# Common Fixed Point Results on Generalized Weak Compatible Mapping in Quasi-Partial b-Metric Space

Authors: Pragati Gautam; Luis Manuel Sánchez Ruiz; Swapnil Verma; Gauri Gupta
Journal: Journal of Mathematics
 (2021)

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## Abstract

The focus of this paper is to acquaint with generalized condition (B) in a quasi-partial b-metric space and to establish coincidence and common fixed point theorems for weakly compatible pairs of mapping. Additionally, with the background of quasi-partial b-metric space, the outcomes obtained are exemplified to prove the existence and uniqueness of fixed point.


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## Body

## 1. Introduction

In the early years of20th century, the French mathematician Fréchet [1] commenced the concept of metric space, and due to its consequences and practicable implementations, the idea has been enlarged, upgraded, and generalized in different directions. In 1922, Banach [2] introduced the very important Banach contraction principle which holds a remarkable position in the field on nonlinear analysis. One such generalization was established by Künzi et al. [3] known as quasi-partial metric space by Karapinar et al. [4, 5]. In 1993, Czerwik [6] introduced the concept of b-metric space. Later, Gupta and Gautam [7, 8] generalized quasi-partial metric space to quasi-partial b-metric space and proved some fixed point results for such spaces. Several authors [9–18] have already proved the fixed point theorem in metric space, partial metric space [19], quasi-partial metric space, quasi-partial b-metric space [7], and many different spaces. After these classical results, some researchers [20–25] introduced the distinctive concepts and used fixed point theorems to demonstrate the uniqueness of a solution of the equations in different metric spaces such as multivalued contractive type mappings, Reich–Rus–Cirić and Hardy–Rogers contraction mappings, and Chatterjea and cyclic Chatterjea contraction.In this paper, we have introduced the generalized condition (B) in quasi-partial b-metric space to obtain coincidence and common fixed points. Moreover, some examples are given to exemplify the concept followed up with pictographic grid.


