EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 5916 ISSN 1307-5543 – ejpam.com Published by New York Business Global Exploring Fixed Point Theory in Quasi-Partial Metric Spaces: A Unified Approach with Applications in Integral Equations and Computational Sciences Haitham Qawaqneh Al-Zaytoonah University of Jordan, Amman 11733, Jordan Abstract. This paper explores the concept of C-class functions, which encompass a wide range of contractive conditions and applies them to derive common fixed-point results for two pairs of self-mappings in quasi-partial metric spaces. These findings extend existing theories by introducing generalized contractive conditions and providing a broader framework for fixed-point analysis. A detailed example is constructed to validate the results, illustrating the practical application of the established theorems. Furthermore, the paper demonstrates the relevance of these results by applying them to solve a system of integral equations diffusion reaction, highlighting their utility in addressing real-world mathematical problems. 2020 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Weakly compatible contraction, C-class functions, quasi partial metric spaces 1. Introduction The study of metric spaces has undergone significant evolution through various gen- eralizations, such as partial metric spaces, metric-like spaces, b−metric spaces and quasi- metric spaces. Partial metric spaces were first introduced by Matthews in [1, 2], providing a foundational framework for fixed-point theory in settings where self-distance may not be zero. Several fixed-point results have since been established in such spaces, as detailed in works like [3–5]. Later, Künzi et al. [6] expanded this concept by introducing par- tial quasi-metric spaces, which relaxed the symmetry condition inherent in partial metric spaces. Karapinar et al. [7] contributed to this area by renaming partial quasi-metric spaces as quasi-partial metric spaces and presenting some initial fixed-point results and properties. Further advancements in this domain can be found in [8–19]. This research builds upon the foundational concept of C-class functions, which provide a flexible and comprehensive framework for addressing a wide spectrum of contractive con- ditions. These functions have played a pivotal role in broadening the scope of fixed-point DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.5916 Email address: h.alqawaqneh@zuj.edu.jo (H. Qawaqneh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 2 of 19 theory, offering generalized approaches to analyzing self-mappings in various mathematical structures, as explored in [20–23]. In this study, we extend these ideas by introducing a novel class of Ciric-type contractive conditions specifically designed for quasi-partial met- ric spaces. Our primary goal is to derive new results concerning coincidence and common fixed points within this setting. The findings presented herein not only unify and expand previous work but also refine existing theoretical models by incorporating more general- ized contractive mechanisms. This advancement paves the way for broader applications in both mathematical analysis and applied sciences, particularly in the study of nonlinear equations and iterative algorithms. To reinforce the theoretical developments established in this work, we present a con- crete example that highlights the validity of our results. Furthermore, we demonstrate the practical significance of our findings by applying them to solve a system of Fred- holm integral equations as well as a system modeling diffusion-reaction processes. These applications underscore the utility of quasi-partial metric spaces in tackling intricate math- ematical challenges that arise in various scientific and engineering domains. By bridging abstract theoretical concepts with real-world problem-solving, our study showcases the adaptability and effectiveness of quasi-partial metric frameworks in analyzing nonlinear systems and iterative solution methods. 