# Fixed-Point Results for Generalizedα-Admissible Hardy-Rogers’ Contractions in Cone b2-Metric Spaces over Banach’s Algebras with Application

Authors: Ziaul Islam; Muhammad Sarwar; Manuel de la Sen
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In the current manuscript, the notion of a coneb2-metric space over Banach’s algebra with parameter b≻¯e is introduced. Furthermore, using α-admissible Hardy-Rogers’ contractive conditions, we have proven fixed-point theorems for self-mappings, which generalize and strengthen many of the conclusions in existing literature. In order to verify our key result, a nontrivial example is given, and as an application, we proved a theorem that shows the existence of a solution of an infinite system of integral equations.


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## Body

## 1. Introduction and Preliminaries

There are many generalizations in the literature about the concept of metric spaces likeb-metric spaces [1], 2-metric spaces [2], Nb-metric spaces [3], and weak partial b-metric spaces [4]. Gähler incorporated the notion of a 2-metric space in [2]. Recall that a 2-metric is not a continuous function of its variables, whereas a standard metric is. This led Dhage to implement the D-metric notion in [5]. In [6, 7] Mustafa and Sims implemented the G-metric notion for overcoming D-metric flaws. Following that, several fixed-point theorems were proven on G-metric spaces (see [8]). The authors in [9] found that fixed-point theorems in G-metric spaces can potentially be deduced from metric or quasimetric spaces in a variety of cases. Different researchers have additionally indicated that the fixed-point results about cone metric spaces can be acquired in a few cases by diminishing them to their standard metric partners; see for instance [10–12]. It is worth noting that a 2-metric space was not considered to be topologically equivalent to an ordinary metric in the generalizations described above.Bakhtin [1] analyzed the phenomenon of a b-metric space. After this theory, Czerwik [13] demonstrated the contraction mapping method in b-metric spaces which generalized the renowned Banach contraction principle in b-metric spaces.Replacing the set of real numbers by an ordered Banach space, Huang and Zhang [14] generalized the concept of metric spaces and defined the cone metric space, where they studied certain fixed-point results for contractive mapping in the context of cone metric space. Later, Mustafa et al. [15] set the space structure b2-metric as a generalization of b-metric and 2-metric spaces. They illustrated some fixed-point theorems in a partially ordered b2-metric space under different contractive conditions and provided some smart examples and an application to integral equations for their main outcomes.Recently, the equivalence of cone metric space and metric space has become an extremely fascinating topic after the work of several researchers discovered that the fixed-point results in a cone metric space are special cases of metric spaces in some cases. They found thatX,∂ is equivalent to any cone metric if the real-valued function ∂∗ is replaced by a nonlinear scalariztion function ξe or by a Minkowski functional qe. To address these shortcomings, Liu and Xu [16] presented the definition of cone metric space over Banach’s algebra.Fernandez et al.[17] presented the concept of cone b2-metric spaces over Banach’s algebra with coefficient b≥1 as an extension of b2-metric spaces and cone metric spaces over Banach’s algebras. They also presented many fixed-point results under different contractive conditions in the said structure. As an application, they discussed the existence of solutions to the integral equation.On the other hand, Hardy and Rogers [18] introduced a new concept of mapping called the Hardy-Rogers contraction which generalize the Banach contraction principle and Reich’s [19] theorem in the setting of metric spaces. Samet et al. [20] initiated the α-admissibility of mappings and gave a result of α‐ψ-contractive mapping which generalized the Banach contraction principle. After that, many researchers worked on the Hardy-Rogers contraction and α-admissibility of mapping in different settings; for examples, see [21–27] and the references therein.Motivated by the work done in [17, 18, 20] we study some results for the generalized α-admissible Hardy-Roger contractions in cone b2-metric spaces over Banach’s algebras. We note that some well-known results in the literature can be deduced by using the presented work.In the sequel, we need the following definitions and results from the existing literature.Definition 1. (see [28]).
LetB be a real Banach algebra, and the multiplication operation is defined according to the following properties (for all s,m,z∈Bandλ∈ℝ):
(𝔞1) smz=smz;
(𝔞2) sm+z=sm+sz and s+mz=sz+mz;
(𝔞3) λsm=λsm=sλm;
(𝔞4) sm≤sm.
We will presume in the course of this article thatB is a real Banach algebra, unless otherwise specified. We call e the unit of B, if there is s∈B, such that es=se=s. In this case, we call B a unital. It is said that an element s∈B is invertible if an inverse element m∈B occurs, such that sm=ms=e. In such case, the inverse of s is unique and is denoted by s−1. In the sequel, we need the following propositions.Proposition 2. (see [28]).
Lete be the unit element of the Banach algebra B and s∈B be arbitrary. If the spectral radius rs<1, that is
(1)rs=limn→∞sn1/n=infsn1/n<1,then, e−s is invertible. In fact
(2)e−s−1=∑k=1∞sk.Remark 3.
From [28] we see that, for all s in the Banach algebra B with unit e, we have rs≤s.Remark 4. (see [29]).
In Proposition2, if we replace “rs<1” by s≤1, then the conclusion remains true.Remark 5. (see [29]).
Ifrs<1, then sn⟶0 as n⟶∞.Definition 6.
Letθ be the zero element of the unital Banach algebra B and CB≠∅. Then, CB⊂B is a cone in B if
(𝔟1) e∈CB;
(𝔟2) CB+CB⊂CB;
(𝔟3) λCB⊂CB for all λ≥0;
(𝔟4) CB·CB⊂CB;
(𝔟5) CB∩−CB=θ.