## 2. Preliminaries

Let us recall some definition.Definition 1 (see [19]).
A partial metric space on a nonempty setX is a function M:X×X⟶ℝ+ satisfying(1)
Mτ,υ=Mυ,τ (symmetry)(2)
if0<Mτ,τ=Mτ,υ=Mυ,υ, then τ=υ (indistancy implies equality)(3)
Mυ,υ≤Mτ,υ, then τ=υ (small self-distances)(4)
Mτ,Υ+Mυ,υ≤Mτ,υ+Mυ,Υ (triangularity)
for allτ,υ,Υ∈X.Definition 2 (see [4]).
A quasi-partial metric on a nonempty setX is a function q:X×X⟶ℝ+ satisfying(1)
Ifqτ,τ=qτ,υ=qυ,υ, then τ=υ (indistancy implies equality)(2)
qτ,τ≤qτ,υ (small self-distances)(3)
qτ,τ≤qυ,τ (small self-distances)(4)
qτ,υ+qΥ,Υ≤qτ,Υ+qΥ,υ (triangularity)
for allτ,υ,Υ∈X.Definition 3 (see [20]).
A quasi-partial b-metric on a nonempty setX is a function qpb:X×X⟶ℝ+ such that for some real number ρ≥1(1)
Ifqpbτ,τ=qpbτ,υ=qpbυ,υ, then τ=υ (indistancy implies equality)(2)
qpbτ,τ≤qpbτ,υ (small self-distances)(3)
qpbτ,τ≤qpbυ,τ (small self-distances)(4)
qpbτ,υ+qpbΥ,Υ≤ρqpbτ,Υ+qpbυ,Υ (triangularity)
for allτ,υ,Υ∈X. The infimum over all reals ρ≥1 satisfying condition (30) is called the coefficient of X,qpb and represented by RX,qpb.Lemma 1 (see [6]).
LetX,qpb be a quasi-partial b-metric space. Then the following hold:(1)
Ifqpbτ,υ=0, then τ=υ(2)
Ifτ≠υ, then qpbτ,υ>0 and qpbυ,τ>0Definition 4 (see [6]).
LetX,qpb be a quasi-partial b-metric. Then(1)
A sequenceτn⊂X converges to τ∈X if and only if(1)qpbτ,τ=limn,m⟶∞qpbτ,τn=limn,m⟶∞qpbτn,τ.(2)
A sequenceτn⊂X is called a Cauchy sequence if and only if(2)limn,m⟶∞qpbτn,τm,limn,m⟶∞qpbτm,τnexist.(3)
The quasi-partial b-metric spaceX,qpb is said to be complete if every Cauchy sequence τn⊂X converges with respect to xqpb to a point τ∈X such that(3)qpbτ,τ=limn,m⟶∞qpbτn,τm=limn,m⟶∞qpbτm,τn.(4)
A mappingf:X⟶X is said to be continuous at τ0∈X if for every ɛ>0, there exists δ>0 such that fBτ0,δ⊂Bfτ0,ɛ.Lemma 2 (see [6]).
LetX,qpb be a quasi-partial b-metric space and X,dqpb be the corresponding b-metric space. Then X,dqpb is complete if X,qpb is complete.Definition 5 (see [26]).
A self-mappingP on a metric space X,d satisfies condition (B), if there exist δ∈0,1 and ω>0 such that for all τ,υ∈X, we have,(4)dPτ,Pυ≤δdPτ,Pυ+ωmindPτ,Pυ,dυ,Pυ,dτ,Pυ,dυ,Pτ.
Following Babu et al. [26], Abbas et al. [27] and Abbas and Illic [28] extended the concept of condition (B) to a pair of mappings. Abbas et al. [27] called it generalized condition (B), and Abbas and Illic [28] called it generalized almost A-contraction.Definition 6 (see [27]).
LetP and Q be two self-mappings on a metric space X,d. The mapping Q satisfies generalized condition (B) associated with P if there exist δ∈0,1 and ω≥0 with ρ≥1 such that(5)dQτ,Qυ≤δmaxdPτ,Pυ,dPτ,Qτ,dPυ,Pυ,12ρdPτ,Qυ+dPυ,Qτ+ωmindPτ,Qτ,dPυ,Qυ,dPτ,Qυ,dPυ,Qτ.
Clearly condition (B) implies generalized condition (B).Definition 7 (see [29]).
LetP and Q be self-mappings on a set X. A point x∈X is called a coincidence point of P and Q if Px=Qx=w, where w is called a point of coincidence of P and Q.Definition 8 (see [30]).
LetX be a nonempty set. Two mappings P, Q:X⟶X are said to be weakly compatible if they commute at their coincidence point, that is, if Pu=Qu for some u∈X, then PQu=QPu.