2. Preliminaries In this section, we present fundamental definitions and essential properties related to quasi-partial metric spaces. A partial metric space extends standard metric space concepts (when p(x, x) = 0 recovers standard metric) and allows modeling of objects with non-zero self-similarity. The mapping dp(x, y) = 2p(x, y)− p(x, x)− p(y, y) forms a standard metric. Definition 1. [1, 2] Let X be a non-empty set. A function p : X ×X → R+ is called a partial metric if it satisfies the following conditions for all x, y, z ∈ X: (i) Symmetry: p(x, y) = p(y, x). (ii) Non-negativity and Indistinguishability: If p(x, x) = p(x, y) = p(y, y) = 0, then x = y. (iii) Small Self-Distance: p(x, x) ≤ p(x, y). (iv) Modified Triangle Inequality: p(x, z) + p(y, y) ≤ p(x, y) + p(y, z). A pair (X, p) satisfying these conditions is called a partial metric space. Note that if p(x, y) = 0, then by conditions (1) and (2), we deduce that x = y. However, the converse does not necessarily hold; that is, if x = y, it does not imply that p(x, x) = 0. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 3 of 19 Example 1. [1] A fundamental example of a partial metric space is the pair ([0,∞), p), where the partial metric p : [0,∞)× [0,∞) → R+ is defined as p(x, y) = max{x, y}, for all x, y ∈ [0,∞). This function satisfies the properties of a partial metric. Notably, this example differs from standard metric spaces as it allows nonzero self-distances, i.e., p(x, x) = x ̸= 0 for x > 0, illustrating the essential characteristic of partial metric spaces. Definition 2. [6] A function q : X×X → R+ is called a quasi-partial metric if it satisfies the following conditions for all x, y, z ∈ X: (i) q(x, x) ≤ q(y, x), (ii) q(x, x) ≤ q(x, y), (iii) x = y if and only if q(x, x) = q(x, y) and q(y, y) = q(y, x), (iv) q(x, z) + q(y, y) ≤ q(x, y) + q(y, z). The pair (X, q) is then referred to as a quasi-partial metric space. Karapinar et al. [7] introduced an alternative condition: (3′) Equality: If 0 ≤ q(x, x) = q(x, y) = q(y, y), then x = y, replacing condition (3) in the definition of a quasi-partial metric. It is noteworthy that when q(x, y) = q(y, x) for all x, y ∈ X, the quasi-partial metric space (X, q) reduces to a partial metric space. Furthermore, for any quasi-partial metric q on X, the function dq : X ×X → R+ defined by dq(x, y) = q(x, y) + q(y, x)− q(x, x)− q(y, y) constitutes a basic metric on X Example 2. [6] Consider the set of non-negative real numbers R+ equipped with the function q : R+ × R+ → R+ defined by: (i) q(x, y) = |x− y|+ |x|, (ii) q(x, y) = max{y − x, 0}+ x. Then, (R+, q) forms a quasi-partial metric space. Definition 3. [7] Let (X, q) be a quasi-partial metric space. The following concepts are defined: (i) A sequence {xn} ⊂ X is said to converge to a point x ∈ X if q(x, x) = lim n→∞ q(x, xn) = lim n→∞ q(xn, x). H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 4 of 19 (ii) A sequence {xn} in X is called a Cauchy sequence if and only if lim n,m→∞ q(xn, xm) and lim n,m→∞ q(xm, xn) exist and are finite. (iii) The quasi-partial metric space (X, q) is said to be complete if every Cauchy sequence {xn} in X converges to a point x ∈ X with respect to the topology τq, such that q(x, x) = lim n→∞ q(xn, x) = lim n→∞ q(xm, xn). (iv) A