# Some Coincidence and Common Fixed-Point Results on Coneb2-Metric Spaces over Banach Algebras with Applications to the Infinite System of Integral Equations

Authors: Ziaul Islam; Muhammad Sarwar; Doaa Filali; Fahd Jarad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, common fixed-point theorems for self-mappings under different types of generalized contractions in the context of the coneb2-metric space over the Banach algebra are discussed. The existence results obtained strengthen the ones mentioned previously in the literature. An example and an application to the infinite system of integral equations are also presented to validate the main results.


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## Body

## 1. Introduction and Preliminaries

Gähler [1] proposed the definition of 2-metric spaces as a generalization of an ordinary metric space. He defined that ds,m,z geometrically represents the area of a triangle with vertices s,m,z∈ℵ. 2-metric is not a continuous function of its variables. This was one of the key drawbacks of the 2-metric space while an ordinary metric is a continuous function.Keeping these drawbacks in mind, Dhage [2], in his PhD thesis, proposed a concept of the D-metric space as a generalized version of the 2-metric space. He also defined an open ball in such spaces and studied other topological properties of the mentioned structure. According to him, Ds,m,z represented the perimeter of a triangle. He stated that the D-metric induced a Hausdorff topology and in the D-metric space, the family of all open balls forms a basis for such topology.Later, Mustafa and Sims [3] illustrate that the topological structure of Dhage’s D-metric is invalid. Then, they revised the D-metric and expanded the notion of a metric in which each triplet of an arbitrary set is given a real number called as the G-metric space [4].In addition, the definition of theD∗-metric space is proposed by Sedghi et al. [5] as an updated version of Dhage’s D-metric space. Later, they analyzed and found that G-metric and D∗-metric have shortcomings. Later, they proposed a new simplified sturcture called the S-metric space [6].On the other hand, by swapping the real numbers with the ordered Banach space and established cone metric space, Huang and Zhang [7] generalized the notion of a metric space and demonstrated some fixed-point results of contractive maps using the normality condition in such spaces. Rezapour and Hamlbarani [8] subsequently ignored the normality assumption and obtained some generalizations of the Huang and Zhang [7] results. However, it should be noted that the equivalence between cone metric spaces and metric spaces has been developed in recent studies by some scholars in the context of the presence of fixed points in the mapping involved. Liu and Xu [9] proposed the concept of a cone metric space over the Banach algebra in order to solve these shortcomings by replacing the Banach space with the Banach algebra. This became an interesting discovery in the study of fixed-point theory since it can be shown that cone metric spaces over the Banach algebra are not equal to metric spaces in terms of the presence of the fixed points of mappings. Among these generalizations, by generalizing the cone 2-metric spaces [10] over the Banach algebra and b2-metric spaces [11], Fernandez et al. [12] examined cone b2-metric spaces over the Banach algebra with the constant b≥1. In the setting of the new structure, they proved some fixed-point theorems under different types of contractive mappings and showed the existence and uniqueness of a solution to a class of system of integral equations as an application.Recently, in 2020, Islam et al. [13] initiated the notion of the cone b2-metric space over the Banach algebra with constant b≽e which is a generalization of the definition of Fernandez et al. [12]. They proved some fixed-point theorems under α-admissible Hardy-Rogers contractions which generalize many of the results from the existence literature, and as an application, they proved results which guarantee the existence of solution of an infinite system of integral equations.In 1973, Hardy and Rogers [14] proposed a new definition of mappings called the contraction of Hardy-Rogers that generalizes the theory of the Banach contraction and the theorem of Reich [15] in a metric space setting. For other related work about the concept of Hardy-Rogers contractions, see, for instance, [16, 17] and the references therein.We recollect certain essential notes, definitions required, and primary results consistent with the literature.Definition 1 (see [18]).
ConsiderU^ the Banach algebra which is real, and the multiplication operation is defined under the below properties (for all s,m,z∈U^,ρ∈ℝ):
(a1) smz=smz
(a2) sm+z=sm+sz and s+mz=sz+mz
(a3) ρsm=ρsm=sρm
(a4) sm≤sm
Unless otherwise stated, we will assume in this article thatU^ is a real Banach algebra. If s∈U^ occurs, we call e the unit of U^, so that es=se=s. We call U^ a unital in this case. If an inverse element m∈U^ exists, the element s∈U^ is said to be invertible, so that sm=ms=e. The inverse of s in such case is unique and is denoted by s−1. We require the following propositions in the sequel.Proposition 2 (see [18]).
Consider the unital Banach algebraU^ with unit e, and let s∈U^ be the arbitrary element. If the spectral radius rs<1, i.e.,
(1)rs=limn⟶∞sn1/n=infsn1/n<1,
thene−s is invertible. In fact,
(2)e−s−1=∑k=1∞sk.Remark 3.
We see from [18] that rs≤s for all s∈U^ with unit e.Remark 4 (see [19]).
In Proposition2, by replacing “rs<1” by s≤1, then the conclusion remains the same.Remark 5 (see [19]).
Ifrs<1, then sn⟶0 as n⟶∞.Definition 6.
Consider the Banach algebraU^ with unit element e, zero element θU^, and CU^≠∅. Then, CU^⊂U^ is a cone in U^ if:
(b1) e∈CU^
(b2) CU^+CU^⊂CU^
(b3) λCU^⊂CU^ for all λ≥0
(b4) CU^·CU^⊂CU^
(b5) CU^∩−CU^=θU^
Define the partial order relation≼ in U^ w.r.t CU^ by s≼m if and only if m−s∈CU^; also, s≺m if s≼m but s≠m while s≪m stands for m−s∈intCU^, where intCU^ is the interior of CU^. CU^ is solid if intCU^≠∅.
If there isM>0 such that for all s,m∈CU^, we have
(3)θU^≼s≼mimpliess≤Mm,
thenCU^ is normal. If M is least and positive in the above, then it is the normal constant of CU^ [7].Definition 7 (see [7, 9]).
Letd:ℵ×ℵ⟶U^ and ℵ≠∅ be the mapping:
(c1) For all s,m∈ℵ, ds,m≽θU^ and ds,m=θU^ if and only if s=m
(c2) For all s,m∈ℵ, ds,m=dm,s
(c3) For all s,m,z∈ℵ, ds,z≼ds,m+dm,z
Then,ℵ,d over the Banach algebra U^ with cone metric d is a cone metric space.
