Common Fixed-Point Results in Ordered Left (Right) Quasi-b-Metric Spaces and Applications

We use the notions of leftand right-complete quasi-b-metric spaces and partial ordered sets to obtain a couple of common fixedpoint results for strictly weakly isotone increasing mappings and relatively weakly increasing mappings, which satisfy a pair of almost generalized contractive conditions. To illustrate our results, throughout the paper, we give several relevant examples. Further, we use our results to establish sufficient conditions for existence and uniqueness of solution of a system of nonlinear matrix equations and a pair of fractional differential equations. Finally, we provide a nontrivial example to validate the sufficient conditions for nonlinear matrix equations with numerical approximations.


Introduction and Preliminaries
We denote by R the set of real numbers; R + � [0, +∞); we denote by N the set of natural numbers and N * � N ∪ 0 { }. Also, for the mappings T, S, R: Ξ ⟶ Ξ, we denote by CFP (T, S) and CFP (T, S, R) the set of all common fixed points of T, S and T, S, R, respectively. e metric fixed-point theory has been extended in many directions by many renowned mathematicians. One important direction of such ones is to revise the underlying metric space to some other spaces by making suitable changes obtained by Czerwik. He introduced the notion of b-metric spaces (see [1]), which is further extended as quasib-metric spaces by Shah and Hussain [2].
Definition 1 (see [2]). Let Ξ( ≠ ∅) be a set and let b ≥ 1 be a given real number. A function d b : Ξ 2 ⟶ R is a quasib-metric on Ξ if, for all ζ, ξ, ς ∈ Ξ, e pair (Ξ, d b ) is then termed as a quasi-b-metric space with constant b.
It is to be noted that every metric space is quasi-metric space, and quasi-metric space is a quasi-b-metric space but the converses need not be true. e above space is further extended with the introduction of right and left quasib-metric spaces (in the line of [3]).
Definition 2 (see [4]). Let (Ξ, d b ) be a quasi-b-metric space and let ϑ n be a sequence in Ξ. en ϑ n is said to be (i) left-complete if every left-Cauchy sequence in Ξ is convergent (ii) right-complete if every right-Cauchy sequence in Ξ is convergent On the other hand, an extension of fixed-point results for various types of contractions in metric spaces is secured by adding an (partial) ordering structure on the underlying structure (Ξ, d). Some early results in this direction were established by Turinici in [5,6]; one may note that their starting points were "amorphous" contributions in the area due to Matkowski [7,8]. ese types of results have been reinvestigated by Ran and Reurings [9] and also by Nieto and Ródríguez-López [10,11]. In 2019, Gu and Shatanawi [12] obtained some common coupled fixed-point results in partial metric spaces and some recent results of Latif et al. [13] and Malhotra et at. [14] are also important. In [15], Nashine et al. used the concept of T-weakly isotone increasing mappings to extendĆirić's [16] result in ordered metric spaces. e main importance of their results is that they obtained their results without considering any kind of commutativity condition. After all such generalizations and extensions, Nashine and Altun [17] introduced a new notion of increasing mapping, which they designated as T-strictly weakly isotone increasing mapping, and then they obtained some results by considering this new type of increasing mappings. After this, in [18], Nashine and Samet introduced relatively weakly increasing mappings and proved some fixed-point results in ordered metric spaces and applied their results to integral equations. In this sequel, we like to recall some useful definitions in the context of a partially ordered set (Ξ, ≺ ).
(2) the pair (S, T) is called weakly increasing if Sζ ≺ TSζ and Tζ ≺ STζ for each ζ ∈ Ξ. (3) S is called T-weakly isotone increasing if, for each ζ ∈ Ξ, we have Sζ ≺ TSζ ≺ STSζ. (4) the mapping S is said to be T-strictly weakly isotone increasing if, for ζ ∈ Ξ satisfying ζ ≺ Sζ, we have Sζ ≺ TSζ ≺ STSζ. (5) T and S are said to be weakly increasing with respect to R if TΞ ⊆ RΞ and SΞ ⊆ RΞ and, for each ζ ∈ Ξ, we have Let (Ξ, d b ) be a b-metric space. en two mappings T, S: Ξ ⟶ Ξ are said to be compatible if lim n⟶∞ d b (TSς n , STς n ) � 0, for each sequence ς n in Ξ with lim n⟶∞ d b (Tς n , μ) � 0 and lim n⟶∞ d b (Sς n , μ) � 0 for some μ ∈ Ξ. If (Ξ, d b ) is a quasi-b-metric space and (Ξ, ≺ ) is a partially ordered set, then the triplet is called an ordered quasi-b-metric space. e space (Ξ, d b , ≺ ) is called regular if whenever ς n is a nondecreasing sequence in Ξ with respect to ≺ and ς n ⟶ ς ∈ Ξ as n ⟶ ∞, then ς n ≺ ς holds.
In the literature of fixed point, one may note that, to find common fixed point of two or more mappings in the setting of different abstract spaces, more specifically in left-and right-complete quasi-b-metric spaces, commutativity condition of the mappings plays crucial roles. So it is a challenging work to obtain common fixed point of two or more mappings in such spaces without considering the commutativity condition. One of the main motivations of the paper is to resolve this issue. To proceed with this, we utilize the approaches of Nashine and Altun [17] and Nashine and Samet [18] to obtain some common fixed-point results in the setting of ordered left-complete and right-complete quasib-metric spaces. Firstly, we establish some common fixedpoint theorems for a pair of mappings using the T-strictly weakly isotone increasing condition and without using any kind of commutativity condition in ordered left-complete quasi-b-metric spaces. Secondly, we obtain a common fixedpoint result for a triplet of mappings satisfying relatively weakly increasing condition and almost generalized contractive conditions in ordered right-complete quasi-b-metric spaces.
Another important motivation of this paper is to show how we can apply our obtained results in at least two different applicable areas. ese are connected to get solutions of a pair of nonlinear matrix equations and also a pair of fractional differential equations. Further, we provide some nontrivial examples to illustrate our obtained results. Finally, our attempts give extensions of the works discussed in [2, 3, 9-11, 15, 18, 21] and other related results in the sense of generalized contractive conditions and generalized weakly increasing mappings in the crucial setting with new applications to the functional equations.

