EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6811 ISSN 1307-5543 – ejpam.com Published by New York Business Global Solving Fractional Differential Equations in b-Metric Spaces Tailored with a Directed Graph Dur-e-Shehwar Sagheer1, Haitham Qawaqneh2, Samina Batul1, Isma Urooj1, Zainab Rahman1, Hassen Aydi3,4,∗ 1 Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan 2 Al-Zaytoonah University of Jordan, Amman 11733, Jordan 3 Université de Sousse, Institut Supérieur d’Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia 4 Department of Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa Abstract. This study introduces an extended class of contractions incorporating auxiliary func- tions. We examine the existence of solutions for Caputo fractional differential equations with integral boundary conditions within the framework of b-metric spaces endowed with a directed graph. The established findings not only generalize but also unify a number of significant results in the existing literature, as demonstrated by supporting theoretical developments and illustrative examples. 2020 Mathematics Subject Classifications: 34A08, 47H10, 54H25 Key Words and Phrases: b-metric space (bMS), fractional differential equation, auxiliary func- tion, directed graph (DG), coincidence point (CP), Caputo fractional differential equation (CFDE) 1. Introduction and Preliminaries Fixed point theory constitutes a dynamic and significant area of mathematical research, with its principles providing foundational tools for a wide array of scientific disciplines. Its application to fractional differential equations, in particular, has been instrumental in advancing analytical techniques for solving complex nonlinear problems. The field has evolved considerably [1, 2] since Banach’s seminal contraction principle [3], leading to numerous extensions in generalized metric spaces. A prominent such generalization is the b-metric space (bMS), introduced independently by Czerwik [4] and Bakhtin [5] , which ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6811 Email addresses: d.e.shehwar@cust.edu.pk (D. e. Shehwar Sagheer), h.alqawaqneh@zuj.edu.jo (H. Qawaqneh), samina.batul@cust.edu.pk (S. Batul), mmt203011@cust.pk (I. Urooj), mmt193039@cust.pk (Z. Rahman), hassen.aydi@isima.rnu.tn (H. Aydi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 2 of 24 has since become a fertile ground for research. Notable contributions include the work of Afshari et al. [6] and Aydi et al. [7] on multivalued mappings, among other significant developments [8–16]. A pivotal expansion of this framework occurred when Jachymski [Jachymski] incorpo- rated the concept of directed graphs into metric fixed point theory, inspiring subsequent research in graph-based frameworks [17–21]. More recently, the integration of auxiliary functions has further enriched the theory. Karapinar et al. [22] demonstrated their effi- cacy in establishing existence and uniqueness results for fractional differential equations, an approach extended by Wongsaijai et al. [23] to metric spaces with graphs. Motivated by these advancements, this research aims to develop a unified approach within b-metric spaces endowed with a directed graph. The primary contributions of this work are fourfold: (i) We introduce a broader class of contractions integrated with auxiliary functions, extending the results of the aforementioned studies to the more general setting of bMSs. Previous research has established foundational results in this area. Karapinar et al. [22] demonstrated the existence of fixed points in the context of metric spaces; however, their work did not incorporate a directed graph framework. Similarly, Wongsaijai et al. [23] extended this line of inquiry by considering a directed graph, but their analysis was confined to a context that did not address the challenges posed by the triangular inequality in b-metric spaces. (ii) We establish significant theoretical results, including theorems on the existence of coincidence points, leveraging the properties of these auxiliary functions. (iii) We pioneer the analysis of fixed points in a context involving two distinct metrics on a non-empty set. (iv) The proposed framework is not merely theoretical; it provides an effective method for predicting solutions to fractional differential equations, which are critical in modeling phenomena across physics and engineering [24–26]. A key advantage of our results is their capacity to yield precise estimates for solutions, enhancing their practical utility. 