In [20], over the Banach algebra with constant b≥1, the cone b-metric space is introduced as a generalization of the cone metric space over the Banach algebra while in Mitrovic and Hussain [16], over the Banach algebra with parameter b≽e, the concept of cone b-metric spaces is introduced.Definition 8 (see [16]).
Letd:ℵ×ℵ⟶U^ and ℵ≠∅ be the mapping:
(e1) For all s,m∈ℵ, θU^≼ds,m and ds,m=θU^ if and only if s=m
(e2) For all s,m∈ℵ, ds,m=dm,s
(e3) There is b∈CU^, b≽e, and for all s,m,z∈ℵ, ds,z≼bds,m+dm,z
Then,ℵ,d over the Banach algebra U^ with cone b-metric d is a cone b-metric space. Note that if we take b=e, then it reduces to the cone metric space over the Banach algebra U^.Definition 9 (see [1]).
Letd:ℵ×ℵ×ℵ⟶ℝ+ and ℵ≠∅:
(f1) There is a point z∈ℵ for s,m∈ℵ such that ds,m,z≠0, if at least two of s,m,z are not equal
(f2) ds,m,z=0 if and only if at least two of s,m,z are equal
(f3) ds,m,z=dPs,m,z for all s,m,z∈ℵ, where Ps,m,z stands for all permutations of s,m,z
(f4) ds,m,z≤ds,m,t+ds,z,t+dm,z,t for all s,m,z,t∈ℵ
Then,ℵ,d is a 2-metric space with 2-metric d.Definition 10 (see [12]).
Letd:ℵ×ℵ×ℵ⟶U^, ℵ≠∅, and b≥1 be a real number:
(g1) There is a point z∈ℵ for s,m∈ℵ such that ds,m,z≠θU^ if at least two of s,m,z are not equal
(g2) ds,m,z=θU^ if and only if at least two of s,m,z are equal
(g3) ds,m,z=dPs,m,z for all s,m,z∈ℵ, where Ps,m,z stands for all permutations of s,m,z
(g4) ds,m,z≼bds,m,t+ds,z,t+dm,z,t for all s,m,z,t∈ℵ
Then,ℵ,d over the Banach algebra U^ with parameter b≥1 is a cone b2-metric space. By taking b=1, it became a cone 2-metric space. We refer the reader to [21] for other details about the cone 2-metric space over the Banach algebra U^.
Islam et al. [13] initiated the concept of the cone b2-metric space over the Banach algebra with parameter b≽e.Definition 11 (see [13]).
Letd:ℵ×ℵ×ℵ⟶U^ and ℵ≠∅:
(h1) There is a point z∈ℵ for s,m∈ℵ such that ds,m,z≠θU^ if at least two of s,m,z are not equal
(h2) ds,m,z=θU^ iff at least two of s,m,z are equal
(h3) ds,m,z=dPs,m,z for all s,m,z∈ℵ, where Ps,m,z stands for all permutations of s,m,z
(h4) ds,m,z≼bds,m,t+ds,z,t+dm,z,t for all s,m,z,t∈ℵ with b∈CU^, b≽e
Then,ℵ,d over the Banach algebra with parameter b≽e is a cone b2-metric space. By taking b=e, it reduces to a cone 2-metric space.Example 12 (see [13]).
LetU^=Cℝ10,1. For each ht∈U^, ht=ht∞+h′t∞. Then, U^ is a Banach algebra with unit e=1 as a constant function, and multiplication is defined pointwise. Let CU^=ht∈U^∣ht≥0,t∈0,1. Then, CU^ is a cone in U^. Let ℵ=k,0∈ℝ2∣∣0≤k≤1∪0,1. For all S,M,Z∈ℵ, define d:ℵ×ℵ×ℵ⟶U^ as
(4)dS,M,Z=dPS,M,Z,Pdenotespermutations,Δ·ht,otherwise,where Δ is the square of the area of the triangle formed by S,M,Z and h:0,1⟶ℝ defined by ht=et. Consider
(5)ds,0,m,0,0,1·et≼ds,0,m,0,z,0·et+ds,0,z,0,0,1·et+dz,0,m,0,0,1·et.
That is,1/4s−m2·et≼1/4s−z2+z−m2·et, showing that d is not a cone 2-metric, because −3/16et∈CU^ for 0≤s,m,z≤1 with z=1/2, m=0, and s=1. But for b≥2∈CU^ is a cone b2-metric space over the Banach algebra U^.Definition 13 (see [13]).
Consider thatℵ,d is a cone b2-metric space over the Banach algebra U^ with b≽e, and let sn be a sequence in ℵ,d; then,
(i1) sn is said to converge to s∈ℵ if for every c≫θU^ there is N∈ℕ such that dsn,s,a≪c for all n≥N. That is,
(6)limn⟶∞sn=sorsn⟶sn⟶∞.
(i2) If for every c≫θU^ there exists N∈ℕ such that dsn,sm,a≪c for all m,n≥N, then we say that sn is a Cauchy sequence
(i3) ℵ,d is complete if every Cauchy sequence is convergent in ℵDefinition 14 (see [22]).
Let a sequencesn be in U^; then, sequence sn is a c-sequence, if for each c≫θU^ there is N∈ℕ such that sn≪c for all n>N.Lemma 15 (see [23]).
Consider the Banach algebraU^ and intCU^≠∅. Also, consider sn a c-sequence in U^ and k∈CU^ where k is arbitrary; then, ksn is a c-sequence.Lemma 16 (see [23]).
Consider the Banach algebraU^ and intCU^≠∅. Let sn and zn be c-sequences in U^. Then, for arbitrary η,ζ∈CU^, we have ηsn+ζzn which is also a c-sequence.Lemma 17 (see [23]).
Consider the Banach algebraU^ and intCU^≠∅. Let sn⊂CU^ such that sn⟶0 as n⟶∞. Then, sn is a c-sequence.Lemma 18 (see [19]).
LetU^ be the Banach algebra, e be their unit element, and m,s∈U^. If m,s commutes, then
(k1) rs+m≤rs+rm
(k2) rsm≤rsrmLemma 19 (see [19]).
Consider the Banach algebraU^ and s∈U^. If 0≤rs<1, then
(7)re−s−1≤1−rs−1.Lemma 20 (see [24]).
Consider the Banach algebraU^, e is their unit element, and CU^≠∅. Let L∈U^ and sn=Ln. If rL<1, then sn is a c-sequence.Lemma 21 (see [25]).
LetCU^⊂U^ be a cone.