Define a partial order relation⪯ in B w.r.t. CB by s⪯m if and only if m−s∈CB and also s≺m if s⪯m but s≠m while s⪻m stands for m−s∈intCB, where intCB is the interior of CB. CB is solid if intCB≠∅.
If there isM>0 such that for all s,m∈CB, we have
(3)θ≤s≤mimpliess≤Mm,then, CB is normal. IfMis the least and positive among those cited above, then it is a normal constant of CB [14].
Onward, we assume thatCB is a cone in B with intCB≠∅, and ⪯ is a partial order with respect to the cone CB.Definition 7. (see [16–14]).
LetX≠∅ and the mapping ∂:X×X⟶B be
(𝔠1) ∂s,m⪰θ for all s,m∈X, and ∂s,m=θ if and only if s=m;
(𝔠2) ∂s,m=∂m,s for all s,m∈X;
(𝔠3) ∂s,z⪯∂s,m+∂m,z for all s,m,z∈X.
Then∂ is a cone metric and X,∂ is a cone metric space over the Banach algebra B.Definition 8. (see [13]).
LetX≠∅ and b≥1 be a real number. Then, the mapping ∂:X×X⟶ℝ+ is a b-metric if, for all s,m,z∈X, the following holds:
(𝔡1) ∂s,m=0 if and only if s=m;
(𝔡2) ∂s,m=∂m,s;
(𝔡3) ∂s,z≤b∂s,m+∂m,z.
Here, the pairX,∂ is a b-metric space.
The coneb-metric space over a Banach algebra with constant b≥1 is introduced in [30]. Mitrovic and Hussain in [26] introduced the cone b-metric space over a Banach algebra with constant b≥e.Definition 9. (see [26]).
LetX≠∅. A function ∂:X×X→B is a cone b-metric if
(𝔢1) θ⪯∂s,m for all s,m∈X, and ∂s,m=θ if and only if s=m;
(𝔢2) ∂s,m=∂m,s for all s,m∈X;
(𝔢3) There exists b∈CB, b⪰e such that ∂s,z⪯b∂s,m+∂m,z for all s,m,z∈X.
Here, the pairX,∂ is a cone b-metric space over B. If b=e, then X,∂ becomes a cone metric space over B.Definition 10. (see [2]).
LetX≠∅, ∂:X×X×X⟶ℝ+ satisfy the following conditions:
(𝔣1) For s,m∈X, there is a point z∈X with at least two of s,m,z which are not equal, then ∂s,m,z≠0;
(𝔣2) ∂s,m,z=0 if at least two of s,m,z are equal;
(𝔣3) For all s,m,z∈X,∂s,m,z=∂Ps,m,z where Ps,m,z stands for all permutations of s,m,z;
(𝔣4) For all s,m,z,t∈X,∂s,m,z≤∂s,m,t+∂s,z,t+∂m,z,t.
Then, the function∂ is a 2-metric and X,∂ is 2-metric space.Definition 11. (see [17]).
LetX≠∅ and b≥1 be a real number. Let ∂:X×X×X⟶B satisfy the following:
(𝔤1) For s,m∈X, there is a point z∈X with at least two of s,m,z which are not equal, then ∂s,m,z≠θ;
(𝔤2) ∂s,m,z=θ if at least two of s,m,z are equal;
(𝔤3) For all s,m,z∈X,∂s,m,z=∂Ps,m,z where Ps,m,z stands for all permutations of s,m,z;
(𝔤4) For all s,m,z,t∈X,∂s,m,z⪯b∂s,m,t+∂s,z,t+∂m,z,t.
Then, the function∂ is a cone b2-metric and X,∂ is a cone b2-metric space over the Banach algebra B with parameter b. It reduces to a cone 2-metric space if we take b=1 mentioned above. For other details about the cone 2-metric space over the Banach algebra B, we refer the reader to [31].Definition 12. (see [32]).
Letsn be a sequence in B, then
(𝔧1) sn is a c-sequence, if for each c⪼θ there exists a natural number N such that sn⪻c for all n>N;
(𝔧2) sn is a θ-sequence, if sn⟶θ as n⟶∞.Lemma 13. (see [33]).
LetB be Banach’s algebra and intCB≠∅. Also, let sn be c-sequences in B, then for arbitrary k∈CB, ksn is a c-sequence.Lemma 14. (see [33]).
LetB be Banach’s algebra and intCB≠∅. Let sn and zn be c-sequences in B. Let η and ζ∈CB be arbitrarily given vectors, then ηsn+ζzn is a c-sequence.Lemma 15. (see [33]).
LetB be Banach’s algebra and intCB≠∅. Let sn⊂CB such that sn⟶0 as n⟶∞. Then, sn is ac-sequence.Lemma 16. (see [28]).
Lete be the unit element of B, and s∈B, then limn→∞sn1/n exists and the spectral radius rs satisfies
(4)rs=limn→∞sn1/n=infsn1/n.
Ifrs<λ, then λe−s is invertible in B, moreover, we have
(5)λe−s−1=∑k=0∞skλk+1.Lemma 17. (see [28]).
Lete be the unit element of B, and s,m∈B. If s,m commute, then
(𝔨1) rs+m≤rs+rm;
(𝔨2) rsm≤rsrm.Lemma 18. (see [34]).
Lete be the unit element ofB, and CB≠∅. Let L∈B, and sn=Ln. If rL<1, then sn is a c-sequence.Lemma 19. (see [34]).
Lete be the unit element of B, and s∈B. Let λ be a complex number, and rs<λ, then
(6)rλe−s−1≤1λ−rs.Lemma 20. (see [35]).
LetCB⊂B be a cone.
