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

Authors: Hemant Kumar Nashine; Sourav Shil; Hiranmoy Garai; Lakshmi Kanta Dey; Vahid Parvaneh
Journal: Journal of Mathematics
 (2020)

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## Abstract

We use the notions of left- and right-complete quasi-b-metric spaces and partial ordered sets to obtain a couple of common fixed-point 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.


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## Body

## 1. Introduction and Preliminaries

We denote byℝ 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.The 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 ofb-metric spaces (see [1]), which is further extended as quasi-b-metric spaces by Shah and Hussain [2].Definition 1 (see [2]).
LetΞ≠∅ be a set and let b≥1 be a given real number. A function db:Ξ2⟶ℝ is a quasi-b-metric on Ξ if, for all ζ,ξ,ς∈Ξ,(M1)dbζ,ξ≥0(M2)dbζ,ξ=0⇔ζ=ξ(M3)dbζ,ς≤bdbζ,ξ+dbξ,ς
The pairΞ,db 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. The above space is further extended with the introduction of right and left quasi-b-metric spaces (in the line of [3]).Definition 2 (see [4]).
LetΞ,db be a quasi-b-metric space and let ϑn be a sequence in Ξ. Then ϑn is said to be(i)
left-Cauchy if, for everyδ>0, we get N=Nδ∈ℕ such that dbϑr,ϑs<δ for all r>s>N(ii)
right-Cauchy if, for everyδ>0, we get N=Nδ∈ℕ such that dbϑr,ϑs<δ for all s>r>NDefinition 3 (see [4]).
LetΞ,db be a quasi-b-metric space. Then Ξ,db is called(i)
left-complete if every left-Cauchy sequence inΞ is convergent(ii)
right-complete if every right-Cauchy sequence inΞ 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 structureΞ,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]. These 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 extend Ćirić’s [16] result in ordered metric spaces. The 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,≺¯.Definition 4 (see [17–20]).
LetΞ,≺¯ be a partially ordered set and let S,T,ℛ:Ξ⟶Ξ be three mappings. Then,(1)
T is called dominating if ζ≺¯Tζ for each ζ∈Ξ.(2)
the pairS,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 mappingS 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(1)Tζ≺¯Sξ,for allξ∈ℛ−1Tζ,Sζ≺¯Tξ,for allξ∈ℛ−1Sζ.
LetΞ,db be a b-metric space. Then two mappings T,S:Ξ⟶Ξ are said to be compatible if limn⟶∞dbTSςn,STςn=0, for each sequence ςn in Ξ with limn⟶∞dbTςn,μ=0 and limn⟶∞dbSςn,μ=0 for some μ∈Ξ. If Ξ,db is a quasi-b-metric space and Ξ,≺¯ is a partially ordered set, then the triplet Ξ,db,≺¯ is called an ordered quasi-b-metric space. The space Ξ,db,≺¯ is called regular if whenever ςn is a nondecreasing sequence in Ξ with respect to ≺¯ and ςn⟶ς∈Ξ as n⟶∞, then ςn≺¯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 quasi-b-metric spaces. Firstly, we establish some common fixed-point 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 fixed-point 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. These 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.