(l1) If s,m∈U^, k∈CU^, and m≼s, then km≼ks
(l2) If m,k∈CU^, rk<1, and m≼km, then m=0
(l3) For any n∈ℕ, rkn<1 with k∈CU^ and rk<1Lemma 22 (see [26]).
Consider the Banach algebraU^ and intCU^≠∅.
(n1) If m,s,z∈U^ and m≼s≪z, then m≪s
(n2) If m∈CU^ and m≪c for c≫θU^, then m=θU^Lemma 23 (see [21]).
Consider the Banach algebraU^ and intCU^≠∅. Let m∈U^, and suppose that k∈CU^ is an arbitrary given vector such that m≪c for any θU^≪c, then km≪c for any θU^≪c.Lemma 24 (see [27]).
Consider the Banach algebraU^ and intCU^≠∅. Let θU^≼m≪c for each θU^≪c; then, m=θU^.Lemma 25 (see [27]).
Consider the Banach algebraU^ and intCU^≠∅. If sn⟶0 as n⟶∞, then for each c≫θU^, there is N∈ℕ with n>N, such that sn≪c.Definition 26 (see [28]).
Letg and g be self-maps of a set ℵ. If m=gs=fs for some s∈ℵ, then for g and f, s is known as a coincidence point and m is known as a point of coincidence of g and f.Definition 27 (see [29]).
The mappingsg,f:ℵ⟶ℵ are said to be weakly compatible, whenever fs=gs and fgs=gfs for any s∈ℵ.Lemma 28 (see [28]).
Let the mappingsg and f be weakly compatible self-maps of a set ℵ. If g and f have a unique point of coincidence m=fs=gs, then m is the unique common fixed point of g and f.


## 2. Main Results

In this section, in the framework of the coneb2-metric space over Banach algebras with parameter b≽e, we prove some common fixed-point results.Proposition 29.
Letℵ,d over the Banach algebra U^ be the complete cone b2-metric space and CU^≠∅ be a cone in U^. If a sequence sn in ℵ converges to s∈ℵ, then we have the following:
(i)
dsn,s,a is a c-sequence for all a∈ℵ(ii)
For anyα∈ℕ, dsn,sn+α,a is a c-sequence for all a∈ℵProof.
Since the proof is easy, so we left it.
Now, we here state and prove our first main results which generalize and extend many of the conclusions from the existence literature.☐Theorem 30.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b≽e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(8)dEiηis,Fjηjm,a≼ϑ1dGkηks,Hlηlm,a+ϑ2dGkηks,Eiηis,a+ϑ3dHlηlm,Fjηjm,a+ϑ4dGkηks,Fjηjm,a+ϑ5dHlηlm,Eiηis,a,where ϑ1,ϑ2,ϑ3,ϑ4,ϑ5∈CU^ with ∑w=13rϑw+2rϑ4rb+2rϑ5rb<1, rϑ2rb+rϑ5rb2<1, rϑ3rb+rϑ4rb2<1, and ϑ1,ϑ2,ϑ3,ϑ4,ϑ5,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
SetEiηi=S2i+1, Fjηj=T2j+2, Gkηk=I2k+3, and Hlηl=J2l+2, i,j,k,l≥1. Then, by (8), we have
(9)dS2i+1s,T2j+2m,a≼ϑ1dI2k+3s,J2l+2m,a+ϑ2dI2k+3s,S2i+1s,a+ϑ3dJ2l+2m,T2j+2m,a+ϑ4dI2k+3s,T2j+2m,a+ϑ5dJ2l+2m,S2i+1s,a.
Chooses0∈ℵ to be arbitrary. Since Eiℵ⊆Hlℵ and Fjℵ⊆Gkℵ for each i,j,k,l≥1, there exists s1,s2∈ℵ such that S1s0=J2s1 and T2s1=I3s2. Continuing this process, we can define sn by S2n+1s2n=J2n+2s2n+1 and T2n+2s2n+1=I2n+3s2n+2.
Denoteμ2n=J2n+2s2n+1=S2n+1s2n and μ2n+1=I2n+3s2n+2=T2n+2s2n+1 for n=0,1,2,⋯. Now, we show that μn is a Cauchy sequence.
From (9), we know that
(10)dμ2n,μ2n+1,a=dS2n+1s2n,T2n+2s2n+1,a≼ϑ1dI2n+1s2n,J2n+2s2n+1,a+ϑ2dI2n+1s2n,S2n+1s2n,a+ϑ3dJ2n+2s2n+1,T2n+2s2n+1,a+ϑ4dI2n+1s2n,T2n+2s2n+1,a+ϑ5dJ2n+2s2n+1,S2n+1s2n,a=ϑ1dμ2n−1,μ2n,a+ϑ2dμ2n−1,μ2n,a+ϑ3dμ2n,μ2n+1,a+ϑ4dμ2n−1,μ2n+1,a+ϑ5dμ2n,μ2n,a≼ϑ1dμ2n−1,μ2n,a+ϑ2dμ2n−1,μ2n,a+ϑ3dμ2n,μ2n+1,a+ϑ4bdμ2n−1,μ2n+1,μ2n+ϑ4bdμ2n−1,μ2n,a+ϑ4bdμ2n,μ2n+1,a.(11)dμ2n,μ2n+1,μ2n−1=dS2n+1s2n,T2n+2s2n+1,μ2n−1≤ϑ1dI2n+1s2n,J2n+2s2n+1,μ2n−1+ϑ2dI2n+1s2n,S2n+1s2n,μ2n−1+ϑ3dJ2n+2s2n+1,T2n+2s2n+1,μ2n−1+ϑ4dI2n+1s2n,T2n+2s2n+1,μ2n−1+ϑ5dJ2n+2s2n+1,S2n+1s2n,μ2n−1=ϑ1dμ2n−1,μ2n,μ2n−1+ϑ2dμ2n−1,μ2n,μ2n−1+ϑ3dμ2n,μ2n+1,μ2n−1+ϑ4dμ2n−1,μ2n+1,μ2n−1+ϑ5dμ2n,μ2n,μ2n−1≼ϑ3dμ2n,μ2n+1,μ2n−1.
