique ﬁxed point in A∩B.

Proof TakeV=Iin Theorem 2.10.

3 Discussion

In this manuscript, we establish ﬁxed point theorems in generating space of aGbd-family

(X,dα), and a b-quasi-metric family (X,dα). One can see that Theorem 2.2 and Theo-rem 2.10 are generalizations of TheoTheo-rem 2.2 obtained in [7] in the setting of a complete

Gbd-family (X,dα), and a complete b-quasi-metric family (X,dα), respectively. Besides, we introduce other definition of weak cyclic contractions (see Definition 2.6), and obtain fixed point theorems for those contractions (Theorems 2.7 and 2.8). After Theorem 2.10, we also introduce other definition of-cyclic contractions(Def. 2.14) and obtain fixed point theorems (Theorems 2.15 and 2.16). Our main results extend and improve some well-known results in the existing literature. However, in recent remarkable work of Lau and Zhang in [19, 20], the authors studied fixed point properties of semigroups of non-expansive mappings, nonlinear mappings and amenability. In relation to Theorem 2.2 and Theorem 2.10, we pose the following open problem at the end.

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4 Conclusions

In this manuscript, we introduce a new class of generalized (ψ,ϕ)-weak cyclic compat-ible contractions (Definition 2.1) and prove a coincidence point and a unique common fixed point theorem for these new contractions in a complete Gbd-family (X,dα). We also introduce a new class of-cyclic compatible contractionsvia altering distance func-tions(Definition 2.9), and prove an existence theorem for this new class of generalized contractions in the generating space of a b-quasi-metric family (X,dα). Our results extend and improve the results obtained in [7] and some well-known results in the literature. We provide examples to illustrate and support our results and we also pose a problem at the end.

Acknowledgements

The authors would like to express their sincere gratitude to Mr. Anand Prabhakar for his invaluable suggestions and motivation and to the referees for their helpful comments and suggestions.

Competing interests

The authors declare that there is no conﬂict of interest regarding the publication of this article.

Authors’ contributions

Each author equally contributed to this paper, read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, School of Advanced Sciences, VIT University, Andhra Pradesh, India.}2_{Department of}

Mathematics, Faculty of Science, Lampang Rajabhat University, Lampang, Thailand. 3_{Department of Mathematics,}

University of Malakand, Chakdara Dir(L), Pakistan.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 3 January 2018 Accepted: 19 April 2018

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