Results for Pair of Mappings
In this section, at first, we prove a common fixed-point result of a pair of mappings involving T-strictly weakly isotone increasing condition. Before this, we state the following important lemma regarding the left-(right-) Cauchyness of a sequence in quasi-b-metric context.
be a quasi-b-metric space and let x n be a sequence in Ξ. en, we have the following: then x n is a left-Cauchy sequence. (2) If there exists r ∈ [0, 1) satisfying then x n is a right-Cauchy sequence.

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Proof.
e proof of this lemma can be done in the line of ( [22], p. 3, Lemma [23]).
From the left-completeness of Ξ, there exists ϱ ∈ Ξ such that ] n ⟶ ϱ as n ⟶ + ∞. Clearly, if S or T is continuous, then ϱ � S9 or ϱ � T9. us, CFP(T, S) ≠ ∅.By the next result, we show that the continuity of S or T in the previous theorem can be replaced by some other conditions. Theorem 2. If one replaces the continuity in eorem 1 by regularity of Ξ, then conclusion of eorem 1 is valid provided αb < 1.
Next, we characterize the common fixed-points set CFP(T, S) in the following theorem. Proof. First, we assume that CFP(T, S) is totally ordered. Let ϱ, σ ∈ CFP(T, S) with ϱ ≠ σ. Consider (4) for ] � σ and ϑ � ϱ, and we get that is, Again, using (5) for ] � u and ϑ � σ and by calculation we get Adding (22) and (23), we get which gives a contradiction. us, CFP(T, S) is singleton. e converse is trivial. Putting S � T in eorem 3, we obtain the following result.

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Hence, from all three cases, we have us, all the conditions of eorem 1 are fulfilled and the pair T, S has a unique common fixed point (which is 0 ∈ CFP(T, S)).