It means thatdμ2n,μ2n+1,μ2n−1=θU^. Therefore, (10) becomes
(12)e−ϑ3−ϑ4bdμ2n,μ2n+1,a≼ϑ1+ϑ2+ϑ4bdμ2n−1,μ2n,a,that is,
(13)dμ2n,μ2n+1,a≼L1dμ2n−1,μ2n,a,where L1=e−ϑ3−ϑ4b−1ϑ1+ϑ2+ϑ4b. Similarly, it is not difficult to show that
(14)dμ2n+2,μ2n+1,a=dS2n+3s2n+2,T2n+2s2n+1,a≼ϑ1dI2n+3s2n+2,J2n+2s2n+1,a+ϑ2dI2n+3s2n+2,S2n+3s2n+2,a+ϑ3dJ2n+2s2n+1,T2n+2s2n+1,a+ϑ4dI2n+3s2n+2,T2n+2s2n+1,a+ϑ5dJ2n+2s2n+1,S2n+3s2n+2,a=ϑ1dμ2n+1,μ2n,a+ϑ2dμ2n+1,μ2n+2,a+ϑ3dμ2n,μ2n+1,a+ϑ4dμ2n+1,μ2n+1,a+ϑ5dμ2n,μ2n+2,a≼ϑ1dμ2n+1,μ2n,a+ϑ2dμ2n+1,μ2n+2,a+ϑ3dμ2n,μ2n+1,a+ϑ5bdμ2n,μ2n+2,μ2n+1+ϑ5bdμ2n,μ2n+1,a+ϑ5bdμ2n+1,μ2n+2,a.
Asdμ2n,μ2n+2,μ2n+1=θU^, therefore (14) becomes
(15)e−ϑ2−ϑ5bdμ2n+2,μ2n+1,a≼ϑ1+ϑ3+ϑ5bdμ2n,μ2n+1,a,that is,
(16)dμ2n+2,μ2n+1,a≼L2dμ2n,μ2n+1,a,where L2=e−ϑ2−ϑ5b−1ϑ1+ϑ3+ϑ5b. Set K=L1L2, and using inequalities (13) and (16), we deduce that
(17)dμ2n+2,μ2n+1,a≼L2dμ2n+1,μ2n,a≼L2L1dμ2n,μ2n−1,a≼L2L1L2dμ2n−1,μ2n−2,a=L2Kdμ2n−1,μ2n−2,a≼⋯≼L2Kndμ1,μ0,a,(18)dμ2n+3,μ2n+2,a≼L1dμ2n+2,μ2n+1,a≼L1L2Kndμ1,μ0,a=Kn+1dμ1,μ0,a.
In this case, for allt<n, similar to (17) and (18), we have
(19)dμ2n+1,μ2n,μ2t+1=L2Kn−t−1dμ2t+1,μ2t+2,μ2t+1=θU^,dμ2n−1,μ2n,μ2t+1=Kn−t−1dμ2t+1,μ2t+2,μ2t+1=θU^,dμ2n+1,μ2n,μ2t=Kn−tdμ2t+1,μ2t,μ2t=θU^,dμ2n−1,μ2n,μ2t=L2Kn−t−1dμ2t+1,μ2t,μ2t=θU^.
Therefore, for allm<n, a∈ℵ, and using the above inequalities, we have
(20)dμ2n+1,μ2m+1,a≼bdμ2n+1,μ2m+1,μ2n+bdμ2n+1,μ2n,a+bdμ2n,μ2m+1,a=bdμ2n+1,μ2n,a+bdμ2n,μ2m+1,a≼bdμ2n+1,μ2n,a+b2dμ2n,μ2m+1,μ2n−1+b2dμ2n,μ2n−1,a+b2dμ2n−1,μ2m+1,a=bdμ2n+1,μ2n,a+b2dμ2n,μ2n−1,a+b2dμ2n−1,μ2m+1,a.
Continuing this process, we get(21)dμ2n+1,μ2m+1,a≤bdμ2n+1,μ2n,a+b2dμ2n,μ2n−1,a+b3dμ2n−1,μ2n−2,a+b4dμ2n−2,μ2n−3,a+⋯+bn−m−4dμ2m+5,μ2m+4,a+bn−m−3dμ2m+4,μ2m+3,a+bn−m−2dμ2m+3,μ2m+2,a+bn−m−1dμ2m+2,μ2m+1,a≤bKndμ1,μ0,a+b2L2Kn−1dμ1,μ0,a+b3Kn−1dμ1,μ0,a+b4L2Kn−2dμ1,μ0,a+⋯+bn−m−4Km+2dμ1,μ0,a+bn−m−3L2Km+1dμ1,μ0,a+bn−m−2Km+1dμ1,μ0,a+bn−m−1L2Kmdμ1,μ0,a≤bKn+b2L2Kn−1+b3Kn−1+b4L2Kn−2+⋯+bn−m−4Km+2+bn−m−3L2Km+1+bn−m−2Km+1+bn−m−1L2Kmdμ1,μ0,a,dμ2n+1,μ2m+1,a=bKn+b3Kn−1+⋯+bn−m−4Km+2+bn−m−2Km+1+b2L2Kn−1+b4L2Kn−2+⋯+bn−m−3L2Km+1+bn−m−1L2Kmdμ1,μ0,a=Km+1bn−m−2e+b−2K+⋯+bm−n+3Kn−m−1dμ1,μ0,a+L2Kmbn−m−1e+b−2K+⋯+bm−n+3Kn−m−1dμ1,μ0,a=Km+1bn−m−2∑r=0∞b−2Kr+L2Kmbn−m−1∑r=0∞b−2Krdμ1,μ0,a.
That is,(22)dμ2n+1,μ2m+1,a≼∑r=0∞b−2KrKm+1bn−m−2+L2Kmbn−m−1dμ1,μ0,a≼e−b−2K−1KmKbn−m−2+L2bn−m−1dμ1,μ0,a=MKmdμ1,μ0,a,where M=e−b−2K−1Kbn−m−2+L2bn−m−1.
Similarly, we have(23)dμ2n,μ2m+1,a≤bdμ2n,μ2n−1,a+b2dμ2n−1,μ2n−2,a+b3dμ2n−2,μ2n−3,a+b4dμ2n−3,μ2n−4,a+⋯+bn−m−4dμ2m+5,μ2m+4,a+bn−m−3dμ2m+4,μ2m+3,a+bn−m−2dμ2m+3,μ2m+2,a+bn−m−1dμ2m+2,μ2m+1,a≤bL2Kn−1+b3L2Kn−2+⋯+bn−m−4L2Km+2+bn−m−2L2Km+1+b2Kn−1+b4Kn−2+⋯+bn−m−3Km+2+bn−m−1Kmdμ1,μ0,a≤bn−m−2L2Km+1e+b−2K+⋯+bm−n+3Kn−m−1dμ1,μ0,a+bn−m−1Kme+b−2K+⋯+bm−n+3Kn−m−1dμ1,μ0,a≤bn−m−2L2Km+1∑r=0∞b−2Kr+bn−m−1Km∑r=0∞b−2Krdμ1,μ0,a≤∑r=0∞b−2Krbn−m−2L2Km+1+bn−m−1Kmdμ1,μ0,a≤e−b−2K−1Kmbn−m−2L2K+bn−m−1dμ1,μ0,a=NKmdμ1,μ0,a,where N=e−b−2K−1bn−m−2L2K+bn−m−1.
