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

In the early years of 20 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.


Introduction
In the early years of 20 th 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][10][11][12][13][14][15][16][17][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][21][22][23][24][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.

Preliminaries
Let us recall some definition.
Definition 1 (see [19]). A partial metric space on a nonempty set X is a function M: X × X ⟶ R + satisfying Definition 2 (see [4]). A quasi-partial metric on a nonempty set X is a function q: X × X ⟶R + satisfying Definition 3 (see [20]). A quasi-partial b-metric on a nonempty set X is a function qp b : X × X ⟶ R + such that for some real number ρ ≥ 1 for all τ, υ, Υ ∈ X. e infimum over all reals ρ ≥ 1 satisfying condition (30) is called the coefficient of (X, qp b ) and represented by R(X, qp b ).
Lemma 1 (see [6]). Let (X, qp b ) be a quasi-partial b-metric space. en the following hold: Definition 4 (see [6]). Let (X, qp b ) be a quasi-partial b-metric. en (1) A sequence τ n ⊂ X converges to τ ∈ X if and only if (1) (3) e quasi-partial b-metric space (X, qp b ) is said to be complete if every Cauchy sequence τ n ⊂ X converges with respect to x qp b to a point τ ∈ X such that (4) A mapping f: X ⟶ X is said to be continuous at Lemma 2 (see [6]). Let (X, qp b ) be a quasi-partial b-metric space and (X, Definition 5 (see [26] 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 P and Q be two self-mappings on a metric space (X, d). e mapping Q satisfies generalized condition (B) associated with P if there exist δ ∈ (0, 1) and Clearly condition (B) implies generalized condition (B). e mapping R 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 Definition 10. Let P, Q, R, S be four self-mappings on a quasi-partial b-metric space (X, qp b ). e pair of mapping (P, R) satisfies generalized condition (B) associated with (Q, S) ((P, R) is generalized almost Theorem 1. Let P, Q, R, S be four self-mappings on quasipartial b-metric space (X, qp b ) 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 then 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. Let τ * ∈ X. Since RX ⊂ QX there exists τ 0 ∈ X such that υ 0 � Qτ 0 � Rτ * . Suppose there exists a point υ 1 ∈ Sτ 0 corresponding to the point υ 0 . Also since SX ⊂ PX there exist τ 1 ∈ X such that υ 1 � Pτ 1 � Sτ 0 . Going this way we get a sequence υ n ∈ X as is condition gives 4 cases. Also, which implies Let Also, which implies, Let μ 2 � ((δ + M)/ρ), ((δ + 2M)/ρ) < 1 then μ 2 < 1. Also, which implies Let Also, which implies Let Using mathematical induction, which tends to 0 as m tends to ∞ So, υ m and its subsequence is convergent Let PX be closed. erefore, τ ∈ PX, that is, there exists Υ ∈ X such that τ � PΥ, and we need to show τ � RΥ By definition, which is a contradiction. Hence, 4 Journal of Mathematics So, PΥ � RΥ, that is, P and R have a coincidence point. Similarly, Q and S have a coincidence point. If we also assume QX is closed, then (P, R) and (Q, S) have a coincidence point. Since (P, R) and (Q, S) are weakly compatible, we can prove there exists a common fixed point for P, Q, R, S by contradiction.
Let P, Q, R, S be selfmappings on quasi-partial b-metric defined by Here, e 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.
Dominance of right-hand side of equation (27) is easily visually checked in Figure 1. us the inequality required in Definition 10 holds for τ, υ ∈ [0, 2].
Dominance of right-hand side of equation (30) is easily visually checked in Figure 4. us the inequality required in Definition 10 holds for τ, υ ∈ [2,4].
If P � Q and R � S, we get a corollary. Corollary 1. Let P and S be self-mappings on quasi-partial b-metric space (X, qp b ). If for all τ, υ ∈ X, P satisfies the following conditions: then P and S have a coincidence point. Also P and S have a common fixed point if (P, S) are weakly compatible.
Proof. Taking P � Q and R � S in eorem 1, the above result can be obtained.  Proof. is can be done following the same steps as the proof of eorem 1.
Dominance of the right-hand side of equation (34) is easily visually checked in Figure 5. us the inequality required in theorem holds for τ, υ ∈ [0, 2].
Dominance of the right-hand side of equation (35) is easily visually checked in Figure 6. us the inequality required in theorem holds for τ ∈ [0, 2], υ > 2.

Journal of Mathematics
Dominance of the right-hand side of equation (36) is easily visually checked in Figure 7. us the inequality required in theorem holds for τ > 2, υ ∈ [0, 2].
Dominance of the right-hand side of equation (37) is easily visually checked in Figure 8. us the inequality required in theorem holds for τ, υ > 2.
If P � Q and R � S, we get a corollary.   Proof. Taking P � Q and R � S in eorem 2, the above result can be obtained.

Conclusion
is 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. e 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.