(𝔩1) If s,m∈B, k∈CB, and s⪯m, then ks⪯km;
(𝔩2) If s,k∈CB are such that rk<1 and s⪯ks, then s=0;
(𝔩3) If k∈CB and rk<1, then for any fixed n∈ℕ, we have rkn<1.Lemma 21. (see [32]).
LetCB≠∅ and CB⊂B.
(𝔪1) Let k∈CB. Then kn is a θ-sequence if and only if rk<1;
(𝔪2) Every θ-sequence in B is a c-sequence;
(𝔪3) CB is normal if and only if each c-sequence in CB is a θ-sequence.Lemma 22. (see [36]).
LetB be a Banach algebra and intCB≠∅. Then the following are always true:
(𝔫1) If s,m,z∈B and s⪯m⪻z, then s⪻c;
(𝔫2) If s∈CB and s⪻c for each c⪼θ, then s=θ.Definition 23. (see [37]).
Let a coneb2-metric space be X,∂ over the Banach algebra B with parameter b, b⪰e, CB be a solid cone, α:X×X×X⟶CB, and d:X⟶X be two mappings. If for any sequence sn∈X, with αsn,sn+1,z≥e for each n∈ℕ and sn⟶s as n⟶∞, it follows that αsn,s,z≥e for all n∈ℕ and for all z∈X; then, we say that X,∂ is α-regular.


## 2. Results and Discussion

We introduced here the notion of coneb2-metric space over Banach’s algebra with parameter b≥e.Definition 24.
LetX≠∅ and ∂:X×X×X⟶B satisfy the following:
(𝔥1) For s,m∈X, there is a point z∈X with at least two of s,m,z which are not equal, then ∂s,m,z≠θ;
(𝔥2) ∂s,m,z=θ if at least two of s,m,z are equal;
(𝔥3) For all s,m,z∈X,∂s,m,z=∂Ps,m,z where Ps,m,z stands for all permutations of s,m,z;
(𝔥4) For all s,m,z,t∈X, there exists b∈CB, b≥e such that ∂s,m,z⪯b∂s,m,t+∂s,z,t+∂m,z,t.
Then, the function∂ is a cone b2-metric and X,∂ is a cone b2-metric space over Banach’s algebra with parameter b. It is reduced to a cone 2-metric space if we take b=e mentioned above.Remark 25.
Note that every cone 2-metric space is a cone b2-metric space with parameter b=e over Banach’s algebra. But the converse is not true.Example 26.
LetB=ℝ2. For each s1,s2∈B, s1,s2=s1+s2. The multiplication is defined by sm=s1,s2m1,m2=s1m1,s1m2+s2m1. Then B is a Banach algebra with unit e=1,0. Let CB=s1,s2∈ℝ2∣s1,s2≥0. Then CB is a cone in B.
LetX=k,0∈ℝ2∣k≥0∪0,2⊂ℝ2. Define ∂:X×X×X⟶B as follows:
(7)∂S,M,Z=0,0,ifatleasttwoofS,M,Zareequal,∂PS,M,Z,Pdenotespermutations,Δ,Δ,otherwise,where ∆=squareoftheareaoftriangleS,M,Z. We have
(8)∂s,0,m,0,0,2⪯∂s,0,m,0,z,0+∂s,0,z,0,0,2+∂z,0,m,0,0,2.
That is,s−m2⪯s−z2+z−m2, which shows that ∂ is not a cone 2-metric, because −9/2,−9/2∈CB for s,m,z≥0 with s=5, m=0, and z=1/2. But the parameter b=p,0⪰e with p≥2 is a cone b2-metric space over the Banach algebra B.Example 27.
LetB=Cℝ10,1. For each ft∈B, ft=ft∞+f′t∞. The multiplication is defined point wise. Then, B is a Banach algebra with unit e=1 a constant function. Let CB=ft∈B∣ft≥0,t∈0,1. Then, CB is a cone in B.
LetX=k,0∈ℝ2∣0≤k≤1∪0,1. Define ∂:X×X×X⟶B as follows:
(9)∂S,M,Z=∂pS,M,Z,pdenotespermutations,Δ·ft,otherwise,for all S,M,Z∈X, where ∆=squareoftheareaoftriangleS,M,Z and f:0,1⟶ℝ is such that ft=et. We have
(10)∂s,0,m,0,0,1·et⪯∂s,0,m,0,z,0·et+∂s,0,z,0,0,1·et+∂z,0,m,0,0,1·et.
That is,1/4s−m2·et⪯1/4s−z2+z−m2·et, which shows that ∂ is not a cone 2-metric, because −3/16et∉CB for 0≤s,m,z≤1 with s=1, m=0, and z=1/2. But the parameter b≥2∈CB is a cone b2-metric space over the Banach algebra B.Definition 28.
Let a coneb2-metric space be X,∂ over the Banach algebra B with parameter b, and let sn be a sequence in X,∂, then
(𝔦1) sn converges to s∈X if for every c⪼θ there exists N∈ℕ such that ∂sn,s,a⪻c for all n≥N. We denote it by
(11)limn→∞sn=s,or
(12)sn⟶sn⟶∞.
(𝔦2) If for c⪼θ there is N∈ℕ such ∂sn,sm,a⪻c for all n,m≥N, then sn is a Cauchy sequence.
(𝔦3) If every Cauchy sequence is convergent in X, then X,∂ is complete.
Next in the framework of coneb2-metric space over Banach’s algebra, we introduce the notion of α-admissibility of mappings [20] and give the consequence of Hardy and Rogers [18] through α-admissibility in cone b2-metric spaces over Banach’s algebras.Definition 29.
LetX≠∅ and CB be a cone in a Banach algebra B. We say d is α-admissible if d:X⟶X and α:X×X×X⟶CB, such that
(13)s,m∈X,αs,m,z⪰e∀z∈X⟹αds,dm,z⪰e∀z∈X.Definition 30.