Similar to the above, one can easily get that(24)dμ2n,μ2m,a≼OKmdμ1,μ0,a,dμ2n+1,μ2m,a≼PKmdμ1,μ0,a,where O=e−b−2K−1bn−mL2+bn−m−1K and P=e−b−2K−1bn−m−1+bn−m−2L2.
From Lemmas18 and 19, we have that
(25)rK=rL1L2≤rL1×rL2=re−ϑ3−ϑ4b−1ϑ1+ϑ2+ϑ4b×re−ϑ2−ϑ5b−1ϑ1+ϑ3+ϑ5b≤re−ϑ3−ϑ4b−1×rϑ1+ϑ2+ϑ4b×re−ϑ2−ϑ5b−1×rϑ1+ϑ3+ϑ5b≤rϑ1+rϑ2+rϑ4rb1−rϑ3−rϑ4rb×rϑ1+rϑ3+rϑ5rb1−rϑ2−rϑ5rb<1.
SincerK<1, therefore in the light of Remark 5 and Lemma 25, Km⟶0 as m⟶∞, and so, for every c∈intCU^, there exists n0∈ℕ such that Km≪c for all n>n0; that is, the sequence Km is a c-sequence. By Lemma 15, the sequences MKmdμ1,μ0,a, NKmdμ1,μ0,a, OKmdμ1,μ0,a, and PKmdμ1,μ0,a are also c-sequences. Therefore, for any c∈U^ with θU^≪c, there exists n1∈ℕ such that, for any n>m>n1, we have dμn,μm,a≪c for all n>n1 and for all a∈ℵ. Thus, from Lemma 24, it means that dμn,μm,a=θU^. This implies that μn is a Cauchy sequence in ℵ.
IfHlℵ is complete for each l=1,2,3,⋯, there exists q∈Hlℵ such that
(26)μ2n=J2n+2s2n+1=S2n+1s2n⟶qn⟶∞.
So we can find ap∈ℵ such that J2n+2p=q (if Eiℵ is complete for each i=1,2,3,⋯, there exists q∈Eiℵ⊆Hlℵ; then, the conclusion remains the same). Now, we show that T2n+2p=q. By (9), we have
(27)dT2n+2p,q,a≼bdT2n+2p,q,S2n+1s2n+bdT2n+2p,S2n+1s2n,a+bdS2n+1s2n,q,a≼bdT2n+2p,q,S2n+1s2n+bdS2n+1s2n,q,a+bϑ1dI2n+1s2n,J2n+2s2n+1,a+bϑ2dI2n+1s2n,S2n+1s2n,a+bϑ3dJ2n+2s2n+1,T2n+2p,a+bϑ4dI2n+1s2n,T2n+2p,a+bϑ5dJ2n+2s2n+1,S2n+1s2n,a,dT2n+2p,q,a=bdT2n+2p,q,μ2n+bdμ2n,q,a+bϑ1dμ2n−1,q,a+bϑ2dμ2n−1,μ2n,a+bϑ3dq,T2n+2p,a+bϑ4dμ2n−1,T2n+2p,a+bϑ5dq,μ2n,a≼bdT2n+2p,q,μ2n+bdμ2n,q,a+bϑ1dμ2n−1,q,a+bϑ2dμ2n−1,μ2n,a+bϑ3dq,T2n+2p,a+b2ϑ4dμ2n−1,T2n+2p,q+b2ϑ4dμ2n−1,q,a+b2ϑ4dq,T2n+2p,a+bϑ5dq,μ2n,a.
That is,(28)e−bϑ3−b2ϑ4dT2n+2p,q,a≼bdμ2n,T2n+2p,a+b2ϑ4dμ2n−1,T2n+2p,q+b+bϑ5dμ2n,q,a+bϑ1+b2ϑ4dμ2n−1,q,a+bϑ2dμ2n−1,μ2n,a.
Therefore, it follows from Proposition29 and Lemmas 15 and 16 that
(29)e−bϑ3−b2ϑ4dT2n+2p,q,a≼zn,where zn is a c-sequence in CU^. In addition, from Proposition 2 and
(30)rbϑ3+b2ϑ4≤rbrϑ3+rb2rϑ4<1,
it means thate−bϑ3+b2ϑ4 is invertible. In this case, we have
(31)e−bϑ3−b2ϑ4dT2n+2p,q,a≪c,
for anyc≫θU^, which together with Lemma 23 implies that θU^≼dT2n+2p,q,a≪c, for any a∈ℵ, n∈ℕ, and c≫θU^ as e−bϑ3+b2ϑ4 is invertible. Therefore, by Lemma 24, we have dT2n+2p,q,a=θU^ for any n∈ℕ. Namely, T2n+2p=q for any n∈ℕ. That is, T2n+2p=q=J2n+2p.
At the same time, asq=T2n+2p∈Fjℵ⊆Gkℵ, there exists u∈ℵ such that I2n+3u=q.
Now, we show thatS2n+1u=q. From (9), we have that
(32)dS2n+1u,q,a=dS2n+1u,T2n+2p,a≼ϑ1dI2n+3u,J2n+2p,a+ϑ2dI2n+3u,S2n+1u,a+ϑ3dJ2n+2p,T2n+2p,a+ϑ4dI2n+3u,T2n+2p,a+ϑ5dJ2n+2p,S2n+1u,a=ϑ1dq,q,a+ϑ2dq,S2n+1u,a+ϑ3dq,q,a+ϑ4dq,q,a+ϑ5dq,S2n+1u,a.
That is,(33)dS2n+1u,q,a≼ϑ2+ϑ5dS2n+1u,q,a.
Hence, by Lemma20, we know that dS2n+1u,q,a=θU^, and so S2n+1u=q. Therefore, S2n+1u=I2n+3u=q and T2n+2p=J2n+2p=q.
Next, if we assumeGkℵ is complete for each k=1,2,3,⋯, there exists q∈Gkℵ such that
(34)μ2n+1=I2n+3s2n+2=T2n+2s2n+1⟶qasn⟶∞.