## 2. Results for Pair of Mappings

In this section, at first, we prove a common fixed-point result of a pair of mappings involvingT-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.Lemma 1.
LetΞ,db be a quasi-b-metric space and let xn be a sequence in Ξ. Then, we have the following:(1)
If there existsr∈0,1 satisfying(2)dbxn+2,xn+1≤rdbxn+1,xn,for alln∈ℕ,thenxn is a left-Cauchy sequence.(2)
If there existsr∈0,1 satisfying(3)dbxn+1,xn+2≤rdbxn,xn+1,for alln∈ℕ,then xn is a right-Cauchy sequence.Proof.
The proof of this lemma can be done in the line of ([22], p. 3, Lemma [23]).Theorem 1.
LetΞ,db,≺¯ be an ordered left-complete quasi-b-metric space with constant b≥1. Suppose that T,S:Ξ⟶Ξ are two mappings such that(4)dbTν,Sϑ≤αΛν,ϑ+Lmindbν,Tν,dbϑ,Sϑ,dbν,Sϑ,dbϑ,Tν,(5)dbSν,Tϑ≤αϒν,ϑ+Lmindbν,Sν,dbϑ,Tϑ,dbν,Tϑ,dbϑ,Sν,for all comparable ν,ϑ∈Ξ, where α∈0,1, L≥0, and(6)Λν,ϑ=maxdbν,ϑ,dbTν,ν,dbSϑ,ϑ+dbTν,ν2,dbTν,ϑ+dbSϑ,ν2b,(7)ϒν,ϑ=maxdbν,ϑ,dbSν,ν,dbTϑ,ϑ+dbSν,ν2,dbSν,ϑ+dbTϑ,ν2b,.
In addition, letS be T-strictly weakly isotone increasing, there exists an ν0∈Ξ such that ν0≺Sν0, and one of S and T is continuous. Then CFPT,S≠∅.Proof.
Letν0 be such that ν0≺Sν0 and construct a sequence νn in Ξ satisfying(8)ν2n+1=Sν2nandν2n+2=Tν2n+1forn∈0,1,….
AsS is T-weakly isotone increasing,(9)ν1=Sν0≺TSν0=Tν1=ν2≺STSν0=STν1=Sν2=ν3,ν3=Sν2≺TSν2=Tν3=ν4≺STSν2=STν3=Sν4=ν5,and, proceeding with this argument, we get(10)ν1≺ν2≺⋯≺νn≺νn+1≺….
Letδn=dbνn+1,νn. Then, for all n∈ℕ, we show that(11)δn<αδn−1.
From (10), we have that νn≺νn+1 for all n∈ℕ. Then, from (4), with ν=ν2n+1 and ϑ=ν2n, we get(12)dbν2n+2,ν2n+1=dbTν2n+1,Sν2n,≤αΛν2n+1,ν2n,+Lmindbν2n+1,Tν2n+1,dbν2n,Sν2n,dbν2n+1,Sν2n,dbν2n,Tν2n+1,⟹dbν2n+2,ν2n+1≤αΛν2n+1,ν2n.
By (8), we have(13)Λν2n+1,ν2n=maxdbν2n+1,ν2n,dbTν2n+1,ν2n+1,dbSν2n,ν2n+dbTν2n+1,ν2n+12,dbTν2n+1,ν2n+dbSν2n,ν2n+12b,=maxdbν2n+1,ν2n,dbν2n+2,ν2n+1,dbν2n+1,ν2n+dbν2n+2,ν2n+12,dbν2n+2,ν2n+dbν2n+1,ν2n+12b,=maxdbν2n+1,ν2n,dbν2n+2,ν2n+1,dbν2n+1,ν2n+dbν2n+2,ν2n+12,dbν2n+2,ν2n2b.(i)
IfΛν2n+1,ν2n=dbν2n+2,ν2n+1, by (12), we have(14)dbν2n+2,ν2n+1≤αdbν2n+2,ν2n+1<dbν2n+2,ν2n+1,a contradiction.(ii)
IfΛν2n+1,ν2n=dbν2n+2,ν2n/2b, we have(15)dbν2n+2,ν2n+1≤αdbν2n+2,ν2n2b,≤α2dbν2n+2,ν2n+1+dbν2n+1,ν2n⟹dbν2n+2,ν2n+1,≤α2−αdbν2n+1,ν2n⟹dbν2n+2,ν2n+1,≤αdbν2n+1,ν2n⟹δ2n+1,≤αδ2n.(iii)
IfΛν2n+1,ν2n=dbν2n+1,ν2n, we have(16)dbν2n+2,ν2n+1≤αdbν2n+1,ν2n⟹δ2n+1≤αδ2n.(iv)
IfΛν2n+1,ν2n=dbν2n+1,ν2n+dbν2n+2,ν2n+1/2, we have(17)dbν2n+2,ν2n+1≤αdbν2n+1,ν2n+dbν2n+2,ν2n+12⟹dbν2n+2,ν2n+1,≤α2−αdbν2n+1,ν2n⟹dbν2n+2,ν2n+1,≤αdbν2n+1,ν2n⟹δ2n+1,≤αδ2n.
Consequently,δ2n+1<αδ2n for all n∈ℕ. Similarly, using (5) with (7), we can show that δ2n<αδ2n−1. Therefore, (11) holds for all n∈ℕ and so, from Lemma 1, we can conclude that νn is a left-Cauchy sequence.
From the left-completeness ofΞ, there exists ϱ∈Ξ such that νn⟶ϱ as n⟶+∞. Clearly, if S or T is continuous, then ϱ=Sϱ or ϱ=Tϱ. Thus, CFPT,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 Theorem1 by regularity of Ξ, then conclusion of Theorem 1 is valid provided αb<1.Proof.
Following the lines of proof of Theorem1, we have that there exists ϱ∈Ξ such that(18)limn⟶+∞νn=limn⟶+∞Sν2n=limn⟶+∞Tν2n+1=ϱ.
Using (5) for ϑ=ν2n+1 and ν=ϱ, we have(19)dbSϱ,ϱ≤bdbSϱ,Tν2n+1+dbTν2n+1,ϱ,≤αbmaxdbϱ,ν2n+1,dbSϱ,ϱ,dbTν2n+1,ν2n+1+dbSϱ,ϱ2,dbSϱ,ν2n+1+dbTν2n+1,ϱ2b,+Lbmindbϱ,Sϱ,dbν2n+1,Tν2n+1,dbϱ,Tν2n+1,dbν2n+1,Sϱ,+bdbTν2n+1,ϱ.
