Journal of Mathematics
Volume 2021 (2021), Article ID 5526801, 10 pages
https://doi.org/10.1155/2021/5526801

Research Article

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

Pragati Gautam,¹ Luis Manuel Sánchez Ruiz,² Swapnil Verma,¹ and Gauri Gupta¹

¹Department of Mathematics, Kamala Nehru College (University of Delhi), August Kranti Marg, New Delhi 110049, India
²ETSID-Departamento de Matematica Aplicada CITG, Universitat Politecnica de Valencia, Valencia E-46022, Spain

Correspondence should be addressed to Luis Manuel Sánchez Ruiz; lmsr@mat.upv.es

Received 21 February 2021; Revised 14 April 2021; Accepted 31 May 2021; Published 11 June 2021

Academic Editor: Huseyin Isik

Copyright © 2021 Pragati Gautam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

In the early years of 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 set is a function satisfying

(1) (symmetry)(2)if , then (indistancy implies equality)(3), then (small self-distances)(4) (triangularity)

for all

Definition 2 (see [4]). A quasi-partial metric on a nonempty set is a function X satisfying

(1)If , then (indistancy implies equality)(2) (small self-distances)(3) (small self-distances)(4) (triangularity)

for all

Definition 3 (see [20]). A quasi-partial b-metric on a nonempty set is a function such that for some real number

(1)If then (indistancy implies equality)(2) (small self-distances)(3) (small self-distances)(4) (triangularity)

for all The infimum over all reals satisfying condition (30) is called the coefficient of and represented by .

Lemma 1 (see [6]). Let be a quasi-partial b-metric space. Then the following hold

(1)If , then (2)If , then and

Definition 4 (see [6]). Let be a quasi-partial b-metric. Then

(1)A sequence converges to if and only if

(2)A sequence is called a Cauchy sequence if and only if

(3)The quasi-partial b-metric space is said to be complete if every Cauchy sequence converges with respect to to a point such that

(4)A mapping is said to be continuous at if for every , there exists such that

Lemma 2 (see [6]). Let be a quasi-partial b-metric space and be the corresponding b-metric space. Then is complete if is complete.

Definition 5 (see [26]). A self-mapping on a metric space satisfies condition (B), if there exist and such that for all , we have,

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]). Let and be two self-mappings on a metric space . The mapping satisfies generalized condition (B) associated with if there exist and with such that

Clearly condition (B) implies generalized condition (B).

Definition 7 (see [29]). Let and be self-mappings on a set . A point is called a coincidence point of and if , where is called a point of coincidence of and .

Definition 8 (see [30]). Let be a nonempty set. Two mappings , are said to be weakly compatible if they commute at their coincidence point, that is, if for some , then .

3. Main Results

Definition 9. Let P and R be two self-mappings on a quasi-partial b-metric space .
The mapping satisfies generalized condition (B) associated with P (R is a generalized almost P contraction) if there exist , and such that for all , we have

Definition 10. Let be four self-mappings on a quasi-partial b-metric space .
The pair of mapping satisfies generalized condition (B) associated with ((P, R) is generalized almost (Q, S) contraction) if there exist and such that for all , we have

Theorem 1. Let be four self-mappings on quasi-partial b-metric space and if we take the mappings in pair as associated with for all , , and , and

(1) and (2) or is closed(3)

then the pairs and have a coincidence point. Also have a unique common fixed point, providing that pairs and are weakly compatible.

Proof. Let Since there exists such that Suppose there exists a point corresponding to the point Also since there exist such that . Going this way we get a sequence as

This condition gives 4 cases

Case 1.

Also,

which implies

 Let and then  Therefore,

Case 2.

Also,

which implies,

 Let then . Therefore,

Case 3.

Also,

which implies

 Let then . Therefore,

Case 4.

Also,

which implies

 Let then . Therefore,  Choose .

