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

In the current manuscript, the notion of a cone b2-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.


Introduction and Preliminaries
There are many generalizations in the literature about the concept of metric spaces like b-metric spaces [1], 2-metric spaces [2], N b -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 fixedpoint 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][11][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 b 2 -metric as a generalization of b-metric and 2-metric spaces. They illustrated some fixed-point theorems in a partially ordered b 2 -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 that ðX, ∂Þ is equivalent to any cone metric if the real-valued function ∂ * is replaced by a nonlinear scalariztion function ξ e or by a Minkowski functional q e . 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 b 2 -metric spaces over Banach's algebra with coefficient b ≥ 1 as an extension of b 2 -metric spaces and cone metric spaces over Banach's algebras. They also presented many fixedpoint 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][22][23][24][25][26][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 b 2 -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]). Let B be a real Banach algebra, and the multiplication operation is defined according to the following properties (for all s, m, z ∈ B and λ ∈ ℝ): (a 1 ) ðsmÞz = sðmzÞ; (a 2 ) sðm + zÞ = sm + sz and ðs + mÞz = sz + mz; (a 3 ) λðsmÞ = ðλsÞm = sðλmÞ; (a 4 ) ksmk ≤ kskkmk. We will presume in the course of this article that B 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]). Let e be the unit element of the Banach algebra B and s ∈ B be arbitrary. If the spectral radius rðsÞ < 1, that is then, e − s is invertible. In fact Remark 3. From [28] we see that, for all s in the Banach algebra B with unit e, we have rðsÞ ≤ ksk.
Definition 6. Let θ be the zero element of the unital Banach algebra B and C B ≠ ∅. Then, If there is M > 0 such that for all s, m ∈ C B , we have then, C B is normal. IfMis the least and positive among those cited above, then it is a normal constant of C B [14]. Onward, we assume that C B is a cone in B with int C B ≠ ∅, and ⪯ is a partial order with respect to the cone C B .
Here, the pair ðX, ∂Þ is a b-metric space. The cone b-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.

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Here, the pair ðX, ∂Þ is a cone b-metric space over B. If b = e, then ðX, ∂Þ becomes a cone metric space over B.
Then, the function ∂ is a cone b 2 -metric and ðX, ∂Þ is a cone b 2 -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 2metric space over the Banach algebra B, we refer the reader to [31].
Definition 12. (see [32]). Let fs n g be a sequence in B, then (j 1 ) fs n g is a c-sequence, if for each c⪼θ there exists a natural number N such that s n ⪻c for all n > N; (j 2 ) fs n g is a θ-sequence, if s n ⟶ θ as n ⟶ ∞.
Lemma 13. (see [33]). Let B be Banach's algebra and int C B ≠ ∅. Also, let fs n g be c-sequences in B, then for arbitrary k ∈ C B , fks n g is a c -sequence.

Lemma 14.
(see [33]). Let B be Banach's algebra and int C B ≠ ∅. Let fs n g and fz n g be c -sequences in B. Let η and ζ ∈ C B be arbitrarily given vectors, then fηs n + ζz n g is a c -sequence.

Lemma 15.
(see [33]). Let B be Banach's algebra and int C B ≠ ∅. Let fs n g ⊂ C B such that ks n k ⟶ 0 as n ⟶ ∞. Then, fs n g is a c -sequence.
Lemma 18. (see [34]). Let e be the unit element of B, and C B ≠ ∅. Let L ∈ B, and s n = L n . If rðLÞ < 1, then fs n g is a c -sequence.

Lemma 19.
(see [34]). Let e be the unit element of B, and s ∈ B. Let λ be a complex number, and rðsÞ < jλj, then Lemma 20. (see [35]). Let C B ⊂ B be a cone.
(l 1 ) If s, m ∈ B, k ∈ C B , and s⪯m, then ks⪯km; (l 2 ) If s, k ∈ C B are such that rðkÞ < 1 and s⪯ks, then s = 0; (l 3 ) If k ∈ C B and rðkÞ < 1, then for any fixed n ∈ ℕ, we have rðk n Þ < 1.
