Common Fixed-Points Technique for the Existence of a Solution to Fractional Integro-Differential Equations via Orthogonal Branciari Metric Spaces

: The idea of symmetry is a built-in feature of the metric function. In this paper, we investigate the existence and uniqueness of a ﬁxed point of certain contraction via orthogonal triangular α -orbital admissible mapping in the context of orthogonal complete Branciari metric spaces endowed with a transitive binary relation. Our results generalize and extend some pioneering results in the literature. Furthermore, the existence criteria of the solutions to fractional integro-differential equations are established to demonstrate the applicability of our results.


Introduction
In 1922, Banach [1] initiated the Banach contraction theorem that every contraction has a unique fixed point in complete metric space.In 2000, Branciari [2] first defined the notion of Branciari metric spaces, where the triangle inequality is replaced by the quadrilateral inequality for all distinct pairwise points.Turinici [3] proved fixed-point results using functional contractions, and Karpinar [4] proved some fixed-point theorems using implicit functions in the Branciari metric space.Samet et al. (2012) [5], who introduced admissible mapping in α-ψ contraction and is frequently used to generalize the results across different contractions.Popescu [6] proposed in 2014 triangular α-orbital admissible mapping, and many authors extended the results in these spaces; see [7][8][9][10][11][12].
Recently, Gordji et al. (reference [13]) introduced the attractive concept of orthogonal sets, followed by orthogonal metric spaces.Subsequently, they extended the fixed-point theorem by Banach to this newly constructed structure.In addition, they utilized their findings to establish the existence of a solution to an ordinary differential equation.Moreover, in [13,14], the authors improved and established a fixed-point result for F-contraction in this context.Many researchers have contributed to the theory from a variety of perspectives since Gordji created the notions of an orthogonal in [15][16][17][18][19][20][21] and references therein.
Fixed point theory is one of the outstanding fields of fractional differential equations; see [22][23][24][25][26] and references therein for more information.Baitiche, Derbazi, Benchohra, and Cabada [23] constructed a class of nonlinear differential equations using the ψ-Caputo fractional derivative in Banach spaces with Dirichlet boundary conditions in 2022.Importantly, Machado et al. [27] introduced a new history of fractional calculus.The majority of articles and publications on fractional calculus concentrate on the solvability of initial linear fractional differential equations in special function types.
The main benefit of fractional nonlinear differential equations is the possibility of explaining the dynamics of complex nonlocal systems with memory.Specifically, fractional nonlinear differential equations are a new field in which improved fixed-point methods may be utilized.Using the Banach contraction, Lakshmikantham and Rao [28] demonstrated the solution to the integro-differential equation.Ahmad et al. [29] established some existence results for fractional integro-differential equations with nonlinear conditions, and Sudsutad, Alzabut, Nontasawatsri, and Thaiprayoon [30] established some fixed-point results with mixed integro-differential boundary conditions as well as a stability analysis for a generalized proportional fractional Langevin equation with a variable coefficient.Sharma and Chandok [31] investigated Ulam's stability of the fixed-point problem via Caputo-type nonlinear fractional integro-differential equation in the setting of orthogonal metric spaces.Acar and Ozkapu [32] established an order for multivalued rational type F-contraction on orthogonal metric spaces.
In this paper, we initiate a new type of contraction map and develop fixed-point theorems in the context of an orthogonal concept of the Branciari metric spaces and triangular α-orbital admissible mappings, while Arshad et al. [12] proved this in the setting of Branciari metric spaces with a triangular α-orbital admissible.In contrast, we proved our solution to the Cauchy problem involving a fractional integro-differential equation employing a more general contraction operator.
This work consists of the following: The purpose of Section 2 is to offer some notations, basic definitions, and related results in orthogonal Branciari metric space.The main results are presented of this study in Section 3, while the application of the main statements is discussed in Section 4. Section 5 concludes with a discussion of the conclusion and proposal.

Preliminaries
Throughout this paper, the set of all natural numbers and the set of all real numbers are denoted by N and R, respectively.
The Branciari metric space concept has been introduced by Branciari [2].

Definition 1 ([2]
).Let L = ∅ and let π : L × L → R + such that, for all ζ = ξ ∈ L, and all p = q ∈ L, each of them distinct from ζ and ξ, (i Then, the pair (L, π) is said to be a Branciari metric space (BMS).
Branciari [2] introduced the following family of function as follows.
Example of an orthogonal set in a wheel graph.
Aiman et al. [21] introduced the concepts of orthogonal Branciari metric spaces and its related properties.
The following example shows that each α-admissible is an orthogonal α-admissible, but the converse is not true.
Example 2. Let L = [0, 1] with usual metric π and let H : L → L be defined by Kirk and Shahzad [10] introduced the following lemma assertion that a Branciari metric space is a Hausdorff topological space with a neighborhood basis.

