Metric Spaces over Banach Algebras with Applications to the Infinite System of Integral Equations

In this article, common fixed-point theorems for self-mappings under different types of generalized contractions in the context of the cone b2-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.


Introduction and Preliminaries
Gähler [1] proposed the definition of 2-metric spaces as a generalization of an ordinary metric space. He defined that dðs, m, zÞ 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 2metric 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, Dðs, m, zÞ 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 the D * -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 b 2 -metric spaces [11], Fernandez et al. [12] examined cone b 2 -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 b 2 -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]). ConsiderÛ the Banach algebra which is real, and the multiplication operation is defined under the below properties (for all s, m, z ∈Û, ρ ∈ ℝ): (a 1 ) ðsmÞz = sðmzÞ (a 2 ) sðm + zÞ = sm + sz and ðs + mÞz = sz + mz (a 3 ) ρðsmÞ = ðρsÞm = sðρmÞ (a 4 ) ksmk ≤ kskkmk Unless otherwise stated, we will assume in this article thatÛ is a real Banach algebra. If s ∈Û occurs, we call e the unit ofÛ, so that es = se = s. We callÛ a unital in this case. If an inverse element m ∈Û exists, the element s ∈Û 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 algebrâ U with unit e, and let s ∈Û be the arbitrary element. If the spectral radius rðsÞ < 1, i.e., then e − s is invertible. In fact, Remark 3. We see from [18] that rðsÞ ≤ ksk for all s ∈Û with unit e.
Definition 6. Consider the Banach algebraÛ with unit element e, zero element θÛ, and CÛ ≠ ∅. Then, CÛ ⊂Û is a cone inÛ if: If there is M > 0 such that for all s, m ∈ CÛ, we have then CÛ is normal. If M is least and positive in the above, then it is the normal constant of CÛ [7].
Definition 7 (see [7,9] 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] Then, ðℵ, dÞ over the Banach algebraÛ 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 algebraÛ. Definition 9 (see [1]). Let d : ℵ × ℵ × ℵ ⟶ ℝ + and ℵ ≠ ∅: (f 1 ) There is a point z ∈ ℵ for s, m ∈ ℵ such that dðs, m , zÞ ≠ 0, if at least two of s, m, z are not equal Definition 10 (see [12] Then, ðℵ, dÞ over the Banach algebraÛ with parameter b ≥ 1 is a cone b 2 -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 alge-braÛ.
Islam et al. [13] initiated the concept of the cone b 2 -metric space over the Banach algebra with parameter b ≽ e.
Definition 13 (see [13]). Consider that ðℵ, dÞ is a cone b 2 -metric space over the Banach algebraÛ with b ≽ e, and let fs n g be a sequence in ðℵ, dÞ; then, (i 1 ) fs n g is said to converge to s ∈ ℵ if for every c ≫ θÛ there is N ∈ ℕ such that dðs n , s, aÞ ≪ c for all n ≥ N. That is, (i 2 ) If for every c ≫ θÛ there exists N ∈ ℕ such that dð s n , s m , aÞ ≪ c for all m, n ≥ N, then we say that fs n g is a Cauchy sequence (i 3 ) ðℵ, dÞ is complete if every Cauchy sequence is convergent in ℵ Definition 14 (see [22]). Let a sequence fs n g be inÛ; then, sequence fs n g is a c -sequence, if for each c ≫ θÛ there is N ∈ ℕ such that s n ≪ c for all n > N.
Lemma 15 (see [23]). Consider the Banach algebraÛ and int CÛ ≠ ∅. Also, consider fs n g a c -sequence inÛ and k ∈ CÛ where k is arbitrary; then, fks n g is a c -sequence.
Lemma 16 (see [23]). Consider the Banach algebraÛ and int CÛ ≠ ∅. Let fs n g and fz n g be c-sequences inÛ. Then, for arbitrary η, ζ ∈ CÛ, we have fηs n + ζz n g which is also a c-sequence.
Lemma 17 (see [23]). Consider the Banach algebraÛ and int CÛ ≠ ∅. Let fs n g ⊂ CÛ such that ks n k ⟶ 0 as n ⟶ ∞. Then, fs n g is a c-sequence.