(m 1 ) Let k ∈ C B . Then fk n g is a θ-sequence if and only if rðkÞ < 1; (m 2 ) Every θ-sequence in B is a c-sequence; (m 3 ) C B is normal if and only if each c-sequence in C B is a θ-sequence.

Lemma 22.
(see [36]). Let B be a Banach algebra and int C B ≠ ∅. Then the following are always true: (n 1 ) If s, m, z ∈ B and s⪯m⪻z, then s⪻c; (n 2 ) If s ∈ C B and s⪻c for each c⪼θ, then s = θ.
Definition 23. (see [37]). Let a cone b 2 -metric space be ðX, ∂Þ over the Banach algebra B with parameter b, ðb ⪰ eÞ, C B be a solid cone, α : X × X × X ⟶ C B , and d : X ⟶ X be two mappings. If for any sequence fs n g ∈ X, with αðs n , s n+1 , zÞ ≥ e for each n ∈ ℕ and s n ⟶ s as n ⟶ ∞, it follows that αðs n , s, zÞ ≥ e for all n ∈ ℕ and for all z ∈ X; then, we say that ðX, ∂Þ is α -regular.

Results and Discussion
We introduced here the notion of cone b 2 -metric space over Banach's algebra with parameter b ≥ e.
Definition 24. Let X ≠ ∅ and ∂ : X × X × X ⟶ B satisfy the following: (h 1 ) For s, m ∈ X, there is a point z ∈ X with at least two of s, m, z which are not equal, then ∂ðs, m, zÞ ≠ θ; (h 2 ) ∂ðs, m, zÞ = θ if at least two of s, m, z are equal; (h 3 ) For all s, m, z ∈ X, ∂ðs, m, zÞ = ∂ðPðs, m, zÞÞ where Pðs, m, zÞ stands for all permutations of s, m, z;
Then, the function ∂ is a cone b 2 -metric and ðX, ∂Þ is a cone b 2 -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 b 2 -metric space with parameter b = e over Banach's algebra. But the converse is not true.
for all S, M, Z ∈ X, where Δ = square of the area of triangle S, M, Z and f : ½0, 1 ⟶ ℝ is such that fðtÞ = e t . We have That is, ð1/4Þðs − mÞ 2 · e t ⪯ð1/4Þððs − zÞ 2 + ðz − mÞ 2 Þ · e t , which shows that ∂ is not a cone 2-metric, because −ð3/16 Definition 28. Let a cone b 2 -metric space be ðX, ∂Þ over the Banach algebra B with parameter b, and let fs n g be a sequence in ðX, ∂Þ, then (i 1 ) fs n g converges to s ∈ X if for every c⪼θ there exists N ∈ ℕ such that ∂ðs n , s, aÞ⪻c for all n ≥ N. We denote it by or (i 2 ) If for c⪼θ there is N ∈ ℕ such ∂ðs n , s m , aÞ⪻c for all n, m ≥ N, then fs n g is a Cauchy sequence.
(i 3 ) If every Cauchy sequence is convergent in X, then ðX, ∂Þ is complete.
Next in the framework of cone b 2 -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 b 2 -metric spaces over Banach's algebras.
Definition 29. Let X ≠ ∅ and C B be a cone in a Banach algebra B. We say d is α-admissible if d : X ⟶ X and α : X × X × X ⟶ C B , such that Definition 30. Let d : X ⟶ X and ðX, ∂Þ is a cone b 2 -metric space over the Banach algebra B. We say d is continuous at point s 0 ∈ X, if for every sequence s n ∈ X we have ds n ⟶ ds 0 as n ⟶ ∞, whenever s n ⟶ s 0 as n ⟶ ∞. d is continuous if it is continuous at every point of X.