Lemma 2 ([8]
).Let there exists a triangular α-orbital admissible self-map H on L and there exists Very recently, Arshad et al. [12] established the following main results in the setting of the Branciari metric space with triangular α-orbital admissible mapping.In this article, inspired by Muhammad's work, we introduce an orthogonal triangular α-orbital admissible mapping and an orthogonal triangular α-orbital attractive mapping via orthogonal generalized contraction.We present an application of our orthogonal generalized contraction to the solution of integro-differential equations.
Clearly, H is orthogonal triangular α-orbital admissible and H is orthogonal α-orbital admissible, but H is not orthogonal triangular α-admissible.
Arshad et al. [12] proved fixed-point results in Branrciari metric spaces via triangular αorbital admissible mappings.Inspired by [12], we prove fixed-point results via orthogonal triangular α-orbital admissible map using a continuity hypothesis.
Combining all cases, we thus have We obtain ζ * = Hζ * , a contradiction by our assumptions.Therefore, H has a periodic point.
Suppose fix(H) = φ.Then s > 1 and π(ζ * , Hζ * ) > 0. Now, which is a contradiction with k ∈ (0, 1).Therefore, we have a non-empty set of fixed points of H; that is, H has at least one fixed point.
Arshad et al. [12] proved fixed-point results in Branrciari metric spaces via triangular α-orbital admissible mappings.Inspired by [12], we prove the fixed point theorem on an orthogonal triangular α-orbital admissible mapping using without a continuity hypothesis.Theorem 2. Let H be a self-map on orthogonal complete Branciari metric space (L, ⊥, π) and a map α : L × L → [0, ∞) such that (i) We can find ϑ ∈ and κ ∈ (0, 1) satisfying where Then, H has a fixed point ζ * ∈ L.
Suppose fix {H} = φ.Then s > 1 and π(ζ * , Hζ * ) > 0. Now, which is a reductio absurdum.Thus, the set of fixed points of H is non-empty; that is, H has at least one fixed point.
Next, we provide an example that shows that Theorem 2 can be used to prove the existence of fixed-point results when such mapping is applicable.
Define the binary relation ⊥ on L by ζ⊥ξ if ζξ ≥ 0. Clearly, (L, ⊥, π) is an orthogonal complete BMS.Define the mapping H : L → L by Obviously, H is an orthogonal triangular α-orbital admissible mapping. Let Furthermore, the hypotheses of Theorem 2 are satisfied and hence, H has a fixed point.
Example 5. Let L = [0, 1) and let the metric on L be the Euclidean metric.Define ζ⊥ξ if ζξ ∈ {ζ, ξ} for all ζ, ξ ∈ L. Let H : L → L be a mapping defined by Then, it is easy to show that H is an O-contraction on L, but it is not a contraction.Now, we define an orthogonal α-orbital attractive map.

ξ) .
(ii) There exists Then, H has a unique fixed point ζ * ∈ L. Let By continuing this process, we obtain and From condition (i) and ( 13), then for every η ≥ 1, we write then inequality (15) turns into which is a contradiction.Therefore, Thus, from (15), we have This implies Thus, we have Setting η → ∞, we obtain which together with ( 2 ) gives lim From condition 3 , we can find r ∈ (0, 1) and l ∈ (0, ∞] satisfying Suppose that l < ∞.In this case, let B = l 2 > 0. From the definition of the limit, there exists η 0 ≥ 1 such that Since |x − l| ≤ ⇐⇒ l − ≤ x ≤ l + and l < ∞, we obtain l − ≤ x; this implies Then where A = 1 B .Now, suppose that l = ∞.Let B > 0 be an arbitrary positive number.From the definition of the limit, there exists η 0 ≥ 1 such that This implies where A = 1 B .Thus, in all cases, there exist A > 0 and η 0 ≥ 1 such that By using (4), we obtain Letting η → ∞ in the inequality (6), we obtain lim Thus, there exists Now, we show that ζ * is a periodic point in H. Conversely, we assume that H η ζ 1 = H ω ζ 1 , ∀ η, ω ≥ 1 such that η = ω.By (i) and Equation ( 14), we have Since ϑ is non-decreasing, from (20), we obtain Let I = {η} η∈N , satisfying In this case, from Equation ( 21), we obtain Taking η → ∞ in the above equation and by (17), we have for large η.

Conclusions and Open Problem
In this paper, we investigated the existence and uniqueness of a fixed point of orthogonal generalized contraction via orthogonal triangular α-orbital admissible mapping in an orthogonal complete Branciari metric space.Khalehoghli, Rahimi, and Eshaghi Gordji [33,34] presented a real generalization of the mentioned Banach's contraction principle by introducing R-metric spaces, where R is an arbitrary relation on L. We note that in a special case, R can be considered as R := [partially ordered relation], R := ⊥[orthogonal relation], etc.If one can find a suitable replacement for a Banach theorem that may determine the value of fixed point, then many problems can be solved in this R-relation.This will provide a structural method for finding a value of a fixed point.It is an interesting open problem to study the fixed-point results on R-complete R-Branciari metric spaces.