Nonlinear Contractions Employing Digraphs and Comparison Functions with an Application to Singular Fractional Differential Equations

: After the initiation of Jachymski’s contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem


Introduction
Fractional differential equations (abbreviated as FDEs) are generalisations of the ordinary differential equations to an arbitrary non-integer order.In the recent past, FDEs have been studied on account of their remarkable growth and relevance to the field of fractional calculus.For an extensive collection on the background of FDE, we refer the readers to consult [1][2][3][4][5] and the references therein.Various researchers (e.g., [6][7][8][9][10][11]) have discussed the existence theory of FDE employing the approaches of fixed point theory.Recall that a typical fractional BVP (abbreviation of 'boundary value problem') in a dependent variable ϑ and independent variable θ can be represented by Fixed point theory plays in metric space (in short, MS) a central role in nonlinear functional analysis.Throughout the foregoing century, BCP has been expanded and generalised by numerous authors.A common generalisation of this finding is to expand the standard contraction to φ-contraction by means of a proper auxiliary function φ : [0, ∞) → [0, ∞).A variety of generalisations has been developed through effectively modifying φ, resulting in a huge number of articles on this topic.Matkowski [12] invented a new class of φ-contraction that incorporated the concept of comparison functions, which has been further studied in ( [13][14][15][16][17]) besides several others.Quite recently, Pant [17] established an interesting non-unique fixed point theorem enlarging the class of φ-contractions in a complete metric space.
In 2008, Jachymski [18] established a very interesting approach in fixed point theory in the setup of graph metric space.Graphs are algebraic structures that subsume the partial ordering.The chief feature of the graphic approach is that the contraction condition is required to hold for merely certain edges of the underlying graph.This approach gave rise to an emerging discipline of research in metric fixed point theory, which led to the appearance of numerous works, e.g., see [19][20][21][22][23][24][25].In 2010, Bojor [19] extended the results of Jachymski [18] to (G, φ)-contraction in the sense of Matkowski [12].
The intent of this manuscript is to expand the outcomes of Bojor [19] adopting the idea of Pant [17] and to prove the fixed point theorems under the enlarged class of (G, φ)contraction in the setup of graph metric space.Employing the findings proved herewith, we study the existence and uniqueness of positive solutions of a particular form of BVP (1), such that the FDE remains singular.

Graph Metric Space
The set of real numbers (resp.natural numbers) are indicated by R (resp.N).By a graph G, we mean the pair (V(G), E(G)), whereas V(G) (known as set of vertices) and a set E(G) (known as set of edges) have a binary relation on V(G).

Definition 1 ([26]
).A graph is named as a digraph (or, directed graph) if every edge remains an ordered pair of vertices.Definition 2 ([26]).The transpose of a graph G, is a graph denoted by G −1 , described as Definition 4 ([26]).For any two vertices v and u in the graph G, a finite sequence {v 0 ,

Definition 5 ([26]
).A graph G is known as connected if any two vertices of G enjoy a path.If G is connected then G is referred as weakly connected.Definition 6 ([18]).Let (V, ϱ) be a MS and Definition 7 ([20]).Given a graph MS (V, ϱ, G), G is referred as a (C)-graph if for every sequence {v n } ⊂ V having the properties:

Given a digraph
, we adopt the succeeding notations: We are now going to demonstrate the following fpt in a graph MS over a class of (G, φ)-contractivity condition.
then R is a WPM.
Since (v 0 , Rv 0 ) ∈ E(G) and R is a G-edge preserving, by easy induction, we have Define If there is some n 0 ∈ N 0 with ϱ n 0 = 0, then by (3), we find On implementing (4) and the contractivity condition (2), we find Using monotonicity of φ in (5), we have or, With n → ∞ in ( 6) and employing the definition of φ, we find Choose ε > 0.Then, owing to (7), we can find n ∈ N 0 allows for Now, we seek to verify that {v n } is Cauchy.Implementing the monotonicity of φ, (5) and ( 8), we find Implementing the monotonicity of φ, transitivity of G, (4), (8), and the contractivity condition (2), we find By easy induction, one finds It turns out that {v n } continues to be Cauchy.Through the completeness of (V, ϱ), there for every k ∈ N 0 .By contractivity condition (2), we have Using Proposition 1 (whether ϱ(v n k , v) is zero or non-zero), the above inequality becomes Taking k → ∞ in the above inequality and using Next, we present the uniqueness theorem corresponding to Theorem 1.
Theorem 2. Let (V, ϱ, G) be a graph MS whereas (V, ϱ) is a complete MS and G is a transitive and weakly connected.Let R : V → V be a G-edge preserving map and V R ̸ = ∅.Also, assume that either, R is orbitally G-continuous, or, G is a (C)-graph.If there exists a comparison function φ then R is a PM.
Proof.In regard to Theorem 1, if v, u ∈ Fix(R), then, for every n ∈ N 0 , we find By the weak connectedness of G, there is a path {w 0 , w 1 , w 2 , . . .w p } between v and u, i.e., As R is G-edge preserving, we find for each 0 ≤ r ≤ p − 1 that The application of the triangle inequality reveals that For every r(0 To substantiate this, on fixing r, assuming first that δ r n 0 = 0 for some n 0 ∈ N 0 , then, R n 0 +1 (w r ) = R n 0 +1 (w r+1 ).Thus, we find δ r n 0 +1 = ϱ(R n 0 +1 w r , R n 0 +1 w r+1 ) = 0; so induc- tively, we find δ r n = 0 for every n ≥ n 0 , so that lim n→∞ δ r n = 0.In contrast, if δ r n > 0 for every n ∈ N 0 , then, by (9) and the contractivity condition (2), we get Using the monotonicity of φ in (11), we get If δ 0 = 0, then by Proposition 1, one gets δ r n = 0 yielding thereby lim n→∞ δ n = 0. Otherwise, in case δ 0 > 0, using the limit in (11) and the property of φ, one gets Thus in each case, one has lim Further, (10) can be written as which yields that v = u, so R has a unique fixed point.

Applications to Fractional BVP
Consider the following fractional BVP: along with the following assumptions: Obviously, the BVP ( 13) is identical to an integral equation given as under (14) where the Green function is As usual, Γ(•) and β(•, •) will denote the special functions: gamma function and beta function, respectively.Motivated by [8,9], we will determine the unique positive solution of (13).

•
G and H both are continuous; .
Proof.Making use of definition of G, we get where Applying the change of variables v = σ/θ so that θdv = dσ in the above integral, we find By ( 15) and ( 16), we obtain Naturally, the function ϕ(θ) remains increasing on [0, 1].Hence, we conclude In keeping with the argument of the proof of Lemma 1, we conclude Finally, we present the main results.
Theorem 4. Along with the assertions of Theorem 3, BVP (13) owns a unique positive solution.