Letd:X⟶X and X,∂ is a cone b2-metric space over the Banach algebra B. We say d is continuous at point s0∈X, if for every sequence sn∈X we have dsn⟶ds0 as n⟶∞, whenever sn⟶s0 as n⟶∞. d is continuous if it is continuous at every point of X.Definition 31.
Let a coneb2-metric space be X,∂ over a Banach algebra B with parameter b, b≥e, let CB be a solid cone, α:X×X×X⟶CB, and let d:X⟶X be two mappings. Then d is the α-admissible Hardy-Rogers contraction with vectors Ak∈CB, k∈1,⋯,5 such that ∑k=15rAk<1. If
(14)αs,m,z⪰e∀z∈X⟹αds,dm,z⪰e∀z∈X,αs,m,z∂ds,dm,z⪯A1∂s,m,z+A2∂s,ds,z+A3∂m,dm,z+A4∂s,dm,z+A5∂m,ds,z,for all s,m,z∈X with αs,m,z⪰e.Next, we ensure the existence of a fixed point for a continuous generalizedα-admissible Hardy-Rogers contraction mapping in the context of a cone b2-metric space over Banach’s algebra.Theorem 32.
Let a complete coneb2-metric space be X,∂ over the Banach algebra B with parameter b, b⪰e, and intCB≠∅. Let dii=1∞ be a family of self-maps from X to itself and vectors Ak∈CB, k∈1,⋯,5 such that
(15)αs,m,z∂dis,djm,z⪯A1∂s,m,z+A2∂s,dis,z+A3∂m,djm,z+A4∂s,djm,z+A5∂m,dis,z,for i,j≥1 and for all s,m,z∈X together with the following:
𝔬1 There is s0∈X such that αs0,dis0,z⪰e for all z∈X;
𝔬2dii=1∞ are continuous for all i≥1;
𝔬3A1,A2,A3,A4,A5,b commute with each other;
𝔬4rA3+A4b<1 and rA1+A2+A4b<1.
Thendii=1∞ share a common fixed point in X.Proof.
Chooses0∈X in such a way that
(16)αs0,dis0,z≥eforallz∈X.
Now, lets1=d1s0. Then, αs0,s1,z⪰e for all z∈X. Again, we put s2=d2s1 and using α-admissibility of di, we have
(17)αd1s0,d2s1,z=αs1,s2,z≥e.
Puttings3=d3s2 and using α-admissibility of di, we have
(18)αd2s1,d3s2,z=αs2,s3,z⪰e.
By induction, we construct a sequencesn in X by sn+1=dn+1sn for n∈ℕ such that
(19)αsn,sn+1,z≥eforallz∈X.
From condition (3) we obtain
(20)αsn−1,sn,z∂dnsn−1,dn+1sn,z⪯A1∂sn−1,sn,z+A2∂sn−1,dsnsn−1,z+A3∂sn,dn+1sn,z+A4∂sn−1,dn+1sn,z+A5∂sn,dnsn−1,z,that is
(21)αsn−1,sn,z∂sn,sn+1,z⪯A1∂sn−1,sn,z+A2∂sn−1,sn,z+A3∂sn,sn+1,z+A4∂sn−1,sn+1,z.
Since,αsn−1,sn,z⪰e, then we obtain for all z∈X and for all n∈ℕ(22)∂sn,sn+1,z≤αsn−1,sn,z∂sn,sn+1,z.
Therefore, (21) becomes
(23)e−A3∂sn,sn+1,z⪯A1+A2∂sn−1,sn,z+A4∂sn−1,sn+1,z⪯A1+A2∂sn−1,sn,z+A4b∂sn−1,sn+1,sn+∂sn−1,sn,z+∂sn,sn+1,z.
Assume that for anyt∈ℕ, we have
(24)αst−1,st,st−1∂dtst−1,dt+1st,st−1⪯A1∂st−1,st,st−1+A2∂st−1,dtst−1,st−1+A3∂st,dt+1st,st−1+A4∂st−1,dt+1st,st−1+A5∂st,dtst−1,st−1,that is
(25)αst−1,st,st−1∂st,st+1,st−1⪯A3∂st,st+1,st−1.
Since,αst−1,st,z⪰e for all z∈X, particularly, if z=st−1 for t∈ℕ, then we have αst−1,st,z≥e, and hence
(26)∂st,st+1,st−1⪯A3∂st,st+1,st−1,which is possible only when ∂st,st+1,st−1=θ by Lemma 20. Therefore, (5) becomes
(27)e−A3−A4b∂sn,sn+1,z⪯A1+A2+A4b∂sn−1,sn,z.
SincerA3+A4b≤rA3+rA4rb<1, from Proposition 2, we have
(28)∂sn+1,sn,z⪯L∂sn,sn−1,z,where L=e−A3−A4b−1A1+A2+A4b.
Similarly,∂sn,sn−1,z⪯L∂sn−1,sn−2,z, and hence we have for all z∈X(29)∂sn+1,sn,z≤Ln∂s1,s0,z.
In this case, for alll<k, proceeding in a similar way as above, we have
(30)∂sk,sk−1,sl⪯Lk−l−1∂sl+1,sl,sl=θ,that is
(31)∂sk,sk−1,sl=θ.