Lettingn⟶∞ in the above equation, we get(20)dbSϱ,ϱ≤maxαbdbSϱ,ϱ,12dbSϱ,ϱ,12bdbSϱ,ϱ=αbdbSϱ,ϱ.
HencedbSϱ,ϱ=0 and so Sϱ=ϱ. Analogously, using (4) for ν=ϱ and ϑ=ν2n, we get Tϱ=ϱ. It follows that ϱ∈CFPT,S; that is, CFPT,S≠∅.
Next, we characterize the common fixed-points setCFPT,S in the following theorem.Theorem 3.
Let all the conditions of Theorems1 and 2 hold. Then CFPT,S is totally ordered if and only if CFPT,S contains exactly one element.Proof.
First, we assume that CFPT,S is totally ordered. Let ϱ,σ∈CFPT,S with ϱ≠σ. Consider (4) for ν=σ and ϑ=ϱ, and we get(21)dbσ,ϱ=dbTσ,Sϱ≤αΛσ,ϱ,+Lmindbσ,Tσ,dbϱ,Sϱ,dbσ,Sϱ,dbϱ,Tσ,=αmaxdbσ,ϱ,dbTσ,σ,dbSϱ,ϱ+dbTσ,ϱ2,dbTσ,ϱ+dbSϱ,σ2b,=αmaxdbσ,ϱ,dbσ,ϱ+dbϱ,σ2b,=α2bdbσ,ϱ+dbϱ,σ,that is,(22)dbσ,ϱ≤α2bdbσ,ϱ+dbϱ,σ.
Again, using (5) for ν=u and ϑ=σ and by calculation we get(23)dbϱ,σ≤α2bdbϱ,σ+dbσ,ϱ.
Adding (22) and (23), we get(24)dbσ,ϱ+dbϱ,σ≤α2dbϱ,σ+dbσ,ϱ,which gives a contradiction. Thus, CFPT,S is singleton. The converse is trivial.
PuttingS=T in Theorem 3, we obtain the following result.Corollary 1.
LetΞ,d,≺¯ be an ordered left-complete quasi-b-metric space with constant b≥1. Suppose that T:Ξ⟶Ξ is a mapping such that(25)dbTν,Tϑ≤αΛ′ν,ϑ+Lmindbν,Tν,dbϑ,Tϑ,dbν,Tϑ,dbϑ,Tν,for all comparable ν,ϑ∈Ξ, where L≥0, α∈0,1, and(26)Λ′ν,ϑ=maxdbν,ϑ,dbTν,ν,dbTϑ,ϑ+dbTν,ν2,dbTν,ϑ+dbTϑ,ν2b,
Also suppose thatTν≺TTν for all ν∈Ξ with ν≺Tν. If there exists an element ν0∈Ξ such that ν0≺Tν0 and either T is continuous at ν0 or Ξ is regular, then T has a fixed point. Moreover, fixT is totally ordered if and only if it is a singleton.Next, we come up with the following example, which illustrates Theorem1.Example 1.
LetΞ=0,5 and define db:Ξ×Ξ⟶ℝ by(27)dbν,ϑ=0,ifν=ϑ;ν−ϑ,ifν>ϑ;2ϑ−ν,ifν<ϑ.
We define a relation “≺¯” on Ξ by ν≺¯ϑ if and only if either ν,y∈0,3 and ν≥ϑ or ν,ϑ∈3,5 and ν=ϑ. Then Ξ,db,≺¯ is an ordered left-complete quasi-b-metric space with constant b=2. Next, we define two mappings T,S:Ξ⟶Ξ by(28)Tν=ν8,ifν∈0,3;38,ifν∈3,5,Sν=ν7,ifν∈0,3;2,ifν∈3,5.
Forν∈0,3 with ν≺Sν, we have Sν=x/7, TSν=ν/56, and STSν=ν/392. So Sν>TSν>STSν and hence Sν>TSν>STSν. Again, for ν∈3,5 and ν≺Sν, we have Sν=2, TSν=1/4, and STSν=1/28. So Sν>TSν>STSν and hence Sν≺TSν≺STSν. Thus, S is T-strictly weakly isotone increasing. It is clear that T is continuous. We choose α=4/10.
Now assume thatν,ϑ∈Ξ is arbitrary such that ν and ϑ are comparable. Then the following cases arise:Case I: letν,ϑ∈0,3 and ν≺¯ϑ; that is, ν≥ϑ. First, we assume that ν>ϑ. Then(29)dbTν,Sϑ≤ν8>ϑ7,ifν8>ϑ7;2ϑ8>ν7,ifν8<ϑ7,andαdbν,ϑ=4/10ν−ϑ and αdbTν,ν=7/10. Therefore,(30)dbTν,Sϑ≤αdbν,ϑ,ifν8>ϑ7;αdbTν,ν,ifν8<ϑ7.Next, we assume thatν=ϑ. Then dbTν,Sϑ=ν/28 and so(31)dbTν,Sϑ≤αdbTν,ν.Thus,(32)dbTν,Sϑ≤αΛν,ϑ.On the similar arguments, we can get(33)dbSν,Tϑ≤αϒν,ϑ.Case II: letν,ϑ∈0,3 and ϑ≺¯ν; that is, ν≤ϑ. Then, we can show in a similar manner that(34)dbTν,Sϑ≤αΛν,ϑ,dbSν,Tϑ≤αϒν,ϑ.Case III: letν,ϑ∈3,5 and ν=ϑ. Then dbTν,Sϑ=13/4 and dbSν,Tϑ=13/8. Also, we have(35)mindbν,Tν,dbϑ,Sϑ,dbν,Sϑ,dbϑ,Tν=minν−38,ν−2=ν−2,mindbν,Sν,dbϑ,Tϑ,dbν,Tϑ,dbϑ,Sν=ν−2.
Therefore,(36)dbTν,Sϑ≤Lmindbν,Tν,dbϑ,Sϑ,dbν,Sϑ,dbϑ,Tν,dbSν,Tϑ≤Lmindbν,Sν,dbϑ,Tϑ,dbν,Tϑ,dbϑ,Sν,where L=10.
Hence, from all three cases, we have(37)dbTν,Sϑ≤αΛν,ϑ+Lmindbν,Tν,dbϑ,Sϑ,dbν,Sϑ,dbϑ,Tν,dbSν,Tϑ≤αΥν,ϑ+Lmindbν,Sν,dbϑ,Tϑ,dbν,Tϑ,dbϑ,Sν.
Thus, all the conditions of Theorem1 are fulfilled and the pair T,S has a unique common fixed point (which is 0∈CFPT,S).