2. Preliminaries In this work, X denotes a non-empty set and Ḡ = (V (Ḡ), E(Ḡ)) will represent a directed graph (DG). Let us examine some fundamental principles and implications that will serve as the stage for our primary findings. Definition 1. Let X 6= ∅ and b ∈ R such that b ≥ 1. A mapping db : X ×X → [0,∞) satisfying the following axioms is called a b-metric on X: d́b(1): db(x, y) = 0 ⇔ x = y; d́b(2): db(x, y) = db(y, x); D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 3 of 24 d́b(3): db(x, y) ≤ b{db(x, z) + db(z, y)}, for all x, y, z ∈ X. The pair (X, db) is said to be a bMS. Example 1. [27] Let X = lp(R) (with p ∈ (0, 1)) be the space of all real sequences having the property lp(R) = {x = xk ⊂ R}, such that ∑∞ k=1 |xk|p <∞ and define a function db : X ×X → R+ as db(x, y) = ( ∞∑ k=1 |xk − yk|p )1/p , then (X, db) is a bMS with b = 2 1 p . Encouraged by the ideas of Charoensawan et al. [28], we present some key concepts, particularly regarding common fixed points and coincidence points in bMSs endowed with a DG. Definition 2. Let (X, db) be a bMS and db be a continuous metric. Let ∆ denote the diagonal of X × X, then (X, db) is considered to be endowed with a directed graph Ḡ = (V (Ḡ), E(Ḡ)) if V (Ḡ) is the set of all the elements of X, and the set E(Ḡ) contains all elements of ∆, excluding parallel edges. Definition 3. Let (X, db) be a bMS together with Ḡ = (V (Ḡ), E(Ḡ)), and db be a continuous metric, then for T, S : X → X , C(T, S) = {x ∈ X : Tx = Sx}, is called the set of CPs of T and S. We also have Cm(T, S) = {x ∈ X : Tx = Sx = x}, denoting the set of common fixed points for the two mappings. Also, we define V (T, S) = {x ∈ X : (Tx, Sx) ∈ E(Ḡ)}. Remark 1. The set E(Ḡ) possesses the transitive property if (x, y), (y, z) ∈ E(Ḡ), then (x, z) ∈ E(Ḡ). Definition 4. Let (X, db) be a bMS together with a DG, where db is a continuous metric functional. A self-mapping T on X is called Gb-continuous at x ∈ X if for any sequence {xn} in X with (xn, xn+1) ∈ E(Ḡ), we have if xn −→ x Txn −→ Tx ∀ n ∈ N, and if this holds for all x ∈ X, then T is called Gb-continuous. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 4 of 24 Remark 2. In a bMS equipped with Ḡ = (V (Ḡ), E(Ḡ)), we say that (X, db, Ḡ) has property A if for any sequence {xn} in X with xn → x ∈ X, we have (xn, xn+1) ∈ E(Ḡ) (xn, x) ∈ E(Ḡ) ∀ n ∈ N. (1) Definition 5. Let (X, db) be a bMS endowed with a DG where db be a continuous metric functional. Also let T, S : X → X be two functions then T is called S-edge preserving with respect to Ḡ, if following condition holds, (Sx, Sy) ∈ E(Ḡ) ⇒ (Tx, Ty) ∈ E(Ḡ) ∀ x ∈ X. Agarwal et al. [29] demonstrated some excellent results for local and global fixed points. They used generalized contractions in the domain of metric spaces having two distance functions simultaneously. Inspired by his work, we present the following ideas, which are going to facilitate us in proving our main results. Let X be a non-empty set and we define db and d′b such that (X, db) and (X, d′b) both are bMSs. Then if we say db ≤ d′b then it means db(x, y) ≤ d′b(x, y) for all x, y ∈ X. If there are two functions T, S : X → X then T is said to be S-non-decreasing if Sx ≤ Sy Tx ≤ Ty ∀ x, y ∈ X, and if S is an identity map then we say T is called a non-decreasing map. Definition 6. Let (X, db) be a bMS endowed with a directed graph Ḡ = (V (Ḡ), E(Ḡ)), where db is a continuous metric.consider a sequence {xn} be a sequence in (X, db). The mappings T, S : X → X are db are called compatible, if we have lim n→∞ db(STxn, TSxn) = 0, whenever lim n→∞ Txn = lim n→∞ Sxn. Definition 7. Let (X, db) and (N , d′b) be two bMSs and T : X → N and S : X → X be two functions. Then T is S-Cauchy on X if for any sequence {xn} in X the sequence {Sxn} is Cauchy in X implies that the sequence {Txn} is Cauchy in (N , d′b). The following definition is analogous to those given in [23] and [22] in bMS. Definition 8. Let (X, db) be a bMS a function and {xn}, {yn} be two sequences in X such that the sequence {db(xn, yn)} is decreasing and convergent. A function ψ : X×X → [0, 1] satisfying the condition, if lim n→∞ ψ(xn, yn) = 1 then lim n→∞ db(xn, yn) = 0, is called an auxiliary function. Throughout the article, such a family of functions will be represented by Ψ = Ψ(X). Moreover, let the function φ : [0,∞) → [0,∞) be increasing and continuous satisfying the property φ(x) = 0x = 0. In the upcoming discussion, denote the collection of all such functions by Φ. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 5 of 24 3. Main Results In the following section, we derive several fixed point results in the framework of bMS implementing rational type contractions. In addition, an example and an application are provided to help readers understand our conclusion more thoroughly. Definition 9. Let (X, db) be a bMS together with a DG, Ḡ = (V (Ḡ), E(Ḡ)) and let T, S : X → X be two functions, where T is S-edge preserving with respect to Ḡ, then (T, S) is called a ψ − φ-contraction if for all x, y ∈ X with (Sx, Sy) ∈ E(Ḡ) there exist two functions φ ∈ Φ and ψ ∈ Ψ, such that φ(db(Tx, Ty)) ≤ ψ(Sx, Sy)φ(δ(M((Sx, Sx))), (2) where δ ≤ 1 bν for all ν > 3 and M : X ×X → [0,∞) for any x, y ∈ X is given as: M(Sx, Sy) =max { db(Sx, Tx)db(Ty, Sy) db(Sx, Sy) , db(Sx, Sy), db(Sx, Tx), db(Sy, Ty), db(Sx, Ty) + db(Sy, Tx) 2b } . (3) Lemma 1. [30] Let (X, db) be a bMS with b ≥ 1, and the sequences {xn}, {yn} ∈ X converge respectively to a and b. Then we have 1 b2 db(a, b) ≤ lim inf n→∞ db(xn, yn) ≤ lim sup n→∞ db(xn, yn) ≤ bdb(a, b). In particular, if a = b, then we have limn→∞ db(xn, yn) = 0. Moreover, for each c ∈ X, we have 1 b db(a, c) ≤ lim inf n→∞ db(xn, c) ≤ lim sup n→∞ db(xn, c) ≤ bdb(a, c). Lemma 2. Let (X, db) be a complete bMS equipped with a DG, and db be continuous metric. Also consider the pair (T, S) is a ψ−φ-contraction satisfying the following axioms: (1) T (X) ⊆ S(X) ; (2) E(Ḡ) is transitive; (3) lim n→∞ db(Sxn, Sxn+1) = 0, then {Sxn} is a Cauchy sequence in (X, db). Proof. Suppose on contrary that {Sxn} is not Cauchy and the sequences {jn}, {kn} ∈ N are such that jn > kn > n, db(Sxjn , Sxkn) ≥ ε, and db(Sxjn−1, Sxkn) < ε. (4) D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 6 of 24 Using the triangular inequality, ε ≤ db(xjn , xkn) ≤ b[db(xjn , xjn−1) + db(xjn−1, xkn)] < bdb(xjn , xjn−1) + bε. By letting limit n→ ∞, ε ≤ lim inf n→∞ db(xjn , xkn) ≤ lim sup n→∞ db(xjn , xkn) ≤ bε. (5) Also, one can easily get ε b ≤ lim inf n→∞ db(xjn , xkn+1) ≤ lim sup n→∞ db(xjn , xkn+1) ≤ b2ε, ε b ≤ lim inf n→∞ db(xjn+1, xkn) ≤ lim sup n→∞ db(xjn+1, xkn) ≤ b2ε, ε b2 ≤ lim inf n→∞ db(xjn+1, xkn+1) ≤ lim sup n→∞ db(xjn+1, xkn+1) ≤ b3ε. Now, as T (X) ⊆ S(X) holds, then we can construct a sequence xn ∈ X such that Sxn = Txn−1 for all n ∈ N then (Sx0, Tx0) = (Sx0, Sx1) ∈ E(Ḡ), and inductively we may write (Sxn−1, Sxn) ∈ E(Ḡ), for any n. For (Sxjn , Sxkn) ∈ E(Ḡ), we may write φ(db(Sxjn+1, Sxkn+1)) = φ(db(Txjn , Txkn)) ≤ ψ(db(Sxjn , Sxkn))φ(δ(M(Sxjn , Sxkn))) ≤ φ(δ(M(Sxjn , Sxkn))), (6) where M(Sxjn , Sxkn) = max { db(Sxjn , Txjn)db(Txkn , Sxkn) db(Sxjn , Sxkn) , db(Sxjn , Sxkn), db(Sxjn , Txjn), db(Sxkn , Txkn), db(Sxjn , Txkn) + db(Sxkn , Txjn) 2b } = max { db(Sxjn , Sxjn+1)db(Sxkn+1, Sxkn) db(Sxjn , Sxkn) , db(Sxjn , Sxkn), db(Sxjn , Sxjn+1), db(Sxkn , Sxkn+1), db(Sxjn , Sxkn+1) + db(Sxkn , Sxjn+1) 2b } . ≤ max { db(Sxjn , Sxjn+1)db(Sxkn+1, Sxkn) db(Sxjn , Sxkn) , db(Sxjn , Sxkn), db(Sxjn , Sxjn+1), db(Sxkn , Sxkn+1), db(Sxjn , Sxkn) + db(Sxkn , Sxkn+1) + db(Sxkn , Sxjn) + db(Sxjn , Sxjn+1) 2 } . After doing simple calculations and using the assumption (3), we get lim sup k→∞ M(Sxkn , Sxjn) ≤ lim sup db(Sxkn , Sxjn). D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 7 of 24 From (6), we obtain ε b2 ≤ lim sup k→∞ db(Sxjn+1, Sxkn+1)) ≤ lim sup k→∞ δ(db(Sxkn , Sxjn)). Also, by using property of φ, we get φ( ε b2 ) ≤ lim sup k→∞ φ(db(Sxjn+1, Sxkn+1)) ≤ lim sup k→∞ φ(δ(db(Sxkn , Sxjn))). Now, by the continuity of φ and using δ ≤ 1 bν we get φ( ε b2 ) ≤ φ( 1 bν (bε)), which is a contradiction as ν > 3. Hence, {Sxn} is a Cauchy sequence in (X, db). Theorem 1. Let (X, db) be a complete bMS with Ḡ = (V (Ḡ), E(Ḡ)) a DG and db be continuous. Let d′b be another continuous function and the pair (T, S) be an ψ − φ- contraction with respect to db together with the following: (i) S : (X, d′b) → (X, d′b) is continuous, and S(X) is closed w.r.t d′b; (ii) T (X) ⊆ S(X) ; (iii) E(Ḡ) is a transitive set; (iv) If db � d′b assume that T : (X, db) → (X, d′b) is S-Cauchy sequence on X; (v) T : (X, d′b) → (X, d′b) is Ḡb-continuous and T, S are d′b-compatible, then V (T, S) 6= ⇔ C(T, S) 6= . (7) Proof. If C(T, S) 6= then let x ∈ C(T, S). This means Tx = Sx. Then (Tx, Sx) = (Sx, Sx) ∈ ∆ ⊂ E(Ḡ). We have, (Sx, Sx) = (Tx, Sx) ∈ E(Ḡ) showing that x ∈ V (T, S) thus, V (T, S) 6= . Now, to verify other side, suppose that V (T, S) 6= and x0 ∈ X with (Sx0, Tx0) ∈ E(Ḡ). Now, as (2) holds then we can construct a sequence xn ∈ X such that Sxn = Txn−1 for all n ∈ N. Note that if Sxn = Sxn−1 for some n, then xn−1 ∈ C(T, S). This leads to a trivial case so suppose that Sxn 6= Sxn−1 for every n ∈ N. Now, as (Sx0, Tx0) = (Sx0, Sx1) ∈ E(Ḡ), and inductively we may write (Sxn−1, Sxn) ∈ E(Ḡ), for any n. Using contraction conditions, φ(db(Sxn+1, Sxn+2)) = φ(db(Txn, Txn+1)) ≤ ψ(db(Sxn, Sxn+1))φ(δ(M(Sxn, Sxn+1)))), (8) D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 8 of 24 where M(Sxn, Sxn+1) = max { db(Sxn, Txn)db(Txn+1, Sxn+1) db(Sxn, Sxn+1) , db(Sxn, Sxn+1), db(Sxn, Txn), db(Sxn+1, Txn+1), db(Sxn, Txn+1) + db(Sxn+1, Txn) 2b } . = max { db(Sxn, Sxn+1)db(Sxn, Sxn+1), db(Sxn+1, Sxn+2), db(Sxn, Sxn+2) + db(Sxn+1, Sxn+1) 2b } = max { db(Sxn, Sxn+1), db(Sxn+1, Sxn+2), db(Sxn, Sxn+2) 2b } ≤ max{db(Sxn, Sxn+1), db(Sxn+1, Sxn+2)}. We will observe both possibilities separately, if M(Sxn, Sxn+1) = db(Sxn+1, Sxn+2). Then, from (8), φ(db(Sxn+1, Sxn+2)) ≤ ψ(db(Sxn, Sxn+1))φ(δ(db(Sxn+1, Sxn+2))) ≤ ψ(db(Sxn, Sxn+1))φ(db(Sxn+1, Sxn+2)) ≤ φ(db(Sxn+1, Sxn+2)), for each n ≥ 0. As by assumption Sxn+1 6= Sxn+2 so db(Sxn+1, Sxn+2) > 0. Then we have φ(db(Sxn+1, Sxn+2)) > 0. Showing that limn→∞ ψ(db(Sxn, Sxn+1)) = 1. Thus db(Sxn, Sxn+1)) = 0. Also, in the same manner we get φ(db(Sxn+1, Sxn+2)) ≤ ψ(db(Sxn, Sxn+1))φ(δ(db(Sxn, Sxn+1))) ≤ ψ(db(Sxn, Sxn+1))φ(db(Sxn, Sxn+1)), (9) Showing that db(Sxn, Sxn+1) is a non-increasing sequence. Also, by using properties of φ, we observe that φ(db(Sxn, Sxn+1)) is a non-increasing sequence and it is also bounded below so that it would be a convergent sequence. Eventually, there exists c ≥ 0 such that (db(Sxn, Sxn+1)) = c. Now, suppose on contrary that lim n→∞ db(Sxn, Sxn+1) > 0. Also, lim n→∞ φ(db(Sxn, Sxn+1)) > 0, then form (9), we have 1 = lim n→∞ φ(db(Sxn+1, Sxn+2)) φ(db(Sxn, Sxn+1)) ≤ lim n→∞ ψ(db(Sxn, Sxn+1)) ≤ 1. Therefore, lim n→∞ ψ(db(Sxn, Sxn+1)) = 1 producing lim n→∞ db(Sxn, Sxn+1) = 0, i.e., c = 0, which is a contradiction. So we get lim n→∞ db(Sxn, Sxn+1) = 0. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 9 of 24 Lemma (2) shows that {Sxn} is a Cauchy sequence in (X, db). Finally, to show that {Sxn} is a Cauchy sequence in (X, d′b) too. Notice that if db ≥ d′b, proof is trivial. So take db � d′b. Now, as {Sxn} is a Cauchy sequence in (X, db) and T is S-Cauchy on X so have {Txn} is a Cauchy sequence in (X, d′b). This means their exists n0 ∈ N such that d′b(Sxn+1, Sxm+1) = d′b(Txn, Txm) < ε ∀m,n ≥ n0. Showing that {Sxn} is a Cauchy sequence in (X, d′b). Now as S(X) is closed with respect to d′b, and (X, d′b) is complete, so there exists x∗ ∈ S(X) such that lim n→∞ Sxn = lim n→∞ Txn = x∗. (10) Also, T is Ǧb-continuous together with d′b-compatibility of T and S. Hence, we conclude lim n→∞ d′b(TSxn, STxn) = 0. (11) Now, consider d′b(Sx ∗, Tx∗) ≤ b{d′b(Sx∗, STxn) + d′b(STxn, Tx ∗)} ≤ bd′b(Sx ∗, STxn) + b2{d′b(STxn, TSxn) + d′b(TSxn, Tx ∗)}. Taking n→ ∞ and by (10), we get d′b(Sx ∗, Tx∗) = 0. Showing that Sx∗ = Tx∗, thus x∗ ∈ C(T, S) ⇒ C(T, S) 6= . Theorem 2. Let (X, db) be a complete bMS equipped with a DG and a continuous metric db, let d′b be another continuous metric. Define two functions T, S : X → X where (T, S) satisfies ψ − φ-contraction with respect to db and we have, (1) S is a continuous mapping and S(X) is closed; (2) T (X) ⊆ S(X); (3) E(Ḡ) is a transitive set; (4) If db � d′b with T : (X, db) → (X, d′b) is g-Cauchy sequence on X; (5) (X, db, Ḡ) have property A (1). As a consequence, V (T, S) 6= ⇔ C(T, S) 6= . (12) Proof. We can observe that all other conditions of this result coincide with Theorem 1, so it is sufficient to check that even with assumption (5), we will get if V (T, S) 6= then C(T, S) 6= . D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 10 of 24 As {Sxn} is a Cauchy sequence and S(X) is closed in X, so for some x∗ ∈ X we have lim n→∞ Sxn = Sx∗ = lim n→∞ Txn. (13) We claim that x∗ ∈ C(T, S), since if, x∗ is not a coincident point of T and S, then Tx∗ 6= Sx∗. So, db(Tx ∗, Sx∗) > 0. (14) Then (Sxn, Sx ∗) ∈ E(Ḡ) for all n because (X, db, Ḡ) attains the property A. Also, db(Sx ∗, Tx∗) ≤ b{db(Sx∗, Txn(k)) + db(Txn(k), Tx ∗)}, or db(Sx ∗, Tx∗)− bdb(Sx ∗, Txn(k)) ≤ bdb(Txn(k), Tx ∗). We have φ(db(Sx ∗, Tx∗)− bdb(Sx ∗, Txn(k))) ≤ φ(bdb(Txn(k), Tx ∗)) ≤ ψ(db(Sxn(k), Sx ∗))φ(δ(bM(Sxn(k), Sx ∗))) ≤ ψ(db(Sxn(k), Sx ∗))φ(M(Sxn(k), Sx ∗)), (15) then M((Sxn(k), Sx ∗) =max { db(Sxn(k), Txn(k))db(Tx ∗, Sx∗) db(Sxn(k), Sx∗) , db(Sxn(k), Sx ∗), db(Sxn(k), Txn(k)), db(Sx ∗, Tx∗), db(Sxn(k), Tx ∗) + db(Sx ∗, Txn(k)) 2b } . Using (13) and taking limit n→ ∞, we will get lim n→∞ M((Sxn(k), Sx ∗) = lim n→∞ db(Sx ∗, Tx∗) > 0. From the condition (15), we have lim n→∞ ψ(db(Sxn(k), Sx ∗) = 1, hence lim n→∞ db(Sxn(k), Sx ∗) = db(Tx ∗, Sx∗) = 0. It is a contradiction to (14). Hence, x∗ ∈ C(T, S) proving that C(T, S) 6= . Theorem 3. If all the conditions of Theorem 1 are adopted and for any x, y ∈ C(T, S) with Sx 6= Sy, and it is true that (Sx, Sy) ∈ E(Ḡ) then V (T, S) 6= ⇔ Cm(T, S) 6= . Proof. From Theorem 1, there exists x ∈ X such that Sx = Tx. Suppose that there exists another element y ∈ X then Sy = Ty. We will prove that Sx = Sy. On the contrary, suppose it is not true then by assumption (Sx, Sy) ∈ E(Ḡ). Now, we may write φ(db(Ts, Sy)) ≤ ψ(db(Sx, Sy))φ(δ(M(Sx, Sy))) ≤ ψ(db(Sx, Sy))φ(M(Sx, Sy)) ≤ φ(M(Sx, Sy)) = φdb(Tx, Ty)). D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 11 of 24 Thus, ψ(Sx, Sy) = 1, hence Sy = Sx. Next, as x is the coincidence point so xn = x. We may construct a sequence such that Txn−1 = Sxn = Sx = Ts for every n ∈ N. Now, let r = Sx then Sr = SSx = STx. Then, lim n→∞ Txn = lim n→∞ Sxn = Tx, in (X, d′b). Also, lim n→∞ d′b(STxn, TSxn) = 0, since S and T are compatible with respect to d′b. This means STx = TSx, so Sr = STx = TSx = Tr, showing that r ∈ C(T, S). Also, from the above calculations Sr = Tr = Sx = r this shows r ∈ Cm(S, T ). 4. Consequences and an Application If we use δ(M((Sx, Sy))) = db(Sx, Sy)), then we will get following result: Corollary 1. Let (X, db) be a complete bMS with Ḡ as a DG, and db be continuous. Let d′b be another continuous function and the functions (T, S) satisfy an ψ − φ-contraction w.r.t db satisfying the axioms given below: (1) S : (X, d′b) → (X, d′b) is continuous with S(X) is closed w.r.t d′b; (2) T (X) ⊆ S(X) ; (3) E(Ḡ) is a transitive set; (4) If db � d′b, assume that T : (X, db) → (X, d′b) is S-Cauchy sequence in X; (5) T : (X, d′b) → (X, d′b) is Ḡb-continuous with T and S also being d′b-compatible; (6) there exist two functions φ ∈ Φ and ψ ∈ Ψ such that φ(db(Ts, T t)) ≤ ψ(Sx, Sy)φ(db(Sx, Sy))), then V (T, S) 6= ⇒ C(T, S) 6= . (16) Example 2. Let X = [0,∞) ⊆ R and define db, d′b : X ×X → X such that db(x, y) = |x− y|2 and d′b(x, y) = R|x− y|2, where R > 1 is any constant. It is easy to check that db and d′b are b metrics on X, also db < d′b. Now suppose that E(Ḡ) = {(x, y) : x = y or x, y ∈ [0, 1] with x ≤ y}. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 12 of 24 And, define two continuous self mappings on X as follow Sx = x2 and Tx = ln ( 1 + x2 4 ) for all x. Now, consider (Sx, Sy) ∈ E(Ḡ), note that if x = y then (Tx, Ty) ∈ E(Ḡ). Also, if (Sx, Sy) ∈ E(Ḡ) and Sx ≤ Sy then Sx = x2, Sy = y2 ∈ [0, 1] and x2 = Sx ≤ Sy = y2. Therefore, we have Tx = ln ( 1 + x2 4 ) ≤ ln ( 1 + y2 4 ) = Ty, showing that (Tx, Ty) ∈ E(Ḡ). Next, define φ(x) = x 4 and ψ : X ×X → [0, 1] as ψ(x, y) =  ln ( 2+ √ |x−y| 2 ) √ |x−y| , if |x− y| > 1 0, if |x− y| ∈ [0, 1). Now, we have φ(db(Tx, Ty) = | ln(1 + x2 4 )− ln(1 + y2 4 )| 2 4 = ( ln(1 + y2 4 )− ln(1 + x2 4 ) )2 4 = ln ( 2+ y2 2 1+x2 4 ) 4 = ln ( 2 + y2 2 −x2 2 1+x2 4 ) 4 ≤ ln(2 + ∣∣y2 2 − x2 2 ∣∣) 4 = ln(2 + 1 2 ∣∣y2 − x2 ∣∣) 4 ∣∣y2 − x2 ∣∣∣∣y2 − x2 ∣∣ ≤ ln(2 + 1 2 ∣∣y2 − x2 ∣∣)∣∣y2 − x2 ∣∣ ∣∣y2 − x2 ∣∣2 4 = ψ(Sx, Sy)φ(db(Sx, Sy), showing that the contraction condition holds. Now, as db < d′b we will show that T : (X, db) → (X, d′b) is S-Cauchy. If for ε > 0 the sequence {xn} ∈ X with {Sxn} being Cauchy in (X, db) there exists n0 ∈ N such that db(Sxn, Sxk) < ε R ∀ n, k ≥ n0. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 13 of 24 Therefore, d′b(Txn, Txk) = R ∣∣∣∣ ln(1 + (xn) 2 4 )− ln(1 + (xk) 2 4 ) ∣∣∣∣2 = R ( ln(1 + (xk) 2 4 )− ln(1 + (xn) 2 4 ) )2 = R ln ( 2 + (xk) 2 2 1 + (xn)2 4 ) = R ln ( 2 + (xk) 2 2 − (xn)2 2 1 + (xn)2 4 ) ≤ R ln ( 2 + ∣∣∣∣(xk)22 − (xn) 2 2 ∣∣∣∣) = R ln(2 + 1 2 ∣∣(xk)2 − (xn) 2 ∣∣)∣∣(xk)2 − (xn)2 ∣∣2 ∣∣(xk)2 − (xn) 2 ∣∣2 ≤ R ∣∣(xk)2 − (xn) 2 ∣∣2 = Rdb(Sxn, Sxk) < R ε R = ε, showing that T is S−Cauchy. Also, by using the fact that T and S are db-compatible, we may write lim n→∞ Sxn = lim n→∞ Txn = x, then ln(1 + x 4 ) = x this implies x = 0. And letting n→ ∞ we have d′b(STxn, TSxn) = R ∣∣∣∣( ln ( 1 + (xn) 2 4 ))2 − ln ( 1 + (xn) 4 4 )∣∣∣∣2 → 0. Finally, we have (S0, T0) = (0, 0) belongs to set of edges, that is, V (T, S) 6= ⇒ C(T, S) 6= . Now, we will prove that our results can be used to analyze the existence of solutions for fractional boundary value problems of the Caputo variety with fractional order γ. Here, we have γ ∈ (n− 1, n] where the integer n is such that n ≥ 2. Let Q(x) be a continuous function and γ be a real number. The γ order Caputo derivative is as follows cDγQ = Idγe−γDdγeQ. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 14 of 24 Here, Iγ is the Riemann-Liouville integral operator defined below IγQ(x) = 1 Γ(γ) ∫ x 0 (x− ξ)γ−1 Q(ξ) dξ, Γ(γ) = ∫ ∞ 0 xγ−1e−x dx. I0 is the identity operator. We will consider a nonlinear CFDE of the form (cDγy)(x) = υ ( x, y(x) ) , x ∈ [0, 1] and γ ∈ (n− 1, n] (17) along with y = y′ = · · · = y(n−2) = 0 at x = 0, and y = ∫ η 0 y(ξ) dξ at x = 1, (18) where υ : [0, 1]× R → R and η ∈ [0, 1]. This is a specific Volterra integral equation and the solution y ∈ C[0, 1] is of the form y(x) = a0 + a1x+ a2x 2 + · · ·+ an−1x n−1 + 1 Γ(γ) ∫ x 0 (x− ξ)γ−1 υ ( ξ, y(ξ) ) dξ, where ai are real numbers. The boundary conditions given in (4.3) mean a0 = a1 = · · · = an−2 = 0 y(x) = an−1x n−1 + Iγυ ( x, y(x) ) and y(1) = ∫ η 0 y(ξ) dξ = an−1 + 1 Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ, ⇒ an−1 = ∫ η 0 y(ξ) dξ − 1 Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ = ∫ η 0 ( an−1ξ n−1 + 1 Γ(γ) ∫ ξ 0 (ξ − ϑ)γ−1 υ ( ϑ, y(ϑ) ) dϑ ) dξ − 1 Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ = ηn n an−1 + 1 Γ(γ) ∫ η 0 ∫ ξ 0 (ξ − ϑ)γ−1 υ ( ϑ, y(ϑ) ) dϑdξ − 1 Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ = n (n− ηn)Γ(γ) ∫ η 0 ∫ ξ 0 (ξ − ϑ)γ−1 υ ( ϑ, y(ϑ) ) dϑdξ − n (n− ηn)Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ. Substituting this value in y(x) gives the solution of the boundary value problem given above to be the same as that of the Volterra integral equation, i.e. y(x) = nxn−1 (n− ηn)Γ(γ) ∫ η 0 ∫ ξ 0 (ξ − ϑ)γ−1 υ ( ϑ, y(ϑ) ) dϑdξ − nxn−1 (n− ηn)Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ + 1 Γ(γ) ∫ x 0 (x− ξ)γ−1 υ ( ξ, y(ξ) ) dξ. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 15 of 24 To transform this into a fixed point problem, construct T : C[0, 1] → C[0, 1] as below T ( y(x) ) = nxn−1 (n− ηn)Γ(γ) ∫ η 0 ∫ ξ 0 (ξ − ϑ)γ−1 υ ( ϑ, y(ϑ) ) dϑdξ − nxn−1 (n− ηn)Γ(γ) ∫ 1 0 (1− ξ)γ−1 υ ( ξ, y(ξ) ) dξ + 1 Γ(γ) ∫ x 0 (x− ξ)γ−1 υ ( ξ, y(ξ) ) dξ. The solution of the boundary value problem described in equations (4.2)-(4.3) is given by Ty = y. We will now show that a solution for this fixed point problem exists by considering( C[0, 1], || · ||∗∞ ) as a base space with a directed graph Ḡ. We define the b-metric || · ||∗∞ as ||Q −R||∗∞ = sup x∈[0,1] ∣∣Q(x)−R(x) ∣∣2, with b = 2. The following conditions are also important for our discussion. I. There exists Q0 ∈ C[0, 1] such that (Q0,TQ0) ∈ E(Ḡ). II. T is edge preserving on C[0, 1], III. For a sequence {Qn} ∈ C[0, 1] converging to Q with (Qn,Qn+1) ∈ E(Ḡ) ∀ n ∈ N, we must have (Qn,Q) is an edge for each n ∈ N, IV. Set of edges is transitive, V. ∃ φ ∈ Φ with φ(r) < r, ∀ r ∈ (0, 1] such that∣∣∣υ(x,Q(x) ) − υ ( x,R(x) )∣∣∣ ≤ K1φ (∣∣Q(x)−R(x) ∣∣), (19) and ∣∣∣υ(x,Q(x) ) − υ ( x,R(x) )∣∣∣2 ≤ K2φ (∣∣Q(x)−R(x) ∣∣2), (20) where K1 ≤ (n− ηn)Γ(γ + 2) nηγ+1 + (γ + 1)(2n− ηn) , K2 ≤ (2γ − 1)(n− ηn)2Γ(γ + 1)Γ(γ + 2) γ(γ + 1) [ n2η2γ + 2γ [ n2 + (n− ηn)2 ]] + 2(2γ − 1)n [ nγ+2 + ηγ+1(n− ηn) + (γ + 1)(n− ηn) ] , and K2 1 ( φ(|| Q −R ||∞) )2 ≤ K2 ( φ(|| Q −R ||∗∞) ) . VI. √∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ √∫ η 0 ∫ ξ 0 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣2 dϑdξ ≥ 1, D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 16 of 24√∫ 1 0 | 1− ξ |2(γ−1) dξ √∫ 1 0 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣2 dξ ≥ 1 and √∫ x 0 | x− ξ |2(γ−1) dξ √∫ x 0 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣2 dξ ≥ 1. Using these conditions, we will now consider the following result. Theorem 4. Consider a complete bMS (X, db) accompanied with a directed graph Ḡ =( V (Ḡ), E(Ḡ) ) and conditions I to VI are satisfied. Then the integral operator T defined above has a fixed point Q∗ ∈ C[0, 1]. Further, this fixed point is a solution for the boundary value problem given in (4.2) - (4.3). Proof. Let S be the identity map defined on C[0, 1]. From III, we see that T is S-edge preserving with respect to Ḡ. From IV, we have (C[0, 1], || · ||∞, Ḡ) has the property I. Finally, V shows that E(Ḡ) has