Definition 31. Let a cone b 2 -metric space be ðX, ∂Þ over a Banach algebra B with parameter b, ðb ≥ eÞ, let C B be a solid cone, α : X × X × X ⟶ C B , and let d : X ⟶ X be two mappings. Then d is the α -admissible Hardy-Rogers contraction with vectors A k ∈ C B , k ∈ f1, ⋯, 5g such that ∑ 5 k=1 r ðA k Þ < 1: If 4 Advances in Mathematical Physics for all s, m, z ∈ X with αðs, m, zÞ ⪰ e.
Next, we ensure the existence of a fixed point for a continuous generalized α-admissible Hardy-Rogers contraction mapping in the context of a cone b 2 -metric space over Banach's algebra.
Theorem 32. Let a complete cone b 2 -metric space be ðX, ∂Þ over the Banach algebra B with parameter b, ðb ⪰ eÞ, and int C B ≠ ∅. Let fd i g ∞ i=1 be a family of self-maps from X to itself and vectors A k ∈ C B , k ∈ f1, ⋯, 5g such that for i, j ≥ 1 and for all s, m, z ∈ X together with the following: o 1 There is s 0 ∈ X such that αðs 0 , d i ðs 0 Þ, zÞ ⪰ e for all z ∈ X; Proof. Choose s 0 ∈ X in such a way that Now, let s 1 = d 1 ðs 0 Þ. Then, αðs 0 , s 1 , zÞ ⪰ e for all z ∈ X. Again, we put s 2 = d 2 ðs 1 Þ and using α-admissibility of d i , we have Putting s 3 = d 3 ðs 2 Þ and using α-admissibility of d i , we have By induction, we construct a sequence fs n g in X by s n+1 = d n+1 ðs n Þ for n ∈ ℕ such that α s n , s n+1 , z ð Þ≥ e for all z ∈ X: that is Since, αðs n−1 , s n , zÞ ⪰ e, then we obtain for all z ∈ X and for all n ∈ ℕ Therefore, (21) becomes Assume that for any t ∈ ℕ, we have that is Since, αðs t−1 , s t , zÞ ⪰ e for all z ∈ X, particularly, if z = s t−1 for t ∈ ℕ, then we have αðs t−1 , s t , zÞ ≥ e, and hence which is possible only when ∂ðs t , s t+1 , s t−1 Þ = θ by Lemma 20. Therefore, (5) becomes Since rðA 3 + A 4 bÞ ≤ rðA 3 Þ + rðA 4 ÞrðbÞ < 1, from Proposition 2, we have 5
Similarly, ∂ðs n , s n−1 , zÞ⪯L∂ðs n−1 , s n−2 , zÞ, and hence we have for all z ∈ X ∂ s n+1 , s n , z ð Þ≤ L n ∂ s 1 , s 0 , z ð Þ : In this case, for all l < k, proceeding in a similar way as above, we have that is Therefore, for all m < n, we have From Lemma 17 and Lemma 19, we have As rðLÞ < 1, so that in the light of Remark 5, we can get to know kb n−m L m ∂ðs 1 , s 0 , zÞk ≤ kb n−m L m kk∂ðs 1 , s 0 , zÞk ⟶ 0 as (n ⟶ ∞), by Lemma 15 we have fb n−m L m ∂ðs 1 , s 0 , zÞg, a c-sequence in X. At last, by using Lemmas 13 and 22, we get that fs n g is a Cauchy sequence in X. In addition, ðX, ∂Þ is complete; therefore, there exists some s * ∈ X such that lim n→∞ s n = s * : Since d i ′ s are continuous for i = 1, 2, 3, ⋯ . Therefore, for s n ⟶ s * , we have d n+1 ðs n Þ ⟶ d n+1 ðs * Þ as n ⟶ ∞. But as s n+1 = d n+1 ðs n Þ ⟶ d n+1 ðs * Þ as n ⟶ ∞, therefore, from the uniqueness of the limit, we get d n+1 ðs * Þ = s * , that is, s * is a common fixed point of fd i g ∞ i=1 .