## 3. Results for Three Mappings

In this section, we obtain a result for three mappings involving weakly increasing condition.Theorem 4.
LetΞ,db,≺¯ be an ordered right-complete quasi-b-metric space with constant b≥1. Let T,S,ℛ:Ξ⟶Ξ be three mappings such that TΞ⊆ℛΞ and SΞ⊆ℛΞ, and(38)dbTν,Sϑ≤αΛ1ν,ϑ+Lmindbℛν,Tν,dbℛϑ,Sϑ,dbℛν,Sϑ,dbℛϑ,Tν,,(39)dbSν,Tϑ≤αY1ν,ϑ+Lmindbℛν,Sν,dbℛϑ,Tϑ,dbℛν,Tϑ,dbℛϑ,Sν,for all comparable ℛν,ℛy∈Ξ, where α∈0,1, L≥0, and(40)Λ1ν,ϑ=maxdbℛν,ℛϑ,dbℛν,Tν,dbℛϑ,Sϑ+dbℛν,Tν2,dbTν,ℛϑ+dbℛν,Sϑ2b,Y1ν,ϑ=maxdbℛν,ℛϑ,dbℛν,Sν,dbℛϑ,Tϑ+dbℛν,Sν2,dbSν,ℛϑ+dbℛν,Tϑ2b.We assume the following hypotheses:(i)
T and S are weakly increasing with respect to ℛ(ii)
T and S are dominating mapsAssume either of the following:(a)
S and ℛ are compatible; ℛ is continuous at ν0(b)
T and ℛ are compatible; ℛ is continuous at ν0ThenCFPS,T,ℛ≠∅. In addition, CFPS,T,ℛ is totally ordered if and only if CFPS,T,ℛ contains exactly one element.Proof.
Start with defining a sequenceνn in Ξ as(41)ℛν2n+1=Sν2nandℛν2n+2=Tν2n+1forn∈0,1,….
We claim that(42)ℛνn≺¯ℛνn+1,∀n∈ℕ∗.
Using hypothesis (i) and (41),(43)ℛν1=Sν0≺¯Tϑ,∀y∈ℛ−1Sν0.
Sinceℛν1=Sν0, ν1∈ℛ−1Sν0, and we get(44)ℛν1=Sν0≺¯Tν1=ℛν2.
Again,(45)ℛν2=Tν1≺¯Sϑ,∀y∈ℛ−1Tν1.
Sinceν2∈ℛ−1Tν1, we get(46)ℛν2=Tν1≺¯Sν2=ℛν3.
Hence, by induction, (42) holds.
LetΔn=dbℛνn,ℛνn+1. Now we claim that, for all n∈ℕ, we have(47)Δn≤αΔn−1.
Using (38) and (42) with xν=ν2n−1 and ϑ=ν2n,(48)dbℛν2n,ℛν2n+1=dbTν2n−1,Sν2n,≤αΛ1ν2n−1,ν2n+Lmindbℛν2n−1,Tν2n−1,dbℛν2n,Sν2n,dbℛν2n−1,Sν2n,dbℛν2n,Tν2n−1,⟹dbℛν2n,ℛν2n+1≤αΛ1ν2n−1,ν2n,where(49)Λ1ν2n−1,ν2n=maxdbℛν2n−1,ℛν2n,dbℛν2n−1,Tν2n−1,dbℛν2n,Sν2n+dbℛν2n−1,Tν2n−12,dbTν2n−1,ℛν2n+dbℛν2n−1,Sν2n2b,=maxdbℛν2n−1,ℛν2n,dbℛν2n−1,ℛν2n,dbℛν2n,ℛν2n+1+dbℛν2n−1,ℛν2n2,dbℛν2n−1,ℛν2n+12b.(i)
IfΛ1ν2n+1,ν2n=dbℛν2n−1,ℛν2n, by (48), we have(50)dbℛν2n,ℛν2n+1≤αdbℛν2n−1,ℛν2n,that is,Δ2n≤αΔ2n−1.(ii)
IfΛ1ν2n−1,ν2n=1/2bdbℛν2n−1,ℛν2n+1, we get(51)dbℛν2n,ℛν2n+1≤α2bdbℛν2n−1,ℛν2n+1,≤α2dbℛν2n−1,ℛν2n+α2dbℛν2n,ℛν2n+1⟹dbℛν2n,ℛν2n+1,≤α2−αdbℛν2n−1,ℛν2n,≤αdbℛν2n−1,ℛν2n⟹Δ2n,≤αΔ2n−1.(iii)
IfΛ1ν2n−1,ν2n=dbℛν2n,ℛν2n+1+dbℛν2n−1,ℛν2n/2, we get(52)≤α2dbℛν2n,ℛν2n+1+α2dbℛν2n−1,ℛν2n⟹Δ2n,≤αΔ2n−1.
Thus, in all cases, we haveΔ2n≤αΔ2n−1 for all n∈ℕ. Similarly, using (39), we can show Δ2n+1≤αΔ2n for all n≥1. Therefore, we conclude that (47) holds for all n∈ℕ; that is,(53)dbℛνn,ℛνn+1≤αdbℛνn−1,ℛνn,for alln∈ℕ.
Therefore, from Lemma1, it follows that ℛνn is a right-Cauchy sequence. From the right-completeness of Ξ, there exists ϱ∈Ξ such that(54)ℛνn⟶ϱasn⟶∞.
We will prove thatϱ is a common fixed point of the three mappings S,T, and ℛ.
We have(55)ℛν2n+1=Sν2n⟶ϱ,asn⟶∞,(56)ℛν2n+2=Tν2n+1⟶ϱ,asn⟶∞.
First, suppose that (a) holds. Then(57)limn⟶∞Sℛν2n+2=limn⟶∞ℛSν2n+2=ℛϱ.
From (54) and the continuity of ℛ, we have(58)ℛℛνn⟶ℛz,asn⟶∞.
Also, we haveν2n+1≺¯Tν2n+1=ℛν2n+2. Then, from (38), we get(59)dbTν2n+1,Sℛν2n+2≤αΛ1ν2n+1,ℛν2n+2+Lmindbℛν2n+1,Tν2n+1,dbℛℛν2n+2,Sℛν2n+2,dbℛν2n+1,Sℛν2n+2,dbℛℛν2n+2,Tν2n+1,where(60)Λ1ν2n+1,ℛν2n+2=maxdbℛν2n+1,ℛℛν2n+2,dbℛν2n+1,Tν2n+1,dbℛℛν2n+2,Sℛν2n+2+dbℛν2n+1,Tν2n+12,dbTν2n+1,ℛℛν2n+2+dbℛν2n+1,Sℛν2n+22b.
Limitingn⟶∞ in (59) and using (55)–(58), we obtain(61)dbϱ,ℛϱ≤αmaxdbϱ,ℛϱ,dbϱ,ℛϱ+dbϱ,ℛϱ2b,=αdbϱ,ℛϱ⟹dbϱ,ℛϱ,=0,i.e.,ℛϱ=ϱ.
Now,ν2n+1≺¯Tν2n+1 and Tν2n+1⟶ϱ as n⟶∞, so by the assumption we have ν2n+1≺¯ϱ. Then, using (38), we get(62)dbTν2n+1,Sϱ≤αΛ1ν2n+1,ϱ+Lmindbℛν2n+1,Tν2n+1,dbℛϱ,Sϱ,dbℛν2n+1,Sϱ,dbℛϱ,Tν2n+1,where(63)Λ1ν2n+1,ϱ=maxdbℛν2n+1,ℛϱ,dbℛν2n+1,Tν2n+1,dbℛϱ,Sϱ+dbℛν2n+1,Tν2n+12,dbTν2n+1,ℛϱ+dbℛν2n+1,Sϱ2b.
Lettingn⟶∞ in (62) and using (55) and (56), we get(64)dbϱ,Sϱ≤αmaxdbϱ,Sϱ,dbℛϱ,Sϱ2,dbϱ,ℛϱ+dbϱ,Sϱ2b,=αmaxdbϱ,Sϱ2,dbϱ,Sϱ2b,=α2dbϱ,Sϱ⟹dbϱ,Sϱ,=0,i.e.,Sϱ=ϱ.