the transitivity property. Using I, ∀ (Q,R) ∈ E(Ḡ), consider∣∣∣T(Q(x) ) − T ( R(x) )∣∣∣2 = ∣∣∣∣∣ nxn−1 (n− ηn)Γ(γ) ∫ η 0 ∫ ξ 0 (ξ − ϑ)γ−1 ( υ ( ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )) dϑdξ − nxn−1 (n− ηn)Γ(γ) ∫ 1 0 (1− ξ)γ−1 ( υ ( ϑ,Q(ξ) ) − υ ( ϑ,R(ξ) )) dξ + 1 Γ(γ) ∫ x 0 (x− ξ)γ−1 ( υ ( ξ,Q(ξ) ) − υ ( ξ,R(ξ) )) dξ ∣∣∣∣∣ 2 . Expanding the square on the right hand side and further manipulation gives∣∣∣T(Q(x) ) − T ( R(x) )∣∣∣2 ≤ n2x2(n−1) (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣ dϑdξ]2 + n2x2(n−1) (n− ηn)2Γ2(γ) [∫ 1 0 | 1− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ]2 + 1 Γ2(γ) [∫ x 0 | x− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ]2 + 2n2x2(n−1) (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣ dϑdξ][∫ 1 0 | 1− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ] D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 17 of 24 + 2nx(n−1) (n− ηn)Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣ dϑdξ][∫ x 0 | x− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ] + 2nx(n−1) (n− ηn)Γ2(γ) [∫ 1 0 | 1− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ][∫ x 0 | x− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ]. The Cauchy-Schwarz inequality states ∫ b a T (x)S(x) dx ≤ √∫ b a T 2(x) dx √∫ b a x2(x) dx . Assuming that the value on the right hand side is not less than 1 and squaring both sides, we get [ ∫ b a T (x)S(x) dx ]2 ≤ ∫ b a T 2(x) dx ∫ b a x2(x) dx. Applying this to the first 3 terms of the right-hand side in the contraction condition above, we get∣∣∣T(Q(x) ) − T ( R(x) )∣∣∣2 ≤ n2x2(n−1) (n− ηn)2Γ2(γ) ∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ ∫ η 0 ∫ ξ 0 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣2dϑdξ + n2x2(n−1) (n− ηn)2Γ2(γ) ∫ 1 0 | 1− ξ |2(γ−1) dξ ∫ 1 0 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣2 dξ + 1 Γ2(γ) ∫ x 0 | x− ξ |2(γ−1) dξ ∫ x 0 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣2 dξ + 2n2x2(n−1) (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣ dϑdξ][∫ 1 0 | 1− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ] + 2nx(n−1) (n− ηn)Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 ∣∣∣υ(ϑ,Q(ϑ) ) − υ ( ϑ,R(ϑ) )∣∣∣ dϑdξ][∫ x 0 | x− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ] + 2nx(n−1) (n− ηn)Γ2(γ) [∫ 1 0 | 1− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ][∫ x 0 | x− ξ |γ−1 ∣∣∣υ(ξ,Q(ξ) ) − υ ( ξ,R(ξ) )∣∣∣ dξ]. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 18 of 24 Using (4.4) and (4.5), we get ∣∣∣T(Q(x) ) − T ( R(x) )∣∣∣2 ≤ n2x2(n−1) (n− ηn)2Γ2(γ) ∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ ∫ η 0 ∫ ξ 0 K2φ ( | Q −R |2 ) dϑdξ + n2x2(n−1) (n− ηn)2Γ2(γ) ∫ 1 0 | 1− ξ |2(γ−1) dξ ∫ 1 0 K2φ ( | Q −R |2 ) dξ + 1 Γ2(γ) ∫ x 0 | x− ξ |2(γ−1) dξ ∫ x 0 K2φ ( | Q −R |2 ) dξ + 2n2x2(n−1) (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 K1φ ( | Q −R | ) dϑdξ ] [∫ 1 0 | 1− ξ |γ−1 K1φ ( | Q −R | ) dξ ] + 2nx(n−1) (n− ηn)Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 K1φ ( | Q −R | ) dϑdξ ] [∫ x 0 | x− ξ |γ−1 K1φ ( | Q −R | ) dξ ] + 2nx(n−1) (n− ηn)Γ2(γ) [∫ 1 0 | 1− ξ |γ−1 K1φ ( | Q −R | ) dξ ] [∫ x 0 | x− ξ |γ−1 K1φ ( | Q −R | ) dξ ] . Since x ∈ [0, 1], we have ∣∣∣T(Q(x) ) − T ( R(x) )∣∣∣2 ≤ ∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ ∫ η 0 ∫ ξ 0 K2φ ( | Q −R |2 ) dϑdξ + n2 (n− ηn)2Γ2(γ) ∫ 1 0 | 1− ξ |2(γ−1) dξ ∫ 1 0 K2φ ( | Q −R |2 ) dξ + 1 Γ2(γ) ∫ x 0 | x− ξ |2(γ−1) dξ ∫ x 0 K2φ ( | Q −R |2 ) dξ + 2n2 (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 K1φ ( | Q −R | ) dϑdξ ] [∫ 1 0 | 1− ξ |γ−1 K1φ ( | Q −R | ) dξ ] + 2n (n− ηn)Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 K1φ ( | Q −R | ) dϑdξ ] [∫ x 0 | x− ξ |γ−1 K1φ ( | Q −R | ) dξ ] + 2n (n− ηn)Γ2(γ) [∫ 1 0 | 1− ξ |γ−1 K1φ ( | Q −R | ) dξ ] [∫ x 0 | x− ξ |γ−1 K1φ ( | Q −R | ) dξ ] ≤ n2 (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ + ∫ 1 0 | 1− ξ |2(γ−1) dξ D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 19 of 24 + (n− ηn)2 n2 ∫ x 0 | x− ξ |2(γ−1) dξ ] [ K2φ ( || Q −R ||∗∞ )] + 2n2 (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 dϑdξ ∫ 1 0 | 1− ξ |γ−1 dξ + (n− ηn) n ∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 dϑdξ ∫ x 0 | x− ξ |γ−1 dξ + (n− ηn) n ∫ 1 0 | 1− ξ |γ−1 dξ ∫ x 0 | x− ξ |γ−1 dξ ][ K2 1 ( φ ( || Q −R ||∞ ))2 ] ≤ 2n2 (n− ηn)2Γ2(γ) [∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ + ∫ 1 0 | 1− ξ |2(γ−1) dξ + (n− ηn)2 n2 ∫ x 0 | x− ξ |2(γ−1) dξ + ∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 