Journal of Function Spaces
Lemma 20 (see [24]). Consider the Banach algebraÛ, e is their unit element, and CÛ ≠ ∅. Let L ∈Û and s n = L n . If rð LÞ < 1, then fs n g is a c-sequence.
(n 1 ) If m, s, z ∈Û and m≼s ≪ z, then m ≪ s (n 2 ) If m ∈ CÛ and m ≪ c for c ≫ θÛ, then m = θÛ Lemma 23 (see [21]). Consider the Banach algebraÛ and int CÛ ≠ ∅. Let m ∈Û, and suppose that k ∈ CÛ is an arbitrary given vector such that m ≪ c for any θÛ ≪ c, then km ≪ c for any θÛ ≪ c.
If m = gs = f s 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 mappings g, f : ℵ ⟶ ℵ are said to be weakly compatible, whenever f s = gs and f gs = g f s for any s ∈ ℵ.
Lemma 28 (see [28]). Let the mappings g and f be weakly compatible self-maps of a set ℵ. If g and f have a unique point of coincidence m = f s = gs, then m is the unique common fixed point of g and f .

Main Results
In this section, in the framework of the cone b 2 -metric space over Banach algebras with parameter b ≽ e, we prove some common fixed-point results.
Proposition 29. Let ðℵ, dÞ over the Banach algebraÛ be the complete cone b 2 -metric space and CÛ ≠ ∅ be a cone inÛ. If a sequence fs n g in ℵ converges to s ∈ ℵ, then we have the following: (i) fdðs n , s, aÞg is a c-sequence for all a ∈ ℵ (ii) For any α ∈ ℕ, fdðs n , s n+α , aÞg is a c-sequence for all a ∈ ℵ Proof. 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. Let ðℵ, dÞ over the Banach algebraÛ be a cone b 2 -metric space with b ≽ e and CÛ ≠ ∅ be a cone inÛ. Let where , and fH l g ∞ l=1 have a unique point of coincidence in ℵ. Moreover, if fF j , H l g and fE i , G k g are weakly compatible, respectively, then Choose s 0 ∈ ℵ to be arbitrary. Since E i ðℵÞ ⊆ H l ðℵÞ and F j ðℵÞ ⊆ G k ðℵÞ for each i, j, k, l ≥ 1, there exists s 1 , s 2 ∈ ℵ such that S 1 ðs 0 Þ = J 2 ðs 1 Þ and T 2 ðs 1 Þ = I 3 ðs 2 Þ. Continuing this process, we can define fs n g by S 2n+1 ðs 2n Þ = J 2n+2 ðs 2n+1 Þ and T 2n+2 ðs 2n+1 Þ = I 2n+3 ðs 2n+2 Þ.
Finally, we show that S 2i+1 and I 2k+3 , T 2j+2 , and J 2l+2 have a unique point of coincidence in ℵ. Assume that there is another point z ∈ ℵ such that T 2n+2 ðxÞ = J 2n+2 = z; then, 8 Journal of Function Spaces That is, Hence, by Lemma 20, we have that dðq, z, aÞ = θÛ, and so q = z; that is, q is the unique point of coincidence of T 2j+2 and J 2l+2 .
Similarly, we also have q which is the unique point of coincidence of S 2i+1 and I 2k+3 by induction.
Also, as q = T 2n+2 ðqÞ = F η n n ðqÞ, so we have F n ðqÞ = F n ð F η n n ðqÞÞ = F η n n ðF n ðqÞÞ = T 2n+2 ðqÞ, that is, T 2n+2 ðF n ðqÞÞ = F n ð qÞ. But T 2n+2 ðqÞ = q is unique; therefore, F n ðqÞ = q for n = 1, 2, 3, ⋯. Similarly, G n ðqÞ = q and H n ðqÞ = q for n = 1, 2, 3, ⋯. Thus, the four families of mappings fE i g ∞ i=1 , fF j g ∞ j=1 , fG k g ∞ k=1 , and fH l g ∞ l=1 have a unique common fixed point. ☐ Remark 31. Theorem 30 of this paper extends and improves Theorem 2.1 of [30] from cone metric spaces to cone b 2 -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 Theorem 30 for four families of mappings in the context of cone b 2 -metric spaces over Banach algebras, which generalize and improve many known results from the existence literature.
where k ∈ CÛ with rðkÞ < 1 and k, b commute. If E i ðℵÞ ⊆ H l ðℵÞ, F j ðℵÞ ⊆ G k ðℵÞ, and one of E i ðℵÞ, G k ðℵÞ, H l ðℵÞ, and F j ðℵÞ are a complete subspace of ℵ for each i, j, k, l ≥ 1, then fE i g ∞ i=1 , fF j g ∞ j=1 , fG k g ∞ k=1 , and fH l g ∞ l=1 have a unique point of coincidence in ℵ. Moreover, if fF j , H l g and fE i , G k g are weakly compatible, respectively, then fE i g ∞ i=1 , fF j g ∞ j=1 , fG k g ∞ k=1 , and fH l g ∞ l=1 have a unique common fixed point.
Proof. Taking η n = 1, E i = F j , and G k , H l 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 Theorem 30.