Again,ν2n≺¯Sν2n and Sν2n⟶ϱ as n⟶∞, so by the assumption we have ν2n≺¯ϱ and hence using (39) gives(65)dbSν2n,Tϱ≤αΥ1ν2n,ϱ+Lmindbℛν2n,Sν2n,dbℛϱ,Tϱ,dbℛν2n,Tϱ,dbℛϱ,Sν2n,where(66)Y1ν2n,ϱ=maxdbℛν2n,ℛϱ,dbℛν2n,Sν2n,dbℛϱ,Tϱ+dbℛν2n,Sν2n2,dbSν2n,ℛϱ+dbℛν2n,Tϱ2b.
Lettingn⟶∞ in (65) and using (55) and (56), we have(67)dbϱ,Tϱ≤αmaxdbϱ,ℛϱ,dbℛϱ,Tϱ2,dbϱ,ℛϱ+dbϱ,Tϱ2b,=αmaxdbϱ,Tϱ2,dbϱ,Tϱ2b,=α2dbϱ,Tϱ⟹dbϱ,Tϱ,=0,i.e.,Tϱ=ϱ.
Therefore,Sϱ=Tϱ=ℛϱ=ϱ. Hence ϱ is a common fixed point of ℛ,S, and T. The proof is similar when (b) holds.
Now, suppose thatCFPS,T,ℛ is totally ordered. We claim that there is a unique CFPS,T,ℛ. Assume to the contrary that ϱ∈CFPS,T,ℛ and σ∈CFPS,T,ℛ but ϱ≠v. Using (38) with ν=ϱ and ϑ=σ, we have(68)dbϱ,σ=dbTϱ,Sσ,≤αΛ1ϱ,σ+Lmindbℛϱ,Tϱ,dbℛv,Sv,dbℛϱ,Sσ,dbℛσ,Tϱ⟹dbϱ,v,≤αΛ1ϱ,σ,where(69)Λ1ϱ,σ=maxdbℛϱ,ℛσ,dbℛϱ,Tϱ,dbℛσ,Sσ+dbℛϱ,Tϱ2,dbTϱ,ℛσ+dbℛϱ,Sσ2b,=maxdbϱ,σ,dbϱ,σb,=dbϱ,σ.
Therefore,(70)dbϱ,σ≤αdbϱ,σ<dbϱ,σ,a contradiction. Hence, ϱ=σ. The converse is trivial.
Takingℛ= the identity mapping on Ξ in Theorem 4, we have the following consequence.Corollary 2.
LetΞ,db,≺¯ be an ordered right-complete quasi-b-metric space with constant b≥1. Let the pair of mappings T,S:Ξ⟶Ξ satisfy(71)dbTν,Sϑ≤αΛ1′ν,ϑ+Lmindbν,Tν,dby,Sϑ,dbν,Sϑ,dbϑ,Tν,dbSν,Tϑ≤αY1′ν,ϑ+Lmindbν,Sν,dby,Tϑ,dbν,Tϑ,dby,Sν,for all comparable ν,ϑ∈Ξ, α∈0,1, L≥0, and(72)Λ1′ν,ϑ=maxdbν,ϑ,dbν,Tν,dby,Sϑ+dbν,Tν2,dbTν,ϑ+dbν,Sϑ2b,Y1′ν,ϑ=maxdbν,ϑ,dbν,Sν,dby,Tϑ+dbν,Sν2,dbSν,ϑ+dbν,Tϑ2b.Further, assume that(i)
T and S are weakly increasing mapping(ii)
T and S are dominating mapsThenCFPS,T≠∅. In addition, CFPS,T is totally ordered if and only if CFPS,T contains exactly one element.Now, we provide the following example to authenticate Theorem4.Example 2.
We considerΞ=0∪ℕ and define db:Ξ×Ξ⟶ℝ by(73)dbν,ϑ=0,ifν=ϑ;2ν+ϑ,if any one ofν,ϑis0and the other is not0;1ν−2ϑ,ifν≠ϑandν,ϑ≠0.
Also, we define a relation≺¯ on Ξ by ν≺¯ϑ if and only if ν≤ϑ. Then, we can easily check that Ξ,db,≺¯ is an ordered right-complete quasi-b-metric space with constant b=2. Next, we define three mappings T,S,ℛ:Ξ⟶Ξ by(74)Tν=0,ifν=0;20ν+2,ifν≠0,Sν=0,ifν=0;40ν+3,ifν≠0,ℛν=0,ifν=0;ν+1,ifν≠0.
Then, clearlyTΞ⊆ℛΞ and SΞ⊆ℛΞ. We choose α=19/20. Now we assume that ν,ϑ∈Ξ are arbitrary such that ℛν and ℛϑ are comparable. Then the following cases arise:Case I: letν,ϑ≠0 and ℛν≺¯ℛϑ. First, we assume that ν≠ϑ. Then dbTν,Sϑ=1/201/ν+2−1/ϑ+3 and dbℛν,Tν=9ν+19/10ν+1ν+2. Therefore, we have(75)dbTν,Sϑ≤αdbℛν,Tν≤αΛ1ν,ϑ.Next, we assume thatν=ϑ. Then dbTν,Sϑ=1/20ν+1ν+3 and dbℛν,Tν=9ν+19/10ν+1ν+2, and so we have(76)dbTν,Sϑ≤αdbℛν,Tν≤αΛ1ν,ϑ.In a similar way, we can show that(77)dbSν,Tϑ≤αY1ν,ϑ.Case II: letν,ϑ≠0 and ℛϑ≺¯ℛν. Then, proceeding in the same way as that in Case I, we can show that(78)dbTν,Sϑ≤αΛ1ν,ϑ,dbSν,Tϑ≤αY1ν,ϑ.Case III: letν=0 and ϑ≠0. Then dbTν,Sϑ=1/20ϑ+3, dbSν,Tϑ=1/10ϑ+2, and dbℛν,ℛϑ=2/ϑ+1. Therefore,(79)dbTν,Sϑ≤αdbℛν,ℛϑ,dbSν,Tϑ≤αdbℛν,ℛϑ,which implies that(80)dbTν,Sϑ≤αΛ1ν,ϑ,dbSν,Tϑ≤αY1ν,ϑ.Case IV: letϑ=0 and ν≠0. This case is similar to the previous case and so is omitted.Case V: letν=0=ϑ. In this case, it is trivial to check that (38) and (39) hold.
So, combining all the five cases, we see that (38) and (39) hold for all comparable ℛν,ℛy∈Ξ.
Now letϑ∈ℛ−1Tν. Then, we have ϑ=20ν+39 if ν≠0 and ϑ=0 if ν=0. Therefore, if ν≠0, then Tν=20ν+2 and Sϑ=4020ν+42, so Tν≺¯Sϑ. If ν=0, then Tν=Sϑ=0 and so Tν≺¯Sϑ. Similarly, for ϑ∈ℛ−1Sν, we can show that Sν≺¯Tϑ. Hence T and S are weakly increasing with respect to ℛ. Further, we can easily show that T and S are dominating maps; S and ℛ are compatible; and ℛ is continuous. Thus, we can apply Theorem 4 to deduce that T,S, and ℛ have a unique common fixed point ϱ=0.