So, we can findu∈ℵ such that I2n+3u=q (if Fjℵ is complete for each j=1,2,3,⋯, there exists q∈Fjℵ⊆Gkℵ; then, the conclusion remains the same).
Now, we show thatS2n+1u=q. By (9), we get that
(35)dS2n+1u,q,a≼bdS2n+1u,q,T2n+2s2n+1+bdS2n+1u,T2n+2s2n+1,a+bdT2n+2s2n+1,q,a≼bdS2n+1u,q,T2n+2s2n+1+bdT2n+2s2n+1,q,a+bϑ1dI2n+3u,J2n+2s2n+1,a+bϑ2dI2n+3u,S2n+1u,a+bϑ3dJ2n+2s2n+1,T2n+2s2n+1,a+bϑ4dI2n+3u,T2n+2s2n+1,a+bϑ5dJ2n+2s2n+1,S2n+1u,a≼bdS2n+1u,q,μ2n+1+bdμ2n+1,q,a+bϑ1dq,μ2n,a+bϑ2dq,S2n+1u,a+bϑ3dμ2n,μ2n+1,a+bϑ4dq,μ2n+1,a+b2ϑ5dμ2n,S2n+1u,q+b2ϑ5dμ2n,q,a+b2ϑ5dq,S2n+1u,a.
That is,(36)e−bϑ3−b2ϑ4dT2n+2p,q,a≤bdμ2n,T2n+2p,a+b2ϑ4dμ2n−1,T2n+2p,q+b+bϑ5dμ2n,q,a+bϑ1+b2ϑ4dμ2n−1,q,a+bϑ2dμ2n−1,μ2n,a.
Therefore, it follows from Proposition29 and Lemmas 15 and 16 that
(37)e−bϑ2−b2ϑ5dS2n+1u,q,a≼zn∗,where zn∗ is a c-sequence in CU^. In addition, from Proposition 2 and
(38)rbϑ3+b2ϑ4≤rbrϑ3+rb2rϑ4<1,
it means thate−bϑ2+b2ϑ5 is invertible. In this case, we have
(39)e−bϑ2−b2ϑ5dS2n+1u,q,a≪c,
for anyc≫θU^, which together with Lemma 23 implies that θU^≼dS2n+1u,q,a≪c for any a∈ℵ, n∈ℕ, and c≫θU^ as e−bϑ2+b2ϑ5 is invertible. Therefore, by Lemma 24, we have dS2n+1u,q,a=θU^ for any n∈ℕ. Namely, S2n+1u=q for any n∈ℕ. That is, S2n+1u=q=I2n+3u.
At the same time, asq=S2n+1u∈Eiℵ⊆Hlℵ, there exists p∈ℵ such that J2n+2p=q. Now, we show that T2n+2p=q. From (9), we have
(40)dT2n+2p,q,a=dS2n+1u,T2n+2p,a≼ϑ1dI2n+3u,J2n+2p,a+ϑ2dI2n+3u,S2n+1u,a+ϑ3dJ2n+2p,T2n+2p,a+ϑ4dI2n+3u,T2n+2p,a+ϑ5dJ2n+2p,S2n+1u,a=ϑ1dq,q,a+ϑ2dq,q,a+ϑ3dq,T2n+2p,a+ϑ4dq,T2n+2p,a+ϑ5dq,q,a.
That is,(41)dT2n+2p,q,a≼ϑ3+ϑ4dT2n+2p,q,a.
Hence, by Lemma20, we know that dT2n+2p,q,a=θU^, and so T2n+2p=q. Therefore, T2n+2p=J2n+2p=q and S2n+1u=I2n+3u=q.
Finally, we show thatS2i+1 and I2k+3, T2j+2, and J2l+2 have a unique point of coincidence in ℵ. Assume that there is another point z∈ℵ such that T2n+2x=J2n+2=z; then,
(42)dq,z,a=dS2n+1u,T2n+2x,a≼ϑ1dI2n+3u,J2n+2x,a+ϑ2dI2n+3u,S2n+1u,a+ϑ3dJ2n+2x,T2n+2x,a+ϑ4dI2n+3u,T2n+2x,a+ϑ5dJ2n+2x,S2n+1u,a=ϑ1dq,z,a+ϑ2dq,q,a+ϑ3dz,z,a+ϑ4dq,z,a+ϑ5dz,q,a.
That is,(43)dq,z,a≼ϑ1+ϑ4+ϑ5dq,z,a.
Hence, by Lemma20, we have that dq,z,a=θU^, and so q=z; that is, q is the unique point of coincidence of T2j+2 and J2l+2.
Similarly, we also haveq which is the unique point of coincidence of S2i+1 and I2k+3 by induction.
So, according to Lemma28, q is the unique common fixed point of S2i+1,I2k+3 and T2j+2,J2l+2 for each i,j,k,l=1,2,3,⋯. Therefore, q is the unique common fixed point of S2i+1,I2k+3,T2j+2, and J2l+2.
Now, it is left to show thatq is the unique common fixed point of Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞.
Asq=S2n+1q=Enηnq, so we have Enq=EnEnηnq=EnηnEnq=S2n+1q, that is, S2n+1Enq=Enq. But S2n+1q=q is unique; therefore, Enq=q for n=1,2,3,⋯.
Also, asq=T2n+2q=Fnηnq, so we have Fnq=FnFnηnq=FnηnFnq=T2n+2q, that is, T2n+2Fnq=Fnq. But T2n+2q=q is unique; therefore, Fnq=q for n=1,2,3,⋯. Similarly, Gnq=q and Hnq=q for n=1,2,3,⋯. Thus, the four families of mappings Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.☐Remark 31.
