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 thatClearly 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 asThis 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 byHere,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 haveDominance of right-hand side of equation (27) is easily visually checked in Figure 1. Thus the inequality required in Definition 10 holds for .
Case 2. For , we haveDominance of right-hand side of equation (28) is easily visually checked in Figure 2. Thus the inequality required in Definition 10 holds for .
Case 3. For we haveDominance of right-hand side of equation (29) is easily visually checked in Figure 3. Thus the inequality required in Definition 10 holds for .
Case 4. For , we haveDominance 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.
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 satisfieswith 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 byHere,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 haveDominance of the right-hand side of equation (34) is easily visually checked in Figure 5. Thus the inequality required in theorem holds for .
Case 2. For we haveDominance of the right-hand side of equation (35) is easily visually checked in Figure 6. Thus the inequality required in theorem holds for .
Case 3. For , we haveDominance of the right-hand side of equation (36) is easily visually checked in Figure 7. Thus the inequality required in theorem holds for , .
Case 4. For , we haveDominance 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.
Corollary 2. Let and be self-mappings on quasi-partial b-metric space . If for all , the pair of mapping satisfiesand 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.