Therefore, for allm<n, we have
(32)∂sn,sm,z⪯b∂sn,sm,sn−1+∂sn,sn−1,z+∂sn−1,sm,z=b∂sn,sn−1,sm+b∂sn,sn−1,z+b∂sn−1,sm,z⪯b∂sn,sn−1,z+bb∂sn−1,sm,sn−2+b∂sn−1,sn−2,z+b∂sn−2,sm,z⪯b∂sn,sn−1,z+b2∂sn−1,sn−2,z+⋯+bn−m∂sm+1,sm,z⪯bLn−1+b2Ln−2+b3Ln−3+⋯+bn−mLm∂s1,s0,z=bn−mLme+b−1L+b−1L2+⋯+b−1Ln−m−1∂s1,s0,z=bn−mLm∑k=0∞b−1Lk∂s1,s0,z=e−b−1L−1bn−mLm∂s1,s0,z.
From Lemma17 and Lemma 19, we have
(33)rL=re−A3−A4b−1A1+A2+A4b≤rA1+rA2+rA4rb1−rA3−rA4rb<1.
AsrL<1, so that in the light of Remark 5, we can get to know bn−mLm∂s1,s0,z≤bn−mLm∂s1,s0,z⟶0 as (n⟶∞), by Lemma 15 we have bn−mLm∂s1,s0,z, a c-sequence in X. At last, by using Lemmas 13 and 22, we get that sn is a Cauchy sequence in X. In addition, X,∂ is complete; therefore, there exists some s∗∈X such that
(34)limn→∞sn=s∗.
Sincedi′s are continuous for i=1,2,3,⋯ .
Therefore, forsn⟶s∗, we have dn+1sn⟶dn+1s∗ as n⟶∞. But as sn+1=dn+1sn⟶dn+1s∗ as n⟶∞, therefore, from the uniqueness of the limit, we get dn+1s∗=s∗, that is, s∗ is a common fixed point of dii=1∞.Remark 33.
Our Theorem32 generalizes Theorem 1 in [38] from a cone b-metric space over a Banach algebra to a cone b2-metric space over a Banach algebra.Theorem 34.
Let a complete coneb2-metric space be X,∂ over a Banach algebra B with parameter b, b≥e, and intCB≠∅. Let di=1∞ be a family of self-maps from X to itself. Assume that mii=1∞ is a nonnegative integer sequence and vectors Ak∈CB, k∈1,⋯,5 such that
(35)∂dimis,djmjm,z⪯A1∂s,m,z+A2∂s,dimis,z+A3∂m,djmjm,z+A4∂s,djmjm,z+A5∂m,dimis,z,or
(36)∂dis,djm,z⪯A1∂s,m,z+A2∂s,dis,z+A3∂m,djm,z+A4∂s,djm,z+A5∂m,dis,z,for i,j≥1 and for all s,m,z∈X together with
𝔭1A1, A2, A3, A4, A5, and b commute with each other;
𝔭2rA3+A4b<1, rA1+A2+A4b<1, rbA2<1, and rA1+A4+A5<1.
Thendii=1∞ share a unique common fixed point in X.Proof.
On takingα=e in Theorem 32, set Θi=dimi for i=1,2,3,⋯. Then (35) becomes
(37)∂Θis,Θjm,z⪯A1∂s,m,z+A2∂s,Θis,z+A3∂m,Θjm,z+A4∂s,Θjm,z+A5∂m,Θis,z.
Chooses0∈X arbitrarily and construct a sequence sn by sn+1=Θsn+1sn for n∈ℕ, then using the same method as the proof of Theorem 32, one can easily show that sn is a Cauchy sequence, and hence from the completeness of X,∂, there exists s∗∈X such that
(38)limn→∞sn=s∗.
Now, we show thats∗ is a fixed point for a family of self-maps Θii=1∞:
(39)∂Θns∗,s∗,z⪯b∂Θns∗,s∗,sm+1+∂Θns∗,sm+1,z+∂sm+1,s∗,z=b∂sm+1,s∗,z+b∂Θns∗,Θm+1sm,z+b∂Θns∗,Θm+1sm,s∗⪯b∂sm+1,s∗,z+bA1∂s∗,sm,z+bA2∂s∗,Θns∗,z+bA3∂sm,sm+1,z+bA4∂s∗,sm+1,z+bA5∂sm,Θns∗,z+bA1∂s∗,sm,s∗+bA2∂s∗,Θns∗,s∗+bA3∂sm,sm+1,s∗+bA4∂s∗,sm+1,s∗+bA5∂sm,Θns∗,s∗⪯b∂sm+1,s∗,z+bA1∂s∗,sm,z+bA2∂s∗,Θns∗,z+bA3∂sm,sm+1,z+bA4∂s∗,sm+1,z+bA5∂sm,Θns∗,z+bA3∂sm,sm+1,s∗+bA5∂sm,Θns∗,s∗.
That is,(40)e−bA2∂Θns∗,s∗,z⪯b+bA4∂sm+1,s∗,z+bA1∂s∗,sm,z+bA3∂sm,sm+1,z+bA5∂sm,Θns∗,z+bA5∂sm,Θns∗,s∗+bA3∂sm,sm+1,s∗.
SincerbA2≤rbrA2<1, so by Proposition 2, we have e−bA2 which is invertible:
(41)∂Θns∗,s∗,z⪯e−bA2−1b+bA4∂sm+1,s∗,z+bA1∂s∗,sm,z+bA3∂sm,sm+1,z+bA5∂sm,Θns∗,z+bA5∂sm,Θns∗,s∗+bA3∂sm,sm+1,s∗.
Keepingn fixed and using Lemma 13 and Lemma 14, the right-hand side of the above inequality is a c-sequence.
Therefore, for anyc∈X with c⪼θ and using Lemma 22, we have ∂Θns∗,s∗,z=θ for all z∈X. Hence, Θns∗=s∗ for all n=1,2,3,⋯, that is, s∗ is a fixed point of Θn.