Theorem30 of this paper extends and improves Theorem 2.1 of [30] from cone metric spaces to cone b2-metric spaces; also, it extends and improves Theorem 3.2 of [17] and Theorem 3.1 of [31] from one family and two families, respectively, to four families of mappings.We obtain a series of new common fixed-point results using Theorem30 for four families of mappings in the context of cone b2-metric spaces over Banach algebras, which generalize and improve many known results from the existence literature.Corollary 32.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b±e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(44)dEiηis,Fjηjm,a≼αdGkηks,Hlηlm,a+βdGkηks,Eiηis,a+dHlηlm,Fjηjm,a+γdGkηks,Fjηjm,a+dHlηlm,Eiηis,a,where α,β,γ∈CU^ with rα+rβ+2rγrb<1, rβrb+rγrb2<1, and α,β,γ,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
Letϑ1=α,ϑ2=ϑ3=β,ϑ4=ϑ5=γ in Theorem 30.☐Corollary 33.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b±e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(45)dEiηis,Fjηjm,a≼ϑ1dGkηks,Hlηlm,a+ϑ2dGkηks,Eiηis,a+ϑ3dHlηlm,Fjηjm,a,where ϑ1,ϑ2,ϑ3∈CU^ with rϑ1+rϑ2+rϑ3<1, rϑ2rb+rϑ3rb<1, and ϑ1,ϑ2,ϑ3,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
Takingϑ4=ϑ5=θU^ in Theorem 30, one can get the desired result.☐Remark 34.
We can have Theorem 3.1 in [21], when Eii=1∞ and Fjj=1∞ are the same mapping and Gkk=1∞ and Hll=1∞ are the identity mappings. Therefore, Theorem 3.1 of [21] is a special case of Corollary 33. Also, Corollary 33 of this paper generalizes Theorem 2.1 of [10] from the cone 2-metric space to the cone b2-metric space and extends Theorem 6.1 in [12].Corollary 35.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b±e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(46)dEiηis,Fjηjm,a≼αdGkηks,Hlηlm,a+βdGkηks,Eiηis,a+dHlηlm,Fjηjm,a,where α,β∈CU^ with rα+2rβ<1, 2rβrb<1, and α,β,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ is a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
One can the result takingϑ1=α,ϑ2=ϑ3=β, and ϑ4=ϑ5=θU^ in Theorem 30.☐Remark 36.
Corollary35 of this paper extends Theorem 6 in [32]; therefore, Theorem 6 in [32] is a special case of Corollary 35.Corollary 37.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b≽e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(47)dEiηis,Fjηjm,a≼αdGkηks,Hlηlm,a+βdGkηks,Fjηjm,a+γdHlηlm,Eiηis,a,where α,β,γ∈CU^ with rα+2rβrb+2rγrb<1, rβrb2+rγrb2<1, and α,β,γ,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
One can get the result takingϑ1=α,ϑ2=ϑ3=θU^ and ϑ4=β,ϑ5=γ in Theorem 30.☐Corollary 38.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b≽e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(48)dEiηis,Fjηjm,a≼kdGkηks,Eiηis,a+ldHlηlm,Fjηjm,a,where k,l∈CU^ with rk+rl<1, rkrb+rlrb<1, and k,l,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
Letϑ1=ϑ4=ϑ5=θU^,ϑ2=k,ϑ3=l in Theorem 30.☐Corollary 39.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b≽e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(49)dEiηis,Fjηjm,a≼kdGkηks,Fjηjm,a+ldHlηlm,Eiηis,a,where k,l∈CU^ with 2rkrb+2rlrb<1, rkrb2+rlrb2<1, and k,l,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
Letϑ1=ϑ2=ϑ3=θU^,ϑ4=k,ϑ5=l in Theorem 30.☐Corollary 40.
Letℵ,d over the Banach algebra U^ be a cone b2-metric space with b≽e and CU^≠∅ be a cone in U^. Let Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ be four families of self-mappings on ℵ. For all i,j,k,l∈ℕ, if a sequence ηnn=1∞ exists of nonnegative integers, such that for all s,m,z∈ℵ,
(50)dEiηis,Fjηjm,a≼kdGkηks,Hlηlm,a,where k∈CU^ with rk<1 and k,b commute. If Eiℵ⊆Hlℵ, Fjℵ⊆Gkℵ, and one of Eiℵ, Gkℵ, Hlℵ, and Fjℵ are a complete subspace of ℵ for each i,j,k,l≥1, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique point of coincidence in ℵ. Moreover, if Fj,Hl and Ei,Gk are weakly compatible, respectively, then Eii=1∞, Fjj=1∞, Gkk=1∞, and Hll=1∞ have a unique common fixed point.Proof.
Letϑ1=k,ϑ2=ϑ3=ϑ4=ϑ5=θU^ in Theorem 30.☐From the above corollary, we obtain the following.Corollary 41.
Letℵ,d over the Banach algebra U^ be a complete cone b2-metric space with b≽e and CU^≠∅ be a cone in U^. Let Eii=1∞ be the family of self-mapping on ℵ. For all i∈ℕ and for all s,m,z∈ℵ,
(51)dEis,Eim,a≼kds,m,a,where k∈CU^ with rk<1 and k,b commute. Then, Eii=1∞ have a unique common fixed point.Proof.
Takingηn=1, Ei=Fj, and Gk,Hl which are identity mappings in Corollary 40, then we can obtain the required result.☐We finish this section with an example that will demonstrate the consequence of Theorem30.Example 42.