Remark 33. Our Theorem 32 generalizes Theorem 1 in [38] from a cone b -metric space over a Banach algebra to a cone b 2 -metric space over a Banach algebra.
Theorem 34. Let a complete cone b 2 -metric space be ðX, ∂Þ over a Banach algebra B with parameter b, ðb ≥ eÞ, and int C B ≠ ∅. Let fdg ∞ i=1 be a family of self-maps from X to itself. Assume that fm i g ∞ i=1 is a nonnegative integer sequence and vectors A k ∈ C B , k ∈ f1, ⋯, 5g such that or for i, j ≥ 1 and for all s, m, z ∈ X together with p 1 A 1 , A 2 , A 3 , A 4 , A 5 , and b commute with each other; p 2 rðA 3 + A 4 bÞ < 1, rðA 1 + A 2 + A 4 bÞ < 1, rðbA 2 Þ < 1, and rðA 1 + A 4 + A 5 Þ < 1.
Then fd i g ∞ i=1 share a unique common fixed point in X.
Proof. On taking α = e in Theorem 32, set Θ i = d m i i for i = 1, 2, 3, ⋯. Then (35) becomes Choose s 0 ∈ X arbitrarily and construct a sequence fs n g by s n+1 = Θ s n+1 ðs n Þ for n ∈ ℕ, then using the same method as the proof of Theorem 32, one can easily show that fs n g is a Cauchy sequence, and hence from the completeness of ðX, ∂Þ, there exists s * ∈ X such that lim n→∞ s n = s * : Now, we show that s * is a fixed point for a family of selfmaps fΘ i g ∞ i=1 : 6 Advances in Mathematical Physics That is, Since rðbA 2 Þ ≤ rðbÞrðA 2 Þ < 1, so by Proposition 2, we have ðe − bA 2 Þ which is invertible: Keeping n fixed and using Lemma 13 and Lemma 14, the right-hand side of the above inequality is a c-sequence.
Assume that o * be another fixed point of Θ n , that is, Θ n ðo * Þ = o * . Then using (37), we have that is, ∂ðs * , o * , zÞ = θ for all z ∈ X. Therefore, s * = o * is the unique fixed point of fΘ n g ∞ n=1 . Thus, we have Θ n ðs * Þ = d m n n ðs * Þ = s * . Also, d n ðs * Þ = d n ðd m n n ðs * ÞÞ = d m n n ðd n ðs * ÞÞ = Θ n ðd n ðs * ÞÞ: That is, d n ðs * Þ = Θ n ðd n ðs * ÞÞ, which implies that d n ðs * Þ is also a fixed point of Θ n . But the fixed point of Θ n is unique which is s * ; therefore, we must accept that d n ðs * Þ = s * .
Since the fixed point of Θ n is unique and is s * , therefore, s * = z * .
Remark 35. Our Theorem 34 generalizes Theorem 3.2 in [27] from a cone 2-metric space over a Banach algebra to a cone b 2 -metric space over Banach's algebra.
From Theorem 34, we obtain the following corollaries.
Corollary 36. Let a complete cone b 2 -metric space be ðX, ∂Þ over the Banach algebra B with parameter b, ðb ⪰ eÞ, and int C B ≠ ∅. Let fd i g ∞ i=1 be a family of self-maps from X to itself. Assume that fm i g ∞ i=1 is a nonnegative integer sequence and vectors A k ∈ C B , k ∈ f1, 2, 3g such that or for all positive integers i, j and for all s, m, z ∈ X with the following conditions: (q 1 ) A 1 , A 2 , A 3 , and b commute with each other; (q 2 ) ∑ 3 k=1 rðA k Þ + 2rðA 2 ÞrðbÞ < 1. Then fd i g ∞ i=1 shares a unique common fixed point in X.