Assume thato∗ be another fixed point of Θn, that is, Θno∗=o∗. Then using (37), we have
(42)∂s∗,o∗,z=∂Θns∗,Θo∗,z⪯A1∂s∗,o∗,z+A2∂s∗,s∗,z+A3∂o∗,o∗,z+A4∂s∗,o∗,z+A5∂o∗,s∗,z=A1+A4+A5∂s∗,o∗,z,that is, ∂s∗,o∗,z=θ for all z∈X.
Therefore,s∗=o∗ is the unique fixed point of Θnn=1∞.
Thus, we haveΘns∗=dnmns∗=s∗.
Also,dns∗=dndnmns∗=dnmndns∗=Θndns∗.
That is,dns∗=Θndns∗, which implies that dns∗ is also a fixed point of Θn. But the fixed point of Θn is unique which is s∗; therefore, we must accept that dns∗=s∗.
For uniqueness, letdnz∗=z∗. That is, dnmnz∗=z∗=Θnz∗.
Since the fixed point ofΘn is unique and is s∗, therefore, s∗=z∗.Remark 35.
Our Theorem34 generalizes Theorem 3.2 in [27] from a cone 2-metric space over a Banach algebra to a cone b2-metric space over Banach’s algebra.
From Theorem34, we obtain the following corollaries.Corollary 36.
Let a complete coneb2-metric space be X,∂ over the Banach algebra B with parameter b, b⪰e, and intCB≠∅. Let dii=1∞ be a family of self-maps from X to itself. Assume that mii=1∞ is a nonnegative integer sequence and vectors Ak∈CB,k∈1,2,3 such that
(43)∂dimis,djmjm,z⪯A1∂s,m,z+A2∂s,dimis,z+A3∂m,djmjm,z,or
(44)∂dis,djm,z⪯A1∂s,m,z+A2∂s,dis,z+A3∂m,djm,z,for all positive integers i,j and for all s,m,z∈X with the following conditions:
(𝔮1) A1, A2, A3, and b commute with each other;
(𝔮2) ∑k=13rAk+2rA2rb<1.
Thendii=1∞ shares a unique common fixed point in X.Proof.
By takingA4=A5=θ in Theorem 34, we can get the required unique fixed point for dii=1∞.Remark 37.
Our Corollary36 generalizes Theorem 6.1 in [17] and Theorem 3.1 in [31]. It also extends Corollary 3.1 in [27] from a cone 2-metric space to a cone b2-metric space over a Banach algebra.Corollary 38.
Let a complete coneb2-metric space be X,∂ over a Banach algebra B with parameter b, b⪰e, and intCB≠∅. Let dii=1∞ be a family of self-maps from X to itself. Assume that mii=1∞ is a nonnegative integer sequence and vectors A1∈CB such that
(45)∂dimis,djmjm,z⪯A1∂s,m,z,or
(46)∂dis,djm,z⪯A1∂s,m,z,for all s,m,z∈X with rA1<1. Then dii=1∞ shares a unique common fixed point in X.Proof.
By takingA2=A3=θ in Corollary 36, we can get the required unique fixed point for dii=1∞.Remark 39.
Corollary38 extends Corollary 3.4 in [27] from a cone 2-metric space to a cone b2-metric space over a Banach algebra and Corollary 6.2 in [17].In the next theorem, the continuity assumption is relaxed.Theorem 40.
Let a complete coneb2-metric space be X,∂ over a Banach algebra B with parameter b, b⪰e, and intCB≠∅. Let dii=1∞ be a family of self-maps from X to itself and vectors Ak∈CB, k∈1,⋯,5 such that
(47)αs,m,z∂dis,djm,z⪯A1∂s,m,z+A2∂s,dis,z+A3∂m,djm,z+A4∂s,djm,z+A5∂m,dis,z,for i,j≥1 and for all s,m,z∈X together with
𝔯1 There is s0∈X such that αs0,dis0,z⪰e for all z∈X;
𝔯2X,∂ is α-regular;
𝔯3A1, A2, A3, A4, A5, and b commute with each other;
𝔯4rbA3+b2A4<1;
Then,dii=1∞ shares a common fixed point in X.Proof.
Chooses0∈X in such a way that αs0,dis0,z⪰e for all z∈X, and construct a sequence sn in X by sn+1=dn+1sn such that αsn,sn+1,z⪰e for all z∈X and n∈ℕ. Then, by using the same method as the proof of Theorem 32, one can get that xn is a Cauchy sequence in X,∂. But, as X,∂ is complete, there exists s∗∈X such that
(48)limn→∞sn=s∗.
Sinceαsn,sn+1,z⪰e and X,∂ is α-regular such that sn⟶s∗ as n⟶∞; therefore, αsn,s∗,z⪰e for all z∈X and n∈ℕ.
Now, we obtain thats∗ is a fixed point of di. Namely, we have
(49)∂dns∗,s∗,z⪯b∂dns∗,s∗,dn+1sn+b∂dns∗,dn+1sn,z+b∂dn+1sn,s∗,z.
As(50)∂dn+1sn,dns∗,s∗⪯αsn,s∗,s∗∂dn+1sn,dns∗,s∗,∂dn+1sn,dns∗,z⪯αsn,s∗,z∂dn+1sn,dns∗,z,and di′s are the α-admissible Hardy-Rogers contraction; therefore, (49) becomes
(51)∂dns∗,s∗,z⪯bαsn,s∗,s∗∂dn+1sn,dns∗,s∗+bαsn,s∗,z∂dn+1sn,dsx∗,z+b∂sn+1,s∗,z⪯bA1∂sn,s∗,s∗+bA2∂sn,sn+1,s∗+bA3∂s∗,dns∗,s∗+bA4∂sn,dns∗,s∗+bA5∂s∗,sn+1,s∗+bA1∂sn,s∗,z+bA5∂s∗,sn+1,z+b∂sn+1,s∗,z⪯bA2∂sn,sn+1,s∗+bA4∂sn,dns∗,s∗+bA1∂sn,s∗,z+bA2∂sn,sn+1,z+bA3∂s∗,dns∗,z+b2A4∂sn,dns∗,s∗+b2A4∂sn,s∗z+b2A4∂s∗,dns∗,z+bA5∂s∗,sn+1,z+b∂sn+1,s∗,z.