LetU^=ℝ2. For each s1,s2∈U^, s1,s2=s1+s2. The multiplication is defined by sm=s1,s2m1,m2=s1m1,s1m2+s2m1. Then, U^ is a Banach algebra with unit element e=1,0. Let CU^=s1,s2∈ℝ2∣s1,s2≥0. Then, CU^ is a cone in U^.Letℵ=s,0∈ℝ2∣s≥0∪0,2⊂ℝ2 and define d:ℵ×ℵ×ℵ⟶U^ as follows:
(52)dS,M,Z=0,0,ifatleasttwoofS,M,Zareequal,dPS,M,Z,Pdenotespermutations,Δ,Δ,otherwise,where Δ is the square of the area of the triangle S,M,Z. We have
(53)ds,0,m,0,0,2≼ds,0,m,0,z,0+ds,0,z,0,0,2+dz,0,m,0,0,2.That is,s−m2≼s−z2+z−m2, which shows that d is not a cone 2-metric, because −9/2,−9/2∈CU^ for s,m,z≥0 with s=5, m=0, and z=1/2. But for the parameter b=2,0≽e is a cone b2-metric space over the Banach algebra U^.Now, we define mappingsEi:ℵ⟶ℵi=1,2,3,⋯ by
(54)Eis,0=161/2i−1321/2i−1ss2i−2,0,Ei0,2=0,0.We have(55)Ei2i−1s,0=Ei2i−2Eis,0=Ei2i−2161/2i−1321/2i−1ss2i−2,0=Ei2i−3162/2i−1322/2i−1s2s2i−2,0=Ei2i−4163/2i−1323/2i−1s3s2i−2,0=⋯⋯=Ei162i−2/2i−1322i−2/2i−1s2i−2s2i−2,0=162i−1/2i−1322i−1/2i−1s2i−1s2i−2,0=14s,0.We define mappingsGk:ℵ⟶ℵ (k=1,2,3,⋯) by
(56)Gks,0=131/2k−112−1/2k−1ss2k−2,0,Gk0,2=0,0.We have(57)Gk2k−1s,0=Gk2k−2Gks,0=Gk2k−2131/2k−112−1/2k−1ss2k−2,0=Gk2k−3132/2i−112−2/2k−1s2s2k−2,0=Gk2k−4133/2k−112−3/2k−1s3s2k−2,0=⋯⋯=Gk132k−2/2k−112−2k−2/2k−1s2k−2s2k−2,0,Gk2k−1s,0=132k−1/2k−112−2k+1/2k−1s2k−1s2k−2,0=23s,0.Similarly, we define mappingsFj,Hl:ℵ⟶ℵ (j,l=1,2,3,⋯) by
(58)Fjs,0=231/2j−13101/2j−1ss2j−2,0,Fj0,2=0,0,Hls,0=231/2l−112−1/2l−1ss2l−2,0,Hl0,2=0,0.Then, it is not difficult to show thatFj2j−1s,0=1/5s,0 and Hl2l−1s,0=1/3s,0. Choose ϑ1=1/10,0, ϑ2=ϑ3=1/8,0, and ϑ4=ϑ5=1/16,0. Clearly,
(59)∑w=13rϑw+2rϑ4rb+2rϑ5rb=110+18+18+21162+21162=3440<1,alsorϑ2rb+rϑ5rb2=21/8+41/16=1/2<1 and rϑ3rb+rϑ4rb2=21/8+41/16=1/2<1.Now, considering the contractive condition (8), we have
(60)ds4,0,s5,0,0,2≼110,0d2s3,0,s3,0,0,2+18,0d2s3,0,s4,0,0,2+18,0ds3,0,s5,0,0,2+116,0d2s3,0,s5,0,0,2+116,0ds3,0,s4,0,0,2,that is,
(61)s4−s52,s4−s52≼110,02s3−s32,2s3−s32+18,02s3−s42,2s3−s42+18,0s3−s52,s3−s52+116,02s3−s52,2s3−s52+116,0+s3−s42,s3−s42=1102s3−s32,1102s3−s32+182s3−s42,182s3−s42+18s3−s52,18s3−s52+1162s3−s52,1162s3−s52+116s3−s42,116s3−s42,which means that
(62)s4−s52≼1102s3−s32+182s3−s42+18s3−s52+1162s3−s52+116s3−s42,s202°110s32+185s122+182s152+1167s152+116s122,that is,
(63)s2400≼s290+25s21152+4s21800+49s23600+s22304,which shows thats2/400≼2827s2/57600, and so 2827s2/57600−s2/400∈CU^, which is true for all s≥0. Hence, condition (8) is true for all s,m,a∈ℵ and i,j,k,l≥1, where ηi=2i−1, ηj=2j−1, ηk=2k−1, and ηl=2l−1. All other conditions of Theorem 30 are satisfied. By Theorem 30, Ei, Fj, Gk, and Hl have a unique common fixed point 0,0 for all i,j,k,l≥1.


## 3. Application to the Infinite System of Integral Equations

We give here a couple of auxiliary facts that will be needed in our further considerations. LetU^=ℝ2 with norm .U^ be a real Banach algebra. Let I=0,T, and denote by CI,U^ the space consisting of all continuous functions defined on interval I with values in the Banach algebra U^. The space CI,U^ will be equipped with s=maxsaU^:a∈I.Letℵ=CI,U^ and define d:ℵ3⟶U^ by
(64)dst,mt,z=minst−mt,mt−z,st−zp,where p≥1 and for all st,mt,z∈ℵ. Then, ℵ,d is a complete cone b2-metric space over the Banach algebra.We consider the infinite system of integral equations of the form(65)sit=git+∫0TMit,wfiw,swdw,where i=1,2,3,⋯. Let Ei:ℵ⟶ℵ. We redefine the above infinite system of integral equations as
(66)Eisit=git+∫0TMit,wfiw,swdw,for allsit∈ℵ and t,w∈I. Clearly, by using Corollary 41, the existence of solution to (65) is equivalent to the existence of a common fixed point of Ei.We assume that(i)
gi:I⟶ℝ are continuous for each i=1,2,3,⋯(ii)
Mi:I×ℝ⟶0,+∞ are continuous and ∫0TMit,wdw≤1 for each i=1,2,3,⋯(iii)
fi:I×ℝ⟶ℝ are continuous for each i=1,2,3,⋯ such that(67)fiw,sw−fiw,mw≤v1/pminsw−mw,mw−z,sw−z,for allsw,mw,z∈ℵ and 0≤v<1.Theorem 43.
Under the assumptions (i)–(iii), the infinite system of integral equations (65) has a unique solution in ℵ.Proof.
TakeU^=ℝ2 with norm s=s1,s2=∣s1∣+∣s2∣, and multiplication is defined by the following way:
(68)sm=s1,s2m1,m2=s1m1,s1m2+s2m1.
LetCU^=s1,s2∈U^:s1,s2≥0. It is clear that CU^ is a normal cone and U^ is a Banach algebra with unit element e=1,0.
Consider the family of mappingEi:ℵ⟶ℵ defined by (66). Let sit,mit,z∈ℵ. From (64), we deduce that
(69)dEisit,Eimit,z=maxa∈0,TminEisit−Eimit,Eisit−z,Eimit−zp≼maxa∈0,TEisit−Eimitp=maxa∈0,T∫0TMit,wfiw,swdw−∫0TMit,wfiw,mwdwp=maxa∈0,T∫0TMit,wfiw,sw−fiw,mwdwp≼maxa∈0,T∫0TMit,wfiw,sw−fiw,mwdwp≼maxa∈0,T∫0TMit,wv1/pminsw−mw,mw−z,sw−zdwp≼∫0Tmaxa∈0,TMit,wv1/pmaxa∈0,Tminsw−mw,mw−z,sw−zp1/pdwp≤∫0Tmaxa∈0,TMit,wv1/pdsit,mit,z1/pdwp.
Therefore,(70)dEisit,Eimit,z≼vdsit,mit,z.
Now, all the assumptions of Corollary41 are fulfilled and the family of mapping Ei has a unique common fixed point in ℵ, which means that the infinite system of integral equations (65) has a unique solution in ℵ.☐



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