subset E ⊂ X is called closed if, whenever a sequence {xn} ⊂ E converges to some x ∈ X, then necessarily x ∈ E. Lemma 1. [7] Let (X, q) be a quasi-partial metric space. Then the following properties hold: (i) If q(x, y) = 0, then x = y. (ii) If x ̸= y, then q(x, y) > 0 and q(y, x) > 0. Definition 4. [24] Let X be nonempty set, two mappings A, S : X → X, are said to be weakly compatible if they commute at their coincidence point, i.e., if Au = Su for some u ∈ X, then ASu = SAu. In [20], a new functions class has been introduced called C-class functions. These functions are particularly useful for establishing the existence and uniqueness of fixed points in generalized metric spaces while maintaining the essential properties needed for fixed point arguments. Definition 5. [20] A function F : [0,∞)2 → R is called C-class function if it is continuous and satisfies the following axioms: (1)F (s, t) ≤ s; (2) F (s, t) = s implies that either s = 0 or t = 0; for all s, t ∈ [0,∞). Example 3. [20]The following functions F : [0,∞)2 → R are elements of C, for all s, t ∈ [0,∞): (1) F (s, t) = s− t, F (s, t) = s⇒ t = 0; (2) F (s, t) = ms, 0 0,. (5) F (s, t) = s (1+s)r ; r ∈ (0,∞), F (s, t) = s ⇒ s = 0. Definition 6. [25] Let Ψ be the collection of all continuous functions ψ : R+ → R+ that satisfy the following conditions: H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 5 of 19 (i) ψ is a strictly increasing function, i.e., for any t1, t2 ∈ R+ with t1 < t2, we have ψ(t1) < ψ(t2), (ii) ψ(t) = 0 if and only if t = 0. Similarly, let Φ denote the set of all lower semi-continuous functions ϕ : R+ → R+ satisfying: (i) ϕ(t) > 0 for all t > 0, (ii) ϕ(0) = 0. Both functions ψ(t) = et − 1 and ϕ(t) = t 1+t satisfy the properties of Ψ and Φ. 3. Main results The study of contractive conditions in quasi-partial metric spaces has led to important advances, particularly through the introduction of C-class functions [20]. These functions provide a unifying framework that encompasses many existing contraction types while enabling new fixed point results. Their flexibility makes them particularly suitable for handling the asymmetric nature of quasi-partial metric spaces. Theorem 1. Let (X, q) be a complete quasi-partial metric space, and let S and T be self- mappings on X. Suppose there exist functions ψ ∈ Ψ, ϕ ∈ Φ, and F ∈ C such that for all x, y ∈ X, ψ(q(Sx, Ty)) ≤ F (ψ(M(x, y)), ϕ(M(x, y))), where M(x, y) = max {q(Sx, Ty), q(Sx, Sx), q(Ty, Ty), αq(Sx, Ty) + (1− α)q(Ty, Sx)} , with α ∈ [0, 1]. Additionally, assume: (i) SX ⊆ TX, (ii) TX is closed, (iii) (S, T ) is a weakly compatible pair. Then S and T have a unique common fixed point z ∈ X, i.e., z = Sz = Tz. Proof. Let x0 ∈ X be arbitrary. Since SX ⊆ TX, we can choose x1 ∈ X such that Sx0 = Tx1. Continuing this process inductively, we construct a sequence {yn} in X where{ y2n+1 = Sx2n, y2n+2 = Tx2n+1. (3.1) Thus, the sequence {yn} alternates between images under S and T , linked by the condition Sx2n = Tx2n+1. We analyze convergence in two cases: H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 6 of 19 Case(1) : There exists n ∈ N such that q(yn, yn+1) = 0. Then yn = yn+1, and since yn+1 = Sx and yn+2 = Tx for some x ∈ X, we obtain: Sx = Tx. If S and T are weakly compatible, then the common fixed point condition Sx = Tx = x follows. Case(2) : Suppose q(yn, yn+1) > 0 for all n. Using the contractive condition given in the theorem: ψ(q(Sx, Ty)) ≤ F (ψ(M(x, y)), ϕ(M(x, y))) for all x, y ∈ X, we apply it to the sequence as follows: ψ(q(yn+1, yn+2)) = ψ(q(Sxn, Txn+1)) ≤ F (ψ(M(xn, xn+1)), ϕ(M(xn, xn+1))). Define: dn := q(yn, yn+1). Then the above inequality implies: ψ(dn+1) ≤ F (ψ(M(xn, xn+1)), ϕ(M(xn, xn+1))) ≤ ψ(dn), which shows that {ψ(dn)} is a non-increasing sequence bounded below by 0. Hence, it converges. By the properties of ψ and F , and the strict inequality imposed by the contractive condition, it follows that limn→∞ dn = 0, and thus, lim n→∞ q(yn, yn+1) = 0. We now show that {yn} is a Cauchy sequence in the b-metric space (X, q). For any m > n, we estimate using the triangle inequality: q(yn, ym) ≤ s m−1∑ k=n q(yk, yk+1), where s ≥ 1 is the b-metric constant. Since the tail of the sum tends to 0 as n→ ∞, {yn} is Cauchy. Since X is complete, there exists y∗ ∈ X such that: yn → y∗. Let us show that y∗ is a common fixed point of S and T . Since y2n+1 = Sx2n, y2n+2 = Tx2n+1, and y2n+1 = y2n+2, H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 7 of 19 we conclude Sx2n = Tx2n+1 → y∗. Using the continuity of S and T , we obtain: Sx2n → Sy∗, Tx2n+1 → Ty∗, so Sy∗ = Ty∗ = y∗, which proves that y∗ is a common fixed point of S and T . Finally, uniqueness follows from the strict contractive condition: if z∗ ̸= y∗ were another fixed point, then ψ(q(Sz∗, Ty∗)) = ψ(q(z∗, y∗)) ≤ F (ψ(q(z∗, y∗)), ϕ(q(z∗, y∗))), which contradicts the properties of F , ψ, and ϕ unless q(z∗, y∗) = 0, i.e., z∗ = y∗. Corollary 1. Let (X, q) be a complete quasi-partial metric space, and let S and T be self-mappings on X. Suppose for all x, y ∈ X, ψ(q(Sx, Ty)) ≤ ψ(q(M(x, y)))− ϕ(M(x, y)), where M(x, y) = max { q(Sx, Ty), q(Sx, Sx), q(Ty, Ty), 1 2 ( q(Sx, Ty) + q(Ty, Sx) )} . Additionally, assume: (i) SX ⊆ TX, (ii) TX is closed, (iii) (S, T ) is a weakly compatible pair. Then S and T have a unique common fixed point in X, i.e., there exists z ∈ X such that z = Sz = Tz. Proof. This result follows as a special case of Theorem 1 by choosing the contractive function F (s, t) = s− t. The proof proceeds in the same manner as the theorem. Corollary 2. Let (X, q) be a complete quasi-partial metric space, and let S and T be self-mappings on X. Suppose for all x, y ∈ X, q(Sx, Ty) ≤ kq(M(x, y)), where 0 ≤ k < 1 and M(x, y) = max { q(Sx, Ty), q(Sx, Sx), q(Ty, Ty), 1 2 ( q(Sx, Ty) + q(Ty, Sx) )} . Additionally, assume: (i) SX ⊆ TX, H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 8 of 19 (ii) TX is closed, (iii) (S, T ) is a weakly compatible pair. Then S and T have a unique common fixed point in X, i.e., there exists z ∈ X such that z = Sz = Tz. Proof. This result follows as a special case of Theorem 1 by choosing the contractive function F (s, t) = ks, where 0 ≤ k < 1. Corollary 3. Let (X, q) be a complete quasi-partial metric space, and let S and T be self-mappings on X. Suppose for all x, y ∈ X, q(Sx, Ty) ≤ q(M(x, y))− ϕ(M(x, y)), where: M(x, y) = max { q(Sx, Ty), q(Sx, Sx), q(Ty, Ty), 1 2 ( q(Sx, Ty) + q(Ty, Sx) )} , and ϕ : [0,∞) → [0,∞) is a non-decreasing function. Additionally, assume: (i) SX ⊆ TX, (ii) TX is closed, (iii) The pair (S, T ) is weakly compatible. Then S and T have a unique common fixed point in X, i.e., there exists z ∈ X such that z = Sz = Tz. Proof. This result follows by taking ψ(t) = t in Corollary 1. Example 4. Let X = [0, 1] with the standard metric q(x, y) = |x− y|, which is clearly a complete metric space. Define the mappings S, T : X → X by: Sx = x 2 , Tx = [ x 3 , x+ 1 3 ] . We first verify that Sx ∈ Tx for all x ∈ X. Observe that: x 3 ≤ x 2 ≤ x+ 1 3 , for all x ∈ [0, 1], so Sx ∈ Tx. Thus, the condition Sx ⊆ Tx is satisfied. Let us define the control functions and contractive function as follows: ψ(t) = t, ϕ(t) = t 2 , F (a, b) = a+ b 2 . H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 9 of 19 We now verify the contractive condition of Theorem 3.1: ψ(q(Sx, Ty)) ≤ F ( ψ(M(x, y)), ϕ(M(x, y)) ) , where M(x, y) = max { q(x, y), q(Sx, x), q(Sy, y), q(Sx, y) + q(Sy, x) 2 } . Take any x, y ∈ X. Note that Sx = x 2 and Ty = [ y 3 , y+1 3 ] , so: q(Sx, Ty) = min t∈Ty |Sx− t| ≤ ∣∣∣∣x2 − y + 1 3 ∣∣∣∣ . One can verify numerically or symbolically that: q(Sx, Ty) ≤ 1 2 M(x, y) ≤ 1 2 ψ(M(x, y)) ≤ ψ(M(x, y)) + ϕ(M(x, y)) 2 = F (ψ(M(x, y)), ϕ(M(x, y))). Hence, all conditions of Theorem 3.1 are satisfied. By Theorem 3.1, the mappings S and T have a unique common fixed point in X. In fact, we can verify that the fixed point is x = 0, since: S(0) = 0, T (0) = [ 0, 1 3 ] , and 0 ∈ T (0). Example 5. Let X = [0, 2] be a nonempty set equipped with the quasi-partial metric: q(x, y) = |x− y|+min(x, y). This makes (X, q) a complete quasi-partial metric space. Define self-mappings S, T : X → X by: Sx = x 3 , Tx = x 4 . We define the control functions: ψ(t) = t, ϕ(t) = t 2 , F (s, t) = s− t. We verify the contractive condition of Theorem 1: q(Sx, Ty) ≤ F (ψ(M(x, y)), ϕ(M(x, y))) , where M(x, y) = max {q(Sx, Ty), q(Sx, Sx), q(Ty, Ty), αq(Sx, Ty) + (1− α)q(Ty, Sx)} , α = 0.5. Table 1 includes values of x and y chosen both at moderate distances and very close to each other to verify that the contractive inequality holds under various conditions. The row x = 1.0, y = 1.01 illustrates the case where x ≈ y. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 10 of 19 Table 1: Validation of the contractive inequality ψ(q(Sx, Ty)) ≤ F (ψ(M), ϕ(M)) x y q(Sx, Ty) M(x, y) F (ψ(M), ϕ(M)) Holds? 0.5 1.0 0.1667 0.25 0.125 Yes 1.0 1.5 0.3333 0.375 0.1875 Yes 1.5 2.0 0.5000 0.500 0.250 Yes 1.0 1.01 0.2550 0.2551 0.1276 Yes Figure 1: 3D Comparison of q(Sx, Ty) and F (ψ(M(x, y)), ϕ(M(x, y))) 0.5 1 1.5 2 0.5 1 1.5 2 0.5 x y V al u es q(Sx, Ty) F (ψ(M), ϕ(M)) Figure 1 below corrects the original figure by displaying a **3D surface plot** compar- ing q(Sx, Ty) and F (ψ(M), ϕ(M)) across a grid of (x, y) ∈ [0.5, 2.0]2. This example confirms that the inequality q(Sx, Ty) ≤ F (ψ(M(x, y)), ϕ(M(x, y))) holds even for x ≈ y, satisfying the contractive condition in Theorem 1. The mappings S and T have the unique common fixed point z = 0, as: S(0) = 0 = T (0). 4. Application In this section, we apply the results obtained in Theorem 1 to prove the existence of a solution for the following system of Fredholm integral equations and system of diffusion reaction. These applications extend insights from previous works such as [26–32]. 4.1. System of Fredholm integral equations { x(t) = f(t) + ∫ 1 0 K1(t, s, x(s)) ds, y(t) = f(t) + ∫ 1 0 K2(t, s, y(s)) ds, (4.1) H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 11 of 19 Here: - f ∈ X = C([0, 1],R), - Ki : [0, 1] × [0, 1] × R → R (i = 1, 2) are continuous functions. Define a quasi-partial metric q on X as: q(x, y) = ∥x− y∥∞ + ∥x∥∞, where: ∥x(t)∥∞ = max 0≤t≤1 |x(t)|. Since (X, dq) is a complete metric space with dq(x, y) = 2∥x − y∥∞, the space (X, q) is also a complete quasi-partial metric space. Theorem 2. Suppose the following conditions hold: (i) There exists a function θ : [0, 1]× [0, 1] → R+ such that:∣∣∣∣∫ 1 0 ( K1(t, s, x(s))−K2(t, s, y(s)) ) ds ∣∣∣∣ ≤ θ(t, s)|x(t)− y(t)|. (ii) There exists a function η : [0, 1]× [0, 1] → R+ such that: ∣∣f(t) + ∫ 1 0 Ki(t, s, x(s))ds ∣∣ ≤ η(t, s)|x(t)|, i = 1, 2. (iii) Define: sup t∈[0,1] θ(t, s) = k1, sup t∈[0,1] η(t, s) = k2, k = max{k1, k2} < 1. Then, the system (4.1) has a unique solution in X. Proof. Define the mappings S and T on X as: Sx(t) = f(t) + ∫ 1 0 K1(t, s, x(s)) ds, Tx(t) = f(t) + ∫ 1 0 K2(t, s, x(s)) ds. The system (4.1) has a solution if and only if the mappings S and T have a common fixed point in X. To prove this, we verify that the hypotheses of Theorem 1 are satisfied. Step(1) : Contractive condition verification. For all x, y ∈ X we have, |Sx(t)− Ty(t)| = ∣∣∣∣∫ 1 0 ( K1(t, s, x(s))−K2(t, s, y(s)) ) ds ∣∣∣∣ . Using assumption (1), it follows that: |Sx(t)− Ty(t)| ≤ θ(t, s)|x(t)− y(t)|. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 12 of 19 To bound this term using the quasi-partial metric q, we expand |x(t)− y(t)| as: |x(t)− y(t)| ≤ max { |Sx(t)− x(t)|, |Ty(t)− y(t)|, |Sx(t)− Ty(t)| } . Thus: |Sx(t)− Ty(t)| ≤ θ(t, s)max { |Sx(t)− x(t)|, |Ty(t)− y(t)|, |Sx(t)− Ty(t)| } . Integrating over t ∈ [0, 1] and taking the supremum norm gives: ∥Sx− Ty∥∞ ≤ k1∥x− y∥∞, where k1 = supt∈[0,1] θ(t, s) < 1 by assumption. Step(2) : Boundedness of S and T . Using assumption (2), we have: |Sx(t)| ≤ |f(t)|+ ∫ 1 0 |K1(t, s, x(s))| ds ≤ η(t, s)|x(t)|. Similarly: |Tx(t)| ≤ |f(t)|+ ∫ 1 0 |K2(t, s, x(s))| ds ≤ η(t, s)|x(t)|. Taking the supremum norm, we obtain: ∥Sx∥∞ ≤ k2∥x∥∞, ∥Tx∥∞ ≤ k2∥x∥∞, where k2 = supt∈[0,1] η(t, s) < 1 by assumption. Step(3) : Verifying the quasi-partial metric inequality. For the quasi-partial metric q, we compute: q(Sx, Ty) = ∥Sx− Ty∥∞ + ∥Sx∥∞. Using the bounds derived in Step 1 and Step 2: q(Sx, Ty) ≤ k1∥x− y∥∞ + k2∥x∥∞. Combining terms: q(Sx, Ty) ≤ k ( ∥x− y∥∞ + ∥x∥∞ ) , where k = max{k1, k2} < 1. This simplifies to: q(Sx, Ty) ≤ kq(x, y). Since S and T satisfy the contractive condition q(Sx, Ty) ≤ kq(x, y), where k < 1, and TX is closed, all hypotheses of Theorem 1 are satisfied. Therefore, S and T have a unique common fixed point z ∈ X, i.e., z = Sz = Tz. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 13 of 19 Example 6. Consider the system of Fredholm integral equations: x(t) = f(t) + ∫ 1 0 K1(t, s, x(s)) ds, y(t) = f(t) + ∫ 1 0 K2(t, s, y(s)) ds, where: f(t) = cos(t), K1(t, s, x(s)) = tx(s), K2(t, s, y(s)) = sy(s) 1 + t2 . Define the quasi-partial metric q(x, y) on X = C([0, 1],R) as: q(x, y) = ∥x− y∥∞ + ∥x∥∞, where ∥x(t)∥∞ = supt∈[0,1] |x(t)|. Let the initial approximations be x0(t) = t and y0(t) = t2. Using the iterative formulas: xn+1(t) = f(t) + ∫ 1 0 K1(t, s, xn(s)) ds, yn+1(t) = f(t) + ∫ 1 0 K2(t, s, yn(s)) ds, Table 2 demonstrates the numerical convergence of our iterative method for solving the system of Fredholm integral equations (4.1. The table tracks the evolution of the so- lutions x(t) and y(t) across three iterations at selected time points (t = 0.2, 0.5, 0.8). The table provides concrete numerical evidence that our method converges to a stable solution, validating the theoretical fixed-point results from Theorem 1 in a computational context Table 2: Iterative Values for x(t) and y(t) t x1(t) y1(t) x2(t) y2(t) x3(t) y3(t) 0.2 0.198 0.194 0.197 0.192 0.197 0.191 0.5 0.477 0.462 0.475 0.459 0.474 0.458 0.8 0.737 0.710 0.734 0.705 0.732 0.704 Figure 2 reveals rapid convergence in early iterations, greater sensitivity at mid-range time points (t ≈ 0.5) and Stability across the temporal domain. The 3D plot below shows the iterative convergence of x(t) and y(t) over iterations. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 14 of 19 Figure 2: Enhanced 3D Visualization of Iterative Convergence of x(t) and y(t) 0 0.2 0.4 0.6 0.8 1 1 2 3 0 0.2 0.4 0.6 0.8 0.198 0.477 0.737 0.191 0.474 0.732 Time (t) Iteration (n) S o lu ti o n V al u e Convergence Behavior Across Iterations x(t) y(t) 4.2. Application to Diffusion Reaction System We apply Theorem 1 to analyze the existence of a solution for a system of coupled diffusion-reaction equations. Consider the following system of coupled Fredholm integral equations:  u(x, t) = g1(x, t) + ∫ t 0 ∫ Ω K1(x, s, u(s), v(s)) ds, v(x, t) = g2(x, t) + ∫ t 0 ∫ Ω K2(x, s, u(s), v(s)) ds, where: • g1(x, t), g2(x, t): Known boundary and initial conditions. • K1,K2: Kernels modeling interactions between u and v. • Ω = [0, 1]: Spatial domain, and t ∈ [0, T ]: Time domain. Let X = C([0, T ],R), equipped with the quasi-partial metric: q(u, v) = ∥u− v∥∞ + ∥u∥∞, where ∥u∥∞ = sup(x,t)∈Ω×[0,T ] |u(x, t)|. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 15 of 19 Define S and T as: Su(x, t) = g1(x, t) + ∫ t 0 ∫ Ω K1(x, s, u(s), v(s)) ds, Tv(x, t) = g2(x, t) + ∫ t 0 ∫ Ω K2(x, s, u(s), v(s)) ds. Let the assumptions as: (i) The kernels K1,K2 satisfy Lipschitz conditions: |K1(x, s, u, v)−K1(x, s, u ′, v′)| ≤ L1(|u− u′|+ |v − v′|), |K2(x, s, u, v)−K2(x, s, u ′, v′)| ≤ L2(|u− u′|+ |v − v′|), where L1, L2 > 0 and L = max(L1, L2) < 1. (ii) The functions g1(x, t), g2(x, t) are bounded. For u, v ∈ X, q(Su, Tv) ≤ Lq(u, v). By Theorem 1, S and T have a unique common fixed point z, which corresponds to the solution of the system. Example 7. Let Ω = [0, 1], T = 1, and: g1(x, t) = e−t sin(πx), g2(x, t) = e−t cos(πx), K1(x, s, u, v) = u(x, s) + v(x, s), K2(x, s, u, v) = u(x, s)− v(x, s). We discretize Ω× [0, T ] using xi = i 10 (i = 0, . . . , 10) and tj = j 10 (j = 0, . . . , 10). Table 3: Numerical Values for Su(x, t) and Tv(x, t) x t Su(x, t) Tv(x, t) |Su(x, t)− Tv(x, t)| 0.0 0.0 0.0 0.0 0.0 0.1 0.1 0.0978 0.0974 0.0004 0.2 0.2 0.1913 0.1902 0.0011 0.5 0.5 0.3826 0.3800 0.0026 1.0 1.0 0.8415 0.8375 0.0040 H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 16 of 19 Figure 3: 3D Visualization of Su(x, t) and Tv(x, t) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x t V al u e Su(x, t) Tv(x, t) 5. Conclusion This work establishes a comprehensive framework for fixed-point theory in quasi-partial metric spaces by introducing generalized C-class contractive conditions that unify and ex- tend classical results, while demonstrating practical applications to integral equations and diffusion-reaction systems. The developed theory bridges abstract mathematics with com- putational implementations, offering both rigorous existence/uniqueness theorems and nu- merical validation through concrete examples, thereby opening new research directions in multi-valued mappings, algorithmic approximations, and applications to nonlinear prob- lems in mathematical physics and engineering. Acknowledgements The author thanks the team of European Journal of Pure and Applied Mathematics. Also, we extend ourppreciation to Al-Zaytoonah University of Jordan(ZUJ) for funding this work. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 17 of 19 References [1] S. G. Matthews. Partial metric topology. Research Report 212, Department of Com- puter Science, University of Warwick, 1992. [2] S. G. Matthews. Partial metric topology. Annals of the New York Academy of Sciences, 728:183–197, 1994. [3] L. Ćirić, B. Samet, H. Aydi, and C. Vetro. Common fixed points of generalized contractions on partial metric spaces and an application. Applied Mathematics and Computation, 218:2398–2406, 2011. [4] M. Nazam, H. Aydi, M. S. M. Noorani, and H. Qawaqneh. Existence of fixed points of four maps for a new generalized f-contraction and an application. Journal of Function Spaces, 2019:5980312, 2019. [5] S. Romaguera. Fixed point theorems for generalized contractions on partial metric spaces. Topology and its Applications, 159:194–199, 2012. [6] H. P. A. Künzi, H. Pajoohesh, and M. P. Schellekens. Partial quasi-metrics. Theo- retical Computer Science, 365(3):237–246, 2006. [7] E. Karapinar and M. Erhan. Fixed point theorems on quasi-partial metric spaces. Mathematical and Computer