Becauselimn→∞∂sn,s∗,z=θ and limn→∞∂sn+1,sn,z=θ for all z∈X, we obtain
(52)∂dns∗,s∗,z≤bA3+b2A4∂dns∗,s∗,z.
Because,rbA3+b2A4<1, from Lemma 20, we claim that ∂dns∗,s∗,z=θ, that is, dns∗=s∗.Example 41.
Consider Example26 which is what we claim a complete cone b2-metric space over the Banach algebra B=ℝ2 with parameter b=2,0≥e. Define Γi:X⟶X by
(53)dis,0=si+4,0,ifs∈0,1,0,0,otherwise,di0,2=0,0foralls,0∈X,i∈ℕ.
Also, defineα:X×X×X⟶CB by
(54)αs,0,m,0,z1,z2=1,0,ifs,m∈0,1,0,0,otherwise,for all z=a1,a2∈X.
ChooseA1=1/2,0, A2=A3=1/4,0, and A4=A5=1/10,0. Clearly rA3+bA4=9/20<1 and rA1+A2+bA4=19/20<1.
Considering the contractive condition(55)αS,M,ZddiS,djM,Z⪯A1dS,M,Z+A2dS,diS,Z+A3dM,djM,Z+A4dS,djM,Z+A5dM,diS,Z,with A1=1/2,0, A2=A3=1/4,0, and A4=A5=1/10,0, we have the following eight cases:
(i)
S∈A, M∈A, and Z∈A(ii)
S∈B, M∈B, and Z∈A(iii)
S∈A, M∈B, and Z∈A(iv)
S∈B, M∈A, and Z∈A(v)
S∈B, M∈B, and Z∈B(vi)
S∈A, M∈A, and Z∈B(vii)
S∈A, M∈B, and Z∈B(viii)
S∈B, M∈A, and Z∈B
All the cases are trivial, except case (vi), in which case we have(56)αs,0,m,0,0,2∂dis,0,djm,0,0,2⪯12,0∂s,0,m,0,0,2+14,0∂s,0,si+4,0,0,2+∂m,0,mj+4,0,0,2+110,0∂s,0,mj+4,0,0,2+∂m,0,si+4,0,0,2.
Since(57)si+4−mj+42⪯12s−m2+143s+isi+42+3m+jmj+42+110js+4s−mj+42+im+4m−si+42,is always true for all i,j∈ℕ and s,m≥0, case (vi) is also satisfied.
The mappingsdi′s are α-admissible. In fact, let S,M∈X such that αs,0,m,0,Z⪰e for all Z∈X. By definition of α, it implies that s,m∈0,1. Therefore, for i,j∈ℕ and s,m∈0,1, we have dis,0=s/i+4,djm,0=m/j+4∈0,1, and so that αdis,0,djm,0,Z⪰e for all Z∈X.
Further, there iss0∈X such that αs0,dis0,Z⪰e for all Z∈X. Indeed, for s0=1,0, we have
(58)α1,0,di1,0,z1,z2=α1,0,1i+4,0,z1,z2⪰eforallz1,z2∈X.
Thus, all the assumptions of Theorem32 are fulfilled, and we conclude the existence of at least one fixed point for each di′s. Indeed, 0,0 is the common fixed point of the family of mapping dii=1∞.Next, we use the following property [20] to guarantee the uniqueness of the fixed point of di′s.(H). DenoteFixdi to be the set of all fixed points of dii=1∞. Assume for all s∗,o∗∈Fixdi, there exists m∈X such that αs∗,m,z≥e and αo∗,m,z⪰e for all z∈X.Theorem 42.
To add condition (H) in Theorem32 (resp., Theorem 40) we obtain uniqueness of the fixed point of each Γii=1∞.Proof.
Using related claims to those in the proof of Theorem32 (resp., Theorem 40), we achieve fixed-point existence. Let (H) be satisfied and s∗,o∗∈Fixdi′s and s∗≠o∗. By condition (H), there exists m∈X such that
(59)αs∗,m,z⪰e,αo∗,m,z⪰e,(60)forallz∈X.
Sincedi′s are α-admissible mappings and s∗,o∗∈Fixdi′s. From (59), we have
(61)αs∗,dinm,z⪰e,αo∗,dinm,z⪰e,forallz∈Xandi,n∈ℕ.
As,αs∗,dinm,z⪰e for all z∈X, therefore, we have
(62)∂s∗,dinm,z⪯αs∗,m,z∂dis∗,didin−1m,z⪯A1∂s∗,din−1m,z+A2∂s∗,dis∗,z+A3∂din−1m,didin−1m,z+A4∂s∗,didin−1m,z+A5∂din−1m,dis∗,z.
That is, we have(63)∂s∗,dinm,z⪯A1∂s∗,din−1m,z+A3∂din−1m,dinm,a+A4∂s∗,dinm,z+A5∂din−1m,s∗,z.
Hence, we have(64)e−A4∂s∗,dinm,z⪯A1+A5∂din−1m,s∗,z+A3∂din−1m,dinm,z.