Results for Three Mappings
In this section, we obtain a result for three mappings involving weakly increasing condition.
Let T, S, R: Ξ ⟶ Ξ be three mappings such that TΞ ⊆ RΞ and SΞ ⊆ RΞ, and for all comparable R], Ry ∈ Ξ, where α ∈ [0, 1), L ≥ 0, and We assume the following hypotheses: (i) T and S are weakly increasing with respect to R (ii) T and S are dominating maps Assume either of the following: Proof. Start with defining a sequence ] n in Ξ as We claim that Using hypothesis (i) and (41), Since R] 1 � S] 0 , ] 1 ∈ R − 1 (S] 0 ), and we get Again, Since Hence, by induction, (42) holds.
erefore, we conclude that (47) holds for all n ∈ N; that is, erefore, from Lemma 1, it follows that R] n is a right-Cauchy sequence. From the right-completeness of Ξ, there exists ϱ ∈ Ξ such that R] n ⟶ ϱ as n ⟶ ∞. (54) We will prove that ϱ is a common fixed point of the three mappings S, T, and R.

Application to Nonlinear Matrix Equations
In this section, we will apply the common fixed-point results in quasi-b-metric spaces of the previous section to obtain variants of the results of Garai and Dey [23] on existence of common solution to systems of NMEs. For other variants on solution to systems of NMEs, one is referred to [24,25]. For a matrix A, any singular value of A will be denoted by s(A), and the sum of these values, that is, the trace norm of A, will be denoted by s + (A) � ‖A‖. We will use the standard partial order on H(n) given by A≽B if and only if A − B is a positive semidefinite matrix. We define a function Utilizing this quasi-b-metric space, we now prove the following theorem regarding the solution (s) of a pair of nonlinear matrix equations.

Theorem 5. Consider the system
where B 1 , B 2 ∈ P(n), A i ∈ M(n), i � 1, . . . , k, and the operators f, g: H(n) ⟶ H(n) are continuous in the trace norm. Let, for some M, N 1 ∈ R, and, for any X ∈ P(n) with ‖X‖ ≤ M, s(f(X)), s(g(X)) ≤ N 1 hold for all singular values of f(X) and g(X), respectively. Assume the following: en system (82) has a solution, and if X is a solution of the system, then X ∈ P(n) with ‖X‖ ≤ M. Further, the iterative sequence X n , where, for j ≥ 0, and X 0 is an arbitrary element of H(n) satisfying ‖X 0 ‖ ≤ M, converges to a unique solution of the system, if X j ≺ X j+1 or X j+1 ≺ X j .
Proof. Let us consider the set Ξ � X ∈ H(n): ‖X‖ ≤ M { }. en, Ξ is right-complete with respect to the metric d b . For any X ∈ Ξ, we have Since ‖X‖ ≤ M, we have s(f(X)) ≤ N 1 for all singular values s(f(X)) of f(X) so, by summing the n singular values of f(X), we get ‖f(X)‖ ≤ nN 1 . Using this in (86), we get Similarly, for any X ∈ Ξ, we can show that erefore, the mappings T, S defined on Ξ by for X ∈ Ξ are self-maps on Ξ. Next, for any X ∈ Ξ, using assumption (3), we have X ≺ T(X) and X ≺ S(X). So the mappings T, S are dominating mappings. Again, for X ∈ Ξ, we have T(X) ∈ Ξ and S(X) ∈ Ξ. So, for any X ∈ Ξ, using assumption (4), we have S(X) ≺ TS(X) and T(X) ≺ ST(X). erefore, T, S are weakly increasing mappings. We choose α � (17/18). Now let X, Y ∈ Ξ be arbitrary such that either X ≺ Y or Y ≺ X. en, we have Journal of Mathematics 13 If R 11 (X, Y) � |s + (X − (1/2)Y)| 2 , then we have If Again if or then we can show that where, To see the convergence of the sequence X n defined in (85), we start with initial values, 16 Journal of Mathematics
is implies that T and S are weakly increasing. Using (III), for all t ∈ J, T and S are dominating operators.
To check contraction conditions, we start with