## 4. 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 sA, 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 Hn given by A≽ℬ if and only if A−ℬ is a positive semidefinite matrix. We define a function db:Hn×Hn⟶ℝ by(81)dbX,Y=0,ifX=Y;X−12Y2,ifX≠Y.ThenHn,db,≺¯ is an ordered right-complete quasi-b-metric space with b=2.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(82)X=ℬ1+∑i=1kAi∗fXAi,X=ℬ2+∑i=1kAi∗gXAi,where ℬ1,ℬ2∈Pn, Ai∈Mn, i=1,…,k, and the operators f,g:Hn⟶Hn are continuous in the trace norm. Let, for some M,N1∈ℝ, and, for any X∈Pn with X≤M, sfX,sgX≤N1 hold for all singular values of fX and gX, respectively. Assume the following:(1)
ℬ1,ℬ2≤M−NN1n, where ∑i=1kAi∗Ai=N(2)
For anyK∈Hn with K≤M, ∑i=1kAi∗fKAi≽O (O stands for null matrix of order n) or ∑i=1kAi∗gKAi≽O holds(3)
For anyK∈Hn with K≤M, K≺¯ℬ1+∑i=1kAi∗fKAiandK≺¯ℬ2+∑i=1kAi∗gKAi hold(4)
For anyK∈Hn with K≤M, if K≺¯ℬ1+∑i=1kAi∗fKAi thenℬ1+∑i=1kAi∗fKAi≺¯ℬ2+∑i=1kAi∗gℬ1+∑i=1kAi∗fKAiAi≺¯ℬ1+∑i=1kAi∗fℬ2+∑i=1kAi∗gℬ1+∑i=1kAi∗fKAiAi hold(5)
There existsL≥0 such that, for any sℬ1−ℬ2,(a)
sℬ1−1/2ℬ2+NsfX−1/2gY≤2/3nℛ11X,Y+Lℛ12X,Y1/2holds for allsfX−1/2gY and X,Y∈Hn with X,Y≤M, X≺¯Y or Y≺¯X, where(83)ℛ11X,Y=maxs+X−12Y2,s+X−12∑i=1kAi∗fXAi−ℬ222,12s+Y−12∑i=1kAi∗gYAi−12ℬ22+s+X−12∑i=1kAi∗fXAi−12ℬ12,14s+∑i=1kAi∗fXAi+ℬ1−12Y2+s+X−12∑i=1kAi∗gYAi−12ℬ22,ℛ12X,Y=mins+X−12∑i=1kAi∗fXAi−12ℬ12,s+Y−12∑i=1kAi∗gYAi−12ℬ22,s+X−12∑i=1kAi∗gYAi−12ℬ22,s+Y−12∑i=1kAi∗fXAi−12ℬ12,.(b)
sℬ1−1/2ℬ2+NsgX−1/2fY≤2/3nℛ21X,Y+Lℛ22X,Y1/2
holds for allsgX−1/2fY and X,Y∈Hn with X,Y≤M, X≺¯Y or Y≺¯X, where(84)ℛ21X,Y=maxs+X−12Y2,s+X−12∑i=1kAi∗gXAi−ℬ222s+Y−12∑i=1kAi∗fYAi−12ℬ12+s+X−12∑i=1kAi∗gXAi−12ℬ2214s+∑i=1kAi∗gXAi+ℬ2−12Y2+s+X−12∑i=1kAi∗fYAi−12ℬ12ℛ22X,Y=mins+X−12∑i=1kAi∗gXAi−12ℬ22,s+Y−12∑i=1kAi∗fYAi−12ℬ12,s+X−12∑i=1kAi∗fYAi−12ℬ12,s+Y−12∑i=1kAi∗gXAi−12ℬ22,Then system (82) has a solution, and if X^ is a solution of the system, then X^∈Pn with X^≤M. Further, the iterative sequence Xn, where, for j≥0,(85)X2j+1=ℬ1+∑i=1kAi∗fX2jAi,X2j+2=ℬ2+∑i=1kAi∗gX2j+1Ai,and X0 is an arbitrary element of Hn satisfying X0≤M, converges to a unique solution of the system, if Xj≺¯Xj+1 or Xj+1≺¯Xj.Proof.
Let us consider the setΞ=X∈Hn:X≤M. Then, Ξ is right-complete with respect to the metric db. For any X∈Ξ, we have(86)ℬ1+∑i=1kAi∗fXAi≤ℬ1+∑i=1kAi∗fXAi,≤ℬ1+∑i=1kAi∗AifX,=ℬ1+NfX.
SinceX≤M, we have sfX≤N1 for all singular values sfX of fX so, by summing the n singular values of fX, we get fX≤nN1. Using this in (86), we get(87)ℬ1+∑i=1kAi∗fXAi≤ℬ1+nNN1≤M.
Similarly, for anyX∈Ξ, we can show that(88)ℬ2+∑i=1kAi∗gXAi≤ℬ2+nNN1≤M.
Therefore, the mappingsT,S defined on Ξ by(89)TX=ℬ1+∑i=1kAi∗fXAi,SX=ℬ2+∑i=1kAi∗gXAi.for X∈Ξ are self-maps on Ξ. Next, for any X∈Ξ, using assumption (3), we have X≺¯TX and X≺¯SX. So the mappings T,S are dominating mappings. Again, for X∈Ξ, we have TX∈Ξ and SX∈Ξ. So, for any X∈Ξ, using assumption (4), we have SX≺¯TSX and TX≺¯STX. Therefore, 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. Then, we have(90)TX−12SY=ℬ1+∑i=1kAi∗fXAi−12ℬ2−12∑i=1kAi∗gYAi,≤ℬ1−ℬ22+∑i=1kAi∗fX−12gYAi,≤ℬ1−12ℬ2+∑i=1kAi∗AifX−12gY,=ℬ1−12ℬ2+NfX−12gY.
Ifℛ11X,Y=s+X−1/2Y2, then we have(91)sℬ1−12ℬ2+NsfX−12gY≤23ns+X−12Y2+Lℛ12X,Y1/2,⟹ℬ1−12ℬ2+NfX−12gY≤23X−12Y2+Lℛ12X,Y1/2,⟹TX−12SY≤23X−12Y2+Lℛ12X,Y1/2,⟹TX−12SY2≤49X−12Y2+Lℛ12X,Y.
Ifℛ11X,Y=s+X−1/2∑i=1kAi∗fXAi−1/2ℬ22, then we can similarly show that(92)TX−12SY2≤49X−12TX2+Lℛ12X,Y.
Again if(93)ℛ11X,Y=12s+Y−12∑i=1kAi∗gYAi−12ℬ22+s+X−12∑i=1kAi∗fXAi−12ℬ12,or(94)ℛ11X,Y=14s+∑i=1kAi∗fXAi+ℬ1−12Y2+s+X−12∑i=1kAi∗gYAi−12ℬ22,then we can show that(95)TX−12SY2≤4912Y−12SY2+X−12TX2+Lℛ12X,Y,or(96)TX−12SY2≤4914TX−12Y2+X−12SY2+Lℛ12X,Y.
Thus, combining equations (91)–(96) and using the definition of the quasi-b-metric db, we get(97)TX−12SY2≤49maxX−12Y2,X−12TX2,12Y−12SY2+X−12TX2,14TX−12Y2+X−12SY2+49Lℛ12X,Y⟹dbTX,SY,≤49maxdbX,Y,dbX,TX,12dbY,SY+dbX,TX,12bdbTX,Y+dbX,SY,+49Lℛ12X,Y⟹dbTX,SY,≤αmaxdbX,Y,dbX,TX,dbY,SY+dbX,TX2,dbTX,Y+dbX,SY2b,+49Lℛ12X,Y⟹dbTX,SY,≤αΛ1′X,Y+LmindbX,TX,dbY,SY,dbX,SY,dbY,TX.
In a similar way, we can show that(98)dbSX,TY≤αY1′X,Y+LmindbX,SX,dbY,TY,dbX,TY,dbY,SX.
Hence, by Corollary2, it is implied that the pair of mappings T,S has a common fixed point in Ξ, say X^. So X^ is a solution to system (82). By assumption (2), we see that any solution of system (82) must be positive definite, and since X^∈Ξ, we have X^≤M. Again, by the order preserving (or order reversing) of the successive elements of the sequence Xn converging to X^, we see that system (82) cannot have more than one solution. Thus, system (82) has a unique solution X^∈Ξ with X^∈Pn and X^≤M.