dϑdξ ∫ 1 0 | 1− ξ |γ−1 dξ + (n− ηn) n ∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 dϑdξ ∫ x 0 | x− ξ |γ−1 dξ + (n− ηn) n ∫ 1 0 | 1− ξ |γ−1 dξ ∫ x 0 | x− ξ |γ−1 dξ ][ K2φ ( || Q −R ||∗∞ )] . Let c = 2n2 (n− ηn)2Γ2(γ) sup x∈(0,1) [∫ η 0 ∫ ξ 0 | ξ − ϑ |2(γ−1) dϑdξ + ∫ 1 0 | 1− ξ |2(γ−1) dξ + (n− ηn)2 n2 ∫ x 0 | x− ξ |2(γ−1) dξ + ∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 dϑdξ ∫ 1 0 | 1− ξ |γ−1 dξ + (n− ηn) n ∫ η 0 ∫ ξ 0 | ξ − ϑ |γ−1 dϑdξ ∫ x 0 | x− ξ |γ−1 dξ + (n− ηn) n ∫ 1 0 | 1− ξ |γ−1 dξ ∫ x 0 | x− ξ |γ−1 dξ ] . Further calculations give c = 2n2 (n− ηn)2Γ2(γ) [ η2γ 2γ(2γ − 1) + 1 2γ − 1 + (n− ηn)2 n2(2γ − 1) + nγ+1 γ2(γ + 1) + ηγ+1(n− ηn) nγ2(γ + 1) + n− ηn nγ2 ] . We see that K2 ≤ 1 c . This gives us ∣∣∣T(Q(x) ) − T ( R(x) )∣∣∣2 ≤ φ ( ||Q −R||∗∞ ) D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 20 of 24 = φ ( db(Q,R) ) . Define an auxiliary function ψ with domain in C[0, 1]× C[0, 1] and range in [0, 1] by ψ(Q,R) =  φ ( ||Q −R||∗∞ ) ||Q −R||∗∞ , if Q 6= R, 0, if Q = R. We see that all conditions of Theorem 3.5 are satisfied, and so there must be a fixed point Q∗ of T in C[0, 1]. We now consider the case where n = 2. This gives K1 ≤ (2− η2)Γ(γ + 2) 2ηγ+1 + (γ + 1)(4− η2) . Since η ∈ [0, 1], putting η = 1 gives K1 ≤ Γ(γ + 2) 5 + 3γ . Similarly, we get K2 ≤ (2γ − 1)Γ(γ + 1)Γ(γ + 2) 2 [ γ(γ + 1)(5γ + 2) + 2(2γ − 1)(γ + 2 + 2γ+2) ] . This gives the following corollary. Corollary 2. Let (C[0, 1], db) be a complete bMS with a DG. We define the transitive set of edges of the graph Ḡ by E(Ḡ) = {(Q,R) : ε ( Q(x),R(x) ) ≥ 0}. Suppose the following conditions are satisfied for all x ∈ [0, 1]. i. There exist ε : R2 → R and φ ∈ Φ with φ(r) < r ∀ r ∈ (0, 1] 3 ∀ Q,R ∈ C[0, 1] with ε ( Q(ξ),R(ξ) ) ≥ 0 ∀ ξ ∈ [0, 1], ∣∣υ(x,Q(x) ) − υ ( x,R(x) )∣∣ ≤ K1φ ( | Q(x)−R(x) | ) ≤ Γ(γ + 2) 5 + 3γ φ ( | Q(x)−R(x) | ) and∣∣υ(x,Q(x) ) − υ ( x,R(x) )∣∣2 ≤ K2φ ( | Q(x)−R(x) |2 ) ≤ (2γ − 1)Γ(γ + 1)Γ(γ + 2) 2 [ γ(γ + 1)(5γ + 2) + 2(2γ − 1)(γ + 2 + 2γ+2) ]φ( | Q(x)−R(x) |2 ) . ii. There exists Q0 ∈ C[0, 1] such that ε ( Q0(x),TQ0(x) ) ≥ 0. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 21 of 24 iii. For all Q,R ∈ C[0, 1], we have ε ( Q(x),R(x) ) ≥ 0ε ( TQ(x),TR(x) ) ≥ 0. iv. Let the sequence {Qn} ∈ C[0, 1] converge to Q ∈ C[0, 1] with ε ( Qn(x),Qn+1(x) ) ≥ 0 ∀ n ∈ N ⇒ ε ( Qn(x),Q(x) ) ≥ 0 ∀ n ∈ N. v. For all Q,R,W ∈ C[0, 1], we have ε ( Q(x),R(x) ) ≥ 0 and ε ( R(x),W(x) ) ≥ 0 ε ( Q(x),W(x) ) ≥ 0 . Then the boundary value problem (cDγy)(x) = υ ( x, y(x) ) , x ∈ [0, 1] and γ ∈ (1, 2] along with y(0) = 0 and y(1) = ∫ η 0 y(ξ)dξ, η ∈ [0, 1] has a solution Q∗ ∈ C[0, 1]. Next, we take the case when E(Ḡ) = C[0, 1] × C[0, 1]. Here, the conditions i - v can be given by the following single condition satisfied for all x ∈ [0, 1]. ∀ Q,R ∈ C[0, 1], ∃ φ ∈ Φ with φ(r) < r ∀ r ∈ (0, 1], 3∣∣υ(x,Q(x) ) − υ ( x,R(x) )∣∣ ≤ K1φ ( | Q(x)−R(x) | ) , ∣∣υ(x,Q(x) ) − υ ( x,R(x) )∣∣2 ≤ K2φ ( | Q(x)−R(x) |2 ) , where K1 ≤ (n− ηn)Γ(γ + 2) nηγ+1 + (γ + 1)(2n− ηn) , and K2 ≤ (2γ − 1)(n− ηn)2Γ(γ + 1)Γ(γ + 2) γ(γ + 1) [ n2η2γ + 2γ [ n2 + (n− ηn)2 ]] + 2(2γ − 1)n [ nγ+2 + ηγ+1(n− ηn) + (γ + 1)(n− ηn) ] . This gives the following corollary. Corollary 3. Let (X, db) be a complete bMS accompanied with a DG, Ḡ = ( V (Ḡ), E(Ḡ) ) with E(Ḡ) as defined above. Then the boundary value problem given by (4.2)-(4.3) has a solution Q∗ ∈ C[0, 1]. D. e. Shehwar Sagheer et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6811 22 of 24 5. Conclusion This manuscript presents the following contributions: • We introduce ψ − φ contraction mappings in the setting of bMSs with a directed graph. For these mappings, we derive conditions under which common fixed points exist and we present supporting examples. We also investigate generalized rational contractions for spaces with two metrics. • We validate our theoretical results by applying them to prove the existence of a solution for a Caputo-type fractional boundary value problem. • Our planned research includes developing iterative solution methods. We also sug- gest that these results can be extended to other fractional problems and to the more general framework of extendedbMSs. Acknowledgements We acknowledge the support of this research from Al-Zaytoonah University. Authors’ Contributions All authors contribute equally in this paper. Conflict of interest The authors declare that they have no conflict of interest. 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