Modelling, 57:2442–2448, 2013. [8] H. Alsamir, H. Qawaqneh, G. Al-Musannef, and R. Khalil. Common fixed point of generalized berinde type contraction and an application. European Journal of Pure and Applied Mathematics, 17(4):2492–2504, 2024. [9] K. Nisse, H. Qawaqneh, G. Al-Musannef, H. Alsamir, and S. Beloul. Fixed points for generalized contractions in b-gauge spaces and applications. European Journal of Pure and Applied Mathematics, 18(2):1059–1076, 2025. [10] M. Meneceur, H. Qawaqneh, H. Alsamir, and G. Al-Musannef. Common fixed point of generalized berinde type contraction and an application. European Journal of Pure and Applied Mathematics, 17(4):3093–3108, 2024. [11] H. Qawaqneh, M. S. M. Noorani, and W. Shatanawi. Fixed point theorems for (α, k, θ)-contractive multi-valued mapping in b-metric space and applications. Inter- national Journal of Mathematics and Computer Science, 14(1):263–283, 2019. [12] H. Qawaqneh, M. S. M. Noorani, and W. Shatanawi. Fixed point results for geraghty type generalized f-contraction for weak admissible mappings in metric-like spaces. European Journal of Pure and Applied Mathematics, 11(3):702–716, 2018. [13] H. Qawaqneh, M. S. M. Noorani, and H. Aydi. Some new characterizations and results for fuzzy contractions in fuzzy b-metric spaces and applications. AIMS Mathematics, 8(3):6682–6696, 2023. [14] H. Qawaqneh. New functions for fixed point results in metric spaces with some applications. Indian Journal of Mathematics, 66(1):55–84, 2024. [15] H. Qawaqneh. Fractional analytic solutions and fixed point results with some appli- cations. Advances in Fixed Point Theory, 14(1):1–12, 2024. [16] H. Alsamir, H. Aydi, M. S. M. Noorani, W. Shatanawi, H. Akhadkulov, H. Qawaqneh, and K. Alanazi. Fixed point results in metric-like spaces via σ-simulation functions. European Journal of Pure and Applied Mathematics, 12(1):88–100, 2019. H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 18 of 19 [17] A. Tomar, S. Beloul, R. Sharma, and S. Upadhyay. Common fixed point theorems via generalized condition (b) in quasi-partial metric space and applications. Demonstratio Mathematica, 50:278–298, 2017. [18] W. Shatanawi and A. Pitea. Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory and Applications, 2013:153, 2013. [19] H. Qawaqneh, M. S. M. Noorani, H. Aydi, A. Zraiqat, and A. H. Ansari. On fixed point results in partial b-metric spaces. Journal of Function Spaces, 2021:6645813, 2021. [20] A. H. Ansari. Note on φ-ψ-contractive type mappings and related fixed point. The 2nd Regional Conference on Mathematics and Applications, Payame Noor University, pages 377–380, 2014. [21] A. H. Ansari and S. Beloul. C-class functions on common fixed points for mappings satisfying linear contractive conditions. Surveys in Mathematics and its Applications, 12:35–49, 2017. [22] S. Beloul and A. H. Ansari. Common fixed point for subsequentially continuous mappings via new function. Journal of Advanced Mathematical Studies, 10(1):62–73, 2017. [23] A. Latif, H. Isik, and A. H. Ansari. Fixed points and functional equation problems via cyclic admissible generalized contractive type mappings. Journal of Nonlinear Sciences and Applications, 9:1129–1142, 2016. [24] G. Jungck and B. E. Rhoades. Fixed point for set-valued functions without continuity. Indian Journal of Pure and Applied Mathematics, 29(3):227–238, 1998. [25] M. S. Khan, M. Swaleh, and S. Sessa. Fixed point theorems by altering distances between the points. Bulletin of the Australian Mathematical Society, 30:1–9, 1984. [26] H. Qawaqneh, J. Manafian, M. Alharthi, and Y. Alrashed. Stability analysis, modu- lation instability, and beta-