## 3. Main Results

Definition 9.
LetP and R be two self-mappings on a quasi-partial b-metric space X,qpb.
The mappingR satisfies generalized condition (B) associated with P (R is a generalized almost P contraction) if there exist δ∈0,1,ρ≥1, and M≥0 such that for all τ,υ∈X, we have(6)qpbRτ,Rυ≤δmaxqpbPτ,Pυ,qpbPτ,Rτ,qpbPυ,Rυ,12ρqpbRτ,Pυ+qpbPτ,Rυ+MminqpbPτ,Rτ,qpbPυ,Rυ,qpbPτ,Rυ,qpbPυ,Rτ.Definition 10.
LetP,Q,R,S be four self-mappings on a quasi-partial b-metric space X,qpb.
The pair of mappingP,R satisfies generalized condition (B) associated with Q,S ((P, R) is generalized almost (Q, S) contraction) if there exist δ∈0,1,ρ≥1 and M≥0 such that for all τ,υ∈X, we have(7)qpbRτ,Sυ≤δmaxqpbPτ,Qυ,qpbPτ,Rτ,qpbQυ,Sυ,12ρqpbRτ,Qυ+qpbPτ,Sυ+MminqpbPτ,Rτ,qpbQυ,Sυ,qpbPτ,Sυ,qpbQυ,Rτ.Theorem 1.
LetP,Q,R,S be four self-mappings on quasi-partial b-metric space X,qpb and if we take the mappings in pair as P,R associated with Q,S for all τ,υ∈X, δ∈0,1, and M≥0, ρ≥1 and(1)
RX⊂QX and SX⊂PX(2)
PX or QX is closed(3)
1/ρδ+2M<1then the pairs P,R and Q,S have a coincidence point. Also P,Q,R,S have a unique common fixed point, providing that pairs P,R and Q,S are weakly compatible.Proof.
Letτ∗∈X. Since RX⊂QX there exists τ0∈X such that υ0=Qτ0=Rτ∗. Suppose there exists a point υ1∈Sτ0 corresponding to the point υ0. Also since SX⊂PX there exist τ1∈X such that υ1=Pτ1=Sτ0. Going this way we get a sequence υn∈X as(8)υ2m+1=Qτ2m+1=Rτ2m,υ2m+2=Pτ2m+2=Rτ2m+1,qpbυ2m+1,υ2m+2=qpbRτ2m,Sτ2m+1≤δmaxqpbPτ2m,Qτ2m+1,qpbPτ2m,Rτ2m,qpbQτ2m+1,Sτ2m+1,12ρqpbRτ2m,Qτ2m+1+qpbPτ2m,Sτ2m+1+MminqpbPτ2m,Rτ2m,qpbQτ2m+1,Sτ2m+1,qpbPτ2m,Rτ2m+1,qpbQτ2m+1,Rτ2m≤δmaxqpbυ2m,υ2m+1,qpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2,12ρqpbυ2m+1,υ2m+1+qpbυ2m,υ2m+2+Mminqpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2=δmaxqpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2+Mminqpbυ2m,υ2m+2,qpbυ2m+1,υ2m+1.
This condition gives 4 cases.Case 1.
(9)maxqpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2=qpbυ2m,υ2m+1.Also,(10)minqpbυ2m,υ2m+2,qpbυ2m+1,υ2m+1=qpbυ2m,υ2m+2.which implies(11)ρqpbυ2m+1,υ2m+2≤δqpbυ2m,υ2m+1+Mqpbυ2m,υ2m+2≤δ+Mqpbυ2m,υ2m+1+ρMqpbυ2m+1,υ2m+2≤δ+M1−Mρqpbυ2m,υ2m+1.Letμ1=δ+M/1−Mρ,δ+2M/ρ<1 and M≥0 then μ1<1.Therefore,qpbυ2m+1,υ2m+2≤μ1qpbυ2m,υ2m+1.Case 2.
(12)maxqpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2=qpbυ2m,υ2m+1.Also,(13)minqpbυ2m,υ2m+2,qpbυ2m+1,υ2m+1=qpbυ2m+1,υ2m+1.which implies,(14)qpbυ2m+1,υ2m+2≤1ρδqpbυ2m,υ2m+1+Mqpbυ2m,υ2m+1≤δ+Mρqpbυ2m,υ2m+1.Letμ2=δ+M/ρ,δ+2M/ρ<1 then μ2<1.Therefore,qpbυ2m+1,υ2m+2≤μ2qpbυ2m,υ2m+1.Case 3.
(15)maxqpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2=qpbυ2m+1,υ2m+2.Also,(16)minqpbυ2m,υ2m+2,qpbυ2m+1,υ2m+1=qpbυ2m,υ2m+2.which implies(17)qpbυ2m+1,υ2m+2≤δqpbυ2m+1,υ2m+2+Mqpbυ2m,υ2m+2≤Mρ1−δ−Mqpbυ2m,υ2m+1.Letμ3=M/ρ1−δ−M,δ+2M/ρ<1 then μ3<1.Therefore,qpbυ2m+1,υ2m+2≤μ3qpbυ2m,υ2m+1.Case 4.