 Using mathematical induction,

 which tends to 0 as m tends to  So, and its subsequence is convergent Let be closed. Therefore, , that is, there exists such that , and we need to show  By definition,

 which is a contradiction. Hence,

 So, , that is, and have a coincidence point. Similarly, and have a coincidence point. If we also assume is closed, then and have a coincidence point. Since and are weakly compatible, we can prove there exists a common fixed point for by contradiction.

Example 1. Let equipped with quasi-partial b-metric . Let be self-mappings on quasi-partial b-metric defined by

Here,

The point 0 is a coincidence point of these mapping. Furthermore, and , that is, the two pairs and are weakly compatible.

Case 1. For , we have

Dominance of right-hand side of equation (27) is easily visually checked in Figure 1. Thus the inequality required in Definition 10 holds for .

Figure 1: Dominance of right-hand side of equation (27) is visually checked for .

Case 2. For , we have

Dominance of right-hand side of equation (28) is easily visually checked in Figure 2. Thus the inequality required in Definition 10 holds for .

Figure 2: Dominance of right-hand side of equation (28) is visually checked for .

Case 3. For we have

Dominance of right-hand side of equation (29) is easily visually checked in Figure 3. Thus the inequality required in Definition 10 holds for .

Figure 3: Dominance of right-hand side of equation (29) is visually checked for .

Case 4. For , we have

Dominance of right-hand side of equation (30) is easily visually checked in Figure 4. Thus the inequality required in Definition 10 holds for .
As a result, all postulates of Theorem 1 are satisfied (, , and ) and 0 is a unique common fixed point of .
If and , we get a corollary.

Figure 4: Dominance of right-hand side of equation (30) is visually checked for .

Corollary 1. Let and be self-mappings on quasi-partial b-metric space . If for all satisfies the following conditions

(1)(2) is closed(3)

then and have a coincidence point. Also and have a common fixed point if are weakly compatible.

Proof. Taking and in Theorem 1, the above result can be obtained.

Theorem 2. Let be self-mappings on a quasi-partial b-metric space . If the pair (P, R) is associated with (Q, S) and satisfies

with δ ∈ (0, 1), M ≥ 0, and p ≥ 1, for all τ, υ ∈X, and

(1) and (2)

then the pairs and have a coincidence point. Also have a common fixed point.

Proof. This can be done following the same steps as the proof of Theorem 1.

Example 2. Let equipped with quasi-partial b-metric . Let be self-mappings on quasi-partial b-metric defined by

Here,

The point 0 is a coincidence point of these mapping. Furthermore, and , that is, the two pairs and , are weakly compatible.

Case 1. For , we have

Dominance of the right-hand side of equation (34) is easily visually checked in Figure 5. Thus the inequality required in theorem holds for .

Figure 5: Dominance of the right-hand side of equation (34) is visually checked for .

Case 2. For we have

Dominance of the right-hand side of equation (35) is easily visually checked in Figure 6. Thus the inequality required in theorem holds for .

Figure 6: Dominance of the right-hand side of equation (35) is visually checked for .

Case 3. For , we have

Dominance of the right-hand side of equation (36) is easily visually checked in Figure 7. Thus the inequality required in theorem holds for , .

Figure 7: Dominance of the right-hand side of equation (36) is visually checked for , .

Case 4. For , we have

Dominance of the right-hand side of equation (37) is easily visually checked in Figure 8. Thus the inequality required in theorem holds for .
As a result, all postulates of Theorem 2 are satisfied (, , and ), and 0 is a unique common fixed point of .
If and , we get a corollary.

Figure 8: Dominance of the right-hand side of equation (37) is visually checked for .

Corollary 2. Let and be self-mappings on quasi-partial b-metric space . If for all , the pair of mapping satisfies

and satisfies the following conditions

(1)(2)

then the pair has a coincidence point. Also and have a common fixed point if are weakly compatible.

Proof. Taking and 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 point x 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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