Proof. By taking A 4 = A 5 = θ in Theorem 34, we can get the required unique fixed point for fd i g ∞ i=1 .
Remark 37. Our Corollary 36 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 b 2 -metric space over a Banach algebra.
Corollary 38. Let a complete cone b 2 -metric space be ðX, ∂Þ over a Banach algebra B with parameter b, ðb ⪰ eÞ, and int C B ≠ ∅. Let fd i g ∞ i=1 be a family of self-maps from X to itself. Assume that fm i g ∞ i=1 is a nonnegative integer sequence and vectors A 1 ∈ C B such that or 7 Advances in Mathematical Physics for all s, m, z ∈ X with rðA 1 Þ < 1. Then fd i g ∞ i=1 shares a unique common fixed point in X.
Proof. By taking A 2 = A 3 = θ in Corollary 36, we can get the required unique fixed point for fd i g ∞ i=1 .
Remark 39. Corollary 38 extends Corollary 3.4 in [27] from a cone 2-metric space to a cone b 2 -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 cone b 2 -metric space be ðX, ∂Þ over a Banach algebra B with parameter b, ðb ⪰ eÞ, and int C B ≠ ∅. Let fd i g ∞ i=1 be a family of self-maps from X to itself and vectors A k ∈ C B , k ∈ f1, ⋯, 5g such that for i, j ≥ 1 and for all s, m, z ∈ X together with r 1 There is s 0 ∈ X such that αðs 0 , d i ðs 0 Þ, zÞ ⪰ e for all z ∈ X; r 2 ðX, ∂Þ is α-regular; r 3 A 1 , A 2 , A 3 , A 4 , A 5 , and b commute with each other; r 4 rðbA 3 + b 2 A 4 Þ < 1; Then, fd i g ∞ i=1 shares a common fixed point in X.
Proof. Choose s 0 ∈ X in such a way that αðs 0 , d i ðs 0 Þ, zÞ ⪰ e for all z ∈ X, and construct a sequence fs n g in X by s n+1 = d n+1 ðs n Þ such that αðs n , s n+1 , zÞ ⪰ e for all z ∈ X and n ∈ ℕ . Then, by using the same method as the proof of Theorem 32, one can get that fx n g is a Cauchy sequence in ðX, ∂Þ. But, as ðX, ∂Þ is complete, there exists s * ∈ X such that lim n→∞ s n = s * : Since αðs n , s n+1 , zÞ ⪰ e and ðX, ∂Þ is α-regular such that s n ⟶ s * as n ⟶ ∞; therefore, αðs n , s * , zÞ ⪰ e for all z ∈ X and n ∈ ℕ. Now, we obtain that s * is a fixed point of d i . Namely, we have Because lim n→∞ ∂ðs n , s * , zÞ = θ and lim n→∞ ∂ðs n+1 , s n , zÞ = θ for all z ∈ X, we obtain Because, rðbA 3 + b 2 A 4 Þ < 1, from Lemma 20, we claim that ∂ðd n ðs * Þ, s * , zÞ = θ, that is, d n ðs * Þ = s * .
Further, there is s 0 ∈ X such that αðs 0 , d i ðs 0 Þ, ZÞ ⪰ e for all Z ∈ X. Indeed, for s 0 = ð1, 0Þ, we have Thus, all the assumptions of Theorem 32 are fulfilled, and we conclude the existence of at least one fixed point for each d i ′ s . Indeed, ð0, 0Þ is the common fixed point of the family of mapping fd i g ∞ i=1 .
Next, we use the following property [20] to guarantee the uniqueness of the fixed point of d i ′ s .
(H). Denote Fixðd i Þ to be the set of all fixed points of Assume for all s * , o * ∈ Fixðd i Þ, there exists m ∈ X such that αðs * , m, zÞ ≥ e and αðo * , m, zÞ ⪰ e for all z ∈ X.
Theorem 42. To add condition (H) in Theorem 32 (resp., Theorem 40) we obtain uniqueness of the fixed point of each