Similarly, we have(65)∂s∗,dinm,z≤A1∂din−1m,s∗,z+A2∂din−1m,dinm,z+A4∂din−1m,s∗,z+A5∂s∗,dinm,z.
That is, we have(66)e−A5∂s∗,dinm,z⪯A1+A4∂din−1m,s∗,z+A2∂din−1m,dinm,z.
Adding up (64) and (66), we have
(67)2e−A4−A5∂s∗,∂inm,z⪯2A1+A4+A5∂∂in−1m,s∗,z+A2+A3∂∂in−1m,dinm,z.
Since,rA4+A5<1, therefore, we have
(68)∂s∗,dinm,z⪯2e−A4−A5−12A1+A4+A5∂din−1m,s∗,z+2e−A4−A5−1A2+A3∂din−1m,dinm,z.
That is, we have(69)∂s∗,dinm,z⪯L1∂din−1m,s∗,z+L2∂din−1m,dinm,z,where L1=2e−A4−A5−12A1+A4+A5 and L2=2e−A4−A5−1A2+A3. Hence, we have
(70)∂s∗,dinm,z⪯L1∂din−1m,s∗,z+L2∂din−1m,dinm,z⪯L12∂din−2m,s∗,z+L22∂din−2m,din−1m,z,⋮⋮⪯L1n∂m,s∗,z+L2n∂m,dim,z.
Since,rL1n<1 and rL2n<1, by Remark 5, it follows that L1n⟶0n⟶∞ and L2n⟶0n→∞, and so
(71)L1n∂m,s∗,z+L2n∂m,dim,z≤L1n∂m,s∗,z+L2n∂m,dim,z≤L1n∂m,s∗,z+L2n∂m,dim,z⟶0n→∞.
Therefore, based on Lemma15, we conclude that for any c∈B with c⪼θ, there exists N∈ℕ such that
(72)∂s∗,dinm,z⪯L1∂din−1m,s∗,z+L2∂din−1m,dinm,z⪻cforalli∈ℕandz∈X.
Hence,dinm⟶s∗n⟶∞. Similarly, we get that dinm⟶o∗n⟶∞. Then, by the uniqueness of the limit, we have s∗=o∗.


## 3. Applications

We give here a couple of auxiliary facts that will be used in our further considerations.LetB with norm ·B be a real infinite-dimensional Banach’s algebra. Let J=0,I and denote C=CJ,B the space consisting of all continuous functions defined on interval J with values in the Banach algebra B=ℝ∞ (the collection of all real sequences).The spaceC will be equipped with sC=maxsaB:a∈J.The purpose of this section is to establish and demonstrate a result on the existence of solutions of a class of an infinite system of integral equations of the form (74).LetX=C, and ∂:X×X×X⟶B be defined by
(73)∂sia,mia,zia=minsia−mia,mia−zia,sia−ziap,where i=1,2,3,⋯ and for all sia, mia, and zia∈X. Then X,∂ is a complete cone b2-metric space over Banach’s algebra. Consider the infinite system of integral equations
(74)sia=gia+∫0IHit,wfiw,s1w,s2w,⋯dw,where i=1,2,3,⋯ and let di:X⟶X be defined by
(75)disia=gia+∫0IHia,wfiw,s1w,s2w,⋯dsforalli=1,2,3,⋯.We assume thatd(1)
gi~J⟶ℝ are continuous(2)
Hi~J×ℝ⟶0,+∞ are continuous and ∫0IHit,w≤1(3)
fi~J×ℝ∞⟶ℝ are continuous such that(76)fiw,s1w,s2w,⋯−fiw,m1w,m2w,⋯≤r1/pminsiw−miw,miw−ziw,siw−ziw,for all ziw∈X, where 0≤r<1.Theorem 43.
Under the assumptions (1)–(3), the infinite system of integral equation (74) has a solution in C.Proof.
TakeB=ℝ∞ with norm o=o1,o2,⋯=∑i=1∞oi and multiplication defined by
(77)oq=o1,o2,⋯q1,q2,⋯=o1q1,o2q2,⋯.
LetCB=o=o1,o2,⋯∈B:o1,o2,⋯≥0. It is clear that CB is a normal cone, and B is a Banach algebra with unit e=1,0,⋯.
Consider the family of mappingdi:X⟶X defined by (75). Let sia, mia, and zia∈X.
From (15), we deduce that
(78)∂disia,dimia,zia=maxa∈0,Imindisia−dimia,dimia−zia,disia−ziap⪯maxa∈0,Idisia−dimiap=maxa∈0,I∫0IHia,wfiw,s1w,s2w,⋯dw−∫0IHia,wfiw,m1w,m2w,⋯dwp=maxa∈0,I∫0IHia,wfiw,s1w,s2w,⋯−fiw,m1s,m2w,⋯dwp⪯maxa∈0,I∫0IHia,wfiw,s1w,s2w,⋯−fiw,m1w,m2w,⋯dwp⪯maxa∈0,I∫0IHia,wr1/pminsiw−miw,miw−ziw,siw−ziwdwp⪯∫0Imaxa∈0,IHia,wr1/pmaxa∈0,Tminsiw−miw,miw−ziw,siw−ziwp1/pdwp⪯∫0Imaxa∈0,IHia,wr1/p∂sia,mia,zia1/pp=r∂sia,mia,ziasupa∈0,I∫0IHia,wp⪯r∂sia,mia,zia.
Therefore, we have(79)∂disia,dimia,zia⪯r∂sia,mia,zia.
Now, all the hypotheses of Corollary38 are satisfied, and the family of mapping dii=0∞ has a unique fixed point in X, which means that the infinite system of integral equations (74) has a solution.



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