(18)maxqpbυ2m,υ2m+1,qpbυ2m+1,υ2m+2=qpbυ2m+1,υ2m+2.Also,(19)minqpbυ2m,υ2m+2,qpbυ2m+1,υ2m+1=qpbυ2m+1,υ2m+1.which implies(20)qpbυ2m+1,υ2m+2≤δqpbυ2m+1,υ2m+2+Mqpbυ2m+1,υ2m+1≤Mρ1−δqpbυ2m,υ2m+1.Letμ4=M/ρ1−δ,δ+2M/ρ<1 then μ4<1.Therefore,qpbυ2m+1,υ2m+2≤μ4qpbυ2m+1,υ2m+2.Chooseμ=maxμ1,μ2,μ3,μ4⇒0<μ<1.(21)⇒qpbυ2m+1,υ2m+2≤μqpbυ2m,υ2m+1.Using mathematical induction,(22)qpbυm,υm+1≤μmqpbυ∗,υ0,which tends to 0 asm tends to ∞So,υm and its subsequence is convergentLetPX be closed. Therefore, τ∈PX, that is, there exists Υ∈X such that τ=PΥ, and we need to show τ=RΥBy definition,(23)qpbRΥ,τ≤δ+MρqpbRΥ,τ,which is a contradiction. Hence,(24)qpbRΥ,τ=0⇒RΥ=τ.So,PΥ=RΥ, that is, P and R have a coincidence point.Similarly,Q and S have a coincidence point.If we also assumeQX is closed, then P,R and Q,S have a coincidence point.SinceP,R and Q,S are weakly compatible, we can prove there exists a common fixed point for P,Q,R,S by contradiction.Example 1.
LetX=0,4 equipped with quasi-partial b-metric qpbτ,υ=τ−υ+τ. Let P,Q,R,S be self-mappings on quasi-partial b-metric defined by(25)Pτ=τ2,τ∈0,2,54,τ∈2,4,Qτ=3τ2,τ∈0,2,32,τ∈2,4Rτ=τ6,τ∈0,2,12,τ∈2,4Sτ=τ4,τ∈0,214,τ∈2,4.
Here,(26)PX=0,12∪54QX=0,32,SX=0,14⊂PXRX=0,16∪12.
The point 0 is a coincidence point of these mapping. Furthermore,PR0=RP0=0 and SQ0=QS0=0, that is, the two pairs P,R and Q,S are weakly compatible.Case 1.
Forτ,υ∈0,2, we have(27)QRτ,Sυ=τ6−υ4+τ6≤85τ2−3υ2+τ2.
Dominance of right-hand side of equation (27) is easily visually checked in Figure 1. Thus the inequality required in Definition 10 holds for τ,υ∈0,2.Figure 1
Dominance of right-hand side of equation (27) is visually checked for τ,υ∈0,2.Case 2.
Forτ∈0,2, υ∈2,4 we have(28)QRτ,Sυ=τ6−14+τ6≤8532−14+32.
Dominance of right-hand side of equation (28) is easily visually checked in Figure 2. Thus the inequality required in Definition 10 holds for τ∈0,2,υ∈2,4.Figure 2
Dominance of right-hand side of equation (28) is visually checked for τ∈0,2,υ∈2,4.Case 3.
Forτ∈2,4,υ∈0,2 we have(29)QRτ,Sυ=12−υ4+12≤8554−3υ2+54.
Dominance of right-hand side of equation (29) is easily visually checked in Figure 3. Thus the inequality required in Definition 10 holds for τ∈2,4,υ∈0,2.Figure 3
Dominance of right-hand side of equation (29) is visually checked for τ∈2,4,υ∈0,2.Case 4.
Forτ,υ∈2,4, we have(30)QRτ,Sυ=12−14+12≤8532−14+32.
Dominance of right-hand side of equation (30) is easily visually checked in Figure 4. Thus the inequality required in Definition 10 holds for τ,υ∈2,4.
As a result, all postulates of Theorem1 are satisfied (δ=4/5, ρ=2≥1, and M=0) and 0 is a unique common fixed point of P,Q,R,S.
IfP=Q and R=S, we get a corollary.Figure 4
Dominance of right-hand side of equation (30) is visually checked for τ,υ∈2,4.Corollary 1.
LetP and S be self-mappings on quasi-partial b-metric space X,qpb. If for all τ,υ∈X,P satisfies the following conditions:(1
)SX⊂PX(2)
PX is closed(3)
δ+2M/ρ<1then P and S have a coincidence point. Also P and S have a common fixed point if P,S are weakly compatible.Proof.
TakingP=Q and R=S in Theorem 1, the above result can be obtained.Theorem 2.
LetP,Q,R,S be self-mappings on a quasi-partial b-metric space X,qpb. If the pair (P, R) is associated with (Q, S) and satisfies(31)qpbRτ,Sυ≤δmaxqpbPτ,Rτ,qpbPτ,Sυ,qpbQυ,Sυ,qpbPτ,Sυ,qpbRτ,Qυ+MminqpbPτ,Rτ,qpbQυ,Sυ,qpbPτ,Sυ,qpbQυ,Rτ,
with δ ∈ (0, 1), M ≥ 0, and p ≥ 1, for all τ, υ ∈X, and(1)
SX⊂PX and RX⊂QX(2)
δ+2M/ρ<1then the pairs P,R and Q,S have a coincidence point. Also P,Q,R,S have a common fixed point.Proof.
This can be done following the same steps as the proof of Theorem1.Example 2.
LetX=0,∞ equipped with quasi-partial b-metric qpbτ,υ=τ−υ+τ. Let P,Q,R,S be self-mappings on quasi-partial b-metric defined by(32)Pτ=2τ,τ∈0,2,4,τ>2,Qτ=τ,τ∈0,2,2,τ>2,Rτ=τ5,τ∈0,2,2,τ>2,Sτ=2τ5,τ∈0,2,1,τ>2.
Here,(33)SX=0,25∪1∪0,2∪4=PX,RX=0,15∪2⊂0,1∪2=QX.
The point 0 is a coincidence point of these mapping. Furthermore,PR0=RP0=0 and SQ0=QS0=0, that is, the two pairs P,R and Q,S, are weakly compatible.Case 1.
Forτ,υ∈0,2, we have(34)QRτ,Sυ=τ5−2υ5+τ5=154τ−2υ+2τ≤1092τ−υ+2τ.
Dominance of the right-hand side of equation (34) is easily visually checked in Figure 5. Thus the inequality required in theorem holds for τ,υ∈0,2.Figure 5
Dominance of the right-hand side of equation (34) is visually checked for τ,υ∈0,2.Case 2.
Forτ∈0,2,υ>2 we have(35)QRτ,Sυ=τ5−1+τ5≤1092−1+2.
Dominance of the right-hand side of equation (35) is easily visually checked in Figure 6. Thus the inequality required in theorem holds for τ∈0,2,υ>2.Figure 6
Dominance of the right-hand side of equation (35) is visually checked for τ∈0,2,υ>2.Case 3.
Forτ>2,υ∈0,2, we have(36)QRτ,Sυ=2−2υ5+2≤1094−2υ5+4.
Dominance of the right-hand side of equation (36) is easily visually checked in Figure 7. Thus the inequality required in theorem holds for τ>2, υ∈0,2.Figure 7
Dominance of the right-hand side of equation (36) is visually checked for τ>2, υ∈0,2.Case 4.
Forτ,υ>2, we have(37)QRτ,Sυ=2−1+2≤309.
Dominance of the right-hand side of equation (37) is easily visually checked in Figure 8. Thus the inequality required in theorem holds for τ,υ>2.
As a result, all postulates of Theorem2 are satisfied (δ=5/9, ρ=2≥1, and M=0), and 0 is a unique common fixed point of P,Q,R,S.
IfP=Q and R=S, we get a corollary.Figure 8
Dominance of the right-hand side of equation (37) is visually checked for τ,υ>2.Corollary 2.
LetP and S be self-mappings on quasi-partial b-metric space X,qpb. If for all τ,υ∈X, the pair of mapping P,S satisfies(38)qpbRτ,Sυ≤δmaxqpbPτ,Rτ,qpbPτ,Sυ,qpbQυ,Sυ,qpbPτ,Sυ,qpbRτ,Qυ+MminqpbPτ,Rτ,qpbQυ,Sυ,qpbPτ,Sυ,qpbQυ,Rτ,and P satisfies the following conditions:(1)
SX⊂PX(2)
δ+2M/ρ<1then the pair P,S has a coincidence point. Also P and S have a common fixed point if P,S are weakly compatible.Proof.
TakingP=Q and R=S in Theorem 2, the above result can be obtained.


## 4. Conclusion

This paper expounds a new notion in quasi-partial b-metric space which is generalized condition (B) that helped to demonstrate coincidence and common fixed point for two weakly compatible pairs of self-mappings. The incentive behind using quasi-partial b-metric space is the fact that the distance from pointx to point y may be different to that from y to x, and the self-distance of a point need not always be zero; also the distance between two points x and z is not equal to the sum of the two distances having a point y in between x and z. Furthermore, the results acquired are validated by explanatory examples.



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