Now we present an example in order to validate the above theorem.Example 3.
Consider system (82) for k=2 and n=4, with fX=X0.2 and gX=X0.3; that is,(99)X=ℬ1+A1∗X0.2A1+A2∗X0.2A2,X=ℬ2+A1∗X0.3A1+A2∗X0.3A2,where(100)A1=0.502413650.803091060.741477381.00089520.803091061.519264541.553157451.684899370.741477381.553157452.249608942.074485921.00089521.684899372.074485922.41733981,A2=0.574896711.045906640.96835830.966803421.045906642.55083091.904158292.056913240.96835831.904158292.092475231.616260380.966803422.056913241.616260381.94523108,ℬ1=1.765350511.199120871.64879132.015991041.199120872.028816061.623003541.72326681.64879131.623003541.8796472.101496682.015991041.72326682.101496682.44522015,ℬ2=−10.643334057848985−23.874210038032118−23.420523624735278−23.908390438513909−23.874210038032118−48.926958696625590−48.676364597578200−50.163509103606913−23.420523624735285−48.676364597578200−49.153540195555415−50.862360333937076−23.908390438513909−50.163509103606913−50.862360333937083−52.535431964591425.
After calculations, we getℬ1=8.11903372,ℬ2=163.0367049758027, N=96.052515868700326, N1=6.183163814689865, and L=3.
LetM=1.038194798043281×103. The conditions of Theorem (99) can be checked numerically, taking various special values for matrices involved. For example, they can be tested (and verified to be true) for(101)X=2.99584232.771466612.545471992.331458342.771466612.701000982.207369322.161962612.545471992.20736932.405322382.035433112.331458342.161962612.035433111.87555389,Y=7.914542017.070302387.306331526.802483677.070302387.645959446.497954646.585409987.306331526.497954647.504610266.767072056.802483676.585409986.767072056.5164748,K=0.2448954632792950.4461630489255490.5364232362397000.5813679229152510.4461630489255490.8204303944670240.9944427798321521.0658595437939280.5364232362397000.9944427798321521.2326318984748341.3040574630802990.5813679229152511.0658595437939281.3040574630802991.398770067769852,where,(102)X=9.97771955,Y=29.58158651,K=3.696727823991004.
To see the convergence of the sequenceXn defined in (85), we start with initial values,(103)X0=2.0461722984036253.4045274416443212.9533948427909262.9428865570170883.4045274416443216.3645692705888735.4493003884290515.4994691368578052.9533948427909265.4493003884290524.8673086226365684.7895980611189002.9428865570170885.4994691368578054.7895980611189004.890358774128345,with X0=18.168408965757408,(104)U0=10.29336432141485612.83887939915023713.60132667882490113.51819185522712612.83887939915023717.87332803105605819.38537872424269519.66228610573708113.60132667882490119.38537872424269522.11121199738573422.24483925839046013.51819185522712619.66228610573708122.24483925839046022.959343292059014,with U0=73.237247641915687,(105)V0=8.3830229.06429379.052739.85519099.06429379.9587616910.072977510.87411339.0527310.072977510.42436411.13533599.855190910.874113311.135335912.02731569,with V0=40.793463379999992 and, after 20 iterations, we have the following approximation of the unique positive definite solution of system (99):(106)X^≈X20=18.83897386271377935.66070081072850036.04610406249382537.65644843112195935.66070081072850772.23939677998643670.69000586894608873.10505302618275936.04610406249381170.69000586894608872.34546035472902874.91496562769182837.65644843112195973.10505302618275974.91496562769182878.209549859295251,with X^=2.416333808567245×102.(107)U^≈U20=18.83897386271376835.66070081072847836.04610406249379737.65644843112194535.66070081072847872.23939677998637970.69000586894603173.10505302618268836.04610406249379770.69000586894605972.34546035472897174.91496562769177137.65644843112193073.10505302618270274.91496562769177178.209549859295194,with U^=2.416333808567243×102.(108)V^≈V20=18.83897386271376535.66070081072848636.04610406249381137.65644843112193035.66070081072847172.23939677998637970.69000586894603173.10505302618270236.04610406249378270.69000586894603172.34546035472897174.91496562769178537.65644843112193073.10505302618270274.91496562769174378.209549859295180,with V^=2.416333808567243×102. Also, the elements of each sequence are order preserving. The graphical representation of convergence is shown below.
Convergence behaviour


## 5. Application to Nonlinear Fractional Differential Equations

Consider a pair of nonlinear fractional differential equations (FDEs, in short):(109)Dβϑt+ℏ1t,ϑt=0,Dβϑt+ℏ2t,ϑt=0,where 0≤t≤1, 1<β, with the two-point boundary conditions(110)ϑ0=0,ϑ1=0,where ℏ1,ℏ2:J=0,1×ℝ⟶ℝ is a continuous function.The Caputo derivative of fractional-orderβ is defined as(111)Dcβϱt=1Γn−β∫0tt−sn−β−1ϱnsds,n−1<β<n,n=β+1,where ϱ:0,∞⟶ℝ is a continuous function, β denotes the integer part of the positive real number β, and Γ is the gamma function.The Riemann-Liouville fractional derivative of orderβ for a continuous function ϱt is defined by(112)Dβϱt=1Γn−βddtn∫0tϱst−sβ−n−1ds,n−1<β<n,n=β+1,provided the right-hand side is point-wise defined on 0,∞.The FDEs (109) are equivalent to the integral equation(113)ϑt=∫01Bt,ζℏjζ,ϑζdζ,for allt∈J,j=1,2,where the Green function is(114)Bt,ζ=t1−ζβ−1−t−ζβ−1,0≤ζ<t≤1t1−ζα−1Γβ,0≤t≤ζ≤1.Following [4], consider that X=C0,1,ℝ endowed with the metric(115)dbν,ϑ=ν−ϑ2∞+ν∞,for allν,ϑ∈Xwithν≠ϑ,0,otherwise.is a right-complete quasi-b-metric space with b=2, where(116)ϑ∞=maxt∈Jϑt,is the usual supremum norm. It is noticed that X,db,2,≺¯ is a right-complete ordered quasi-b-metric space, where ≺¯ denotes ν≺¯ϑ if νt≤ϑt for all t∈J.Theorem 6.
LetT,S:X⟶X be two operators defined by(117)Tϑt=∫01Bt,ζℏ1ζ,ϑζdζ,Sϑt=∫01Bt,ζℏ2ζ,ϑζdζ,where t∈J. Assume the following:(i)
ℏ1,ℏ2:J×ℝ⟶ℝ is a continuous function, which is nondecreasing in the second variable(ii)
For allt∈J,(118)ℏ1t,ϑt≤ℏ2t,∫01Bt,τℏ1τ,ϑτdτ,ℏ2t,ϑt≤ℏ1t,∫01Bt,τℏ2,ϑτdτ,(iii)
For allt∈J,(119)ϑt≤∫01Bt,ζℏjζ,ϑζdζ,(iv)
For allt,s∈J, i≠j=1,2,(120)0≤ℏit,νs−ℏjt,ϑs≤1bνs−ϑs,ν,ϑ∈X,ν≺¯ϑ,ℏjt,νs≤1bνs.
Then, the integral equations (113) have a solution u∗∈CJ,ℝ.Proof.
Using (II), for allt∈J, we have(121)Tϑt=∫01Bt,sℏ1s,ϑsds,≤∫01Bt,sℏ2s,∫01Bs,τℏ1τ,ϑτdτds,=∫01Bt,sℏ2s,Tϑsds,=STϑt.
Similarly,(122)Sϑt=∫01Bt,sℏ2s,ϑsds,≤∫01Bt,sℏ1s,∫01Bs,τℏ2τ,ϑτdτds,=∫01Bt,sℏ1s,Sϑsds,=TSϑt.
Then, we haveTϑ≺¯STϑ and Sϑ≺¯TSϑ for all ϑ∈CJ,ℝ. This 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(123)∫01Bt,ζdζ=∫0ct1−ζβ−1−t−ζβ−1dζ+∫c1t1−ζα−1Γβdζ,=cβ+1Γβc1−cβ,that is, for c∈0,1,(124)∫01Bt,ζdζ=cβ+1Γβc1−cβ=λ,say.
Now(125)Tϑt≤∫0tBt,sℏ1t,ϑsds≤∫01Bt,sλbϑsds=λbϑ∞,Sϑt≤∫0tBt,sℏ2t,ϑsds≤∫01Bt,sλbϑsds=λbϑ∞.
Also(126)Tνt−Sϑt≤∫01ℏ1t,νs−ℏ2t,ϑsds,≤∫01Bt,s1bνs−ϑsds=1b∫01Bt,sνs−ϑs2ds,≤λbν−ϑ2∞,Sνt−Tϑt≤∫01ℏ2t,νs−ℏ1t,ϑsds≤∫01Bt,s1bνs−ϑsds,=1b∫01Bt,sνs−ϑs2ds≤λbν−ϑ2∞.
Therefore, we get(127)Tϑ∞≤λbϑ∞,Tν−Sϑ2∞≤λ2b2ν−ϑ2∞,Sϑ∞≤λbϑ∞,Sν−Tϑ2∞≤λ2b2ν−ϑ2∞.
Hence(128)dbTν,Sϑ≤λbdbν,ϑ,dbSν,Tϑ≤λbdbν,ϑ,for all comparable ν,ϑ∈X such that Tν≠Sϑ. Also, it is an obvious fact that the above inequality holds true if Tν=Sϑ. Therefore, for α=λ/b and L=0, Corollary 2 implies that there is a unique common fixed point u∗∈X of the operators T,S; that is, u∗ is also a solution to the integral equation (117) and the FDEs (109). To see the uniqueness of solution, let ω∗∈X be another solution of integral equation (117); then, using condition (II), we have(129)u∗t−ω∗t=∫01Bt,ζℏ1ζ,u∗ζdζ−∫01Bt,ζℏ2ζ,ω∗ζdζ,≤∫01Bt,ζℏ1ζ,u∗ζ−ℏ2ζ,ω∗ζdζ,≤1b∫01Bt,ζu∗ζ−ω∗ζdζ,≤λbu∗ζ−ω∗ζ,that is, u∗−ω∗∞≤λ/bu∗−ω∗∞, a contradiction, as λ∈0,1,b≥1.


## 6. Concluding Remark

We obtain some common fixed-point results for a pair of mappings and also for a triplet of mappings in quasi-b-metric spaces using the left- and right-completeness in the absence of commutativity condition. We utilize these results to obtain solutions of a pair of nonlinear matrix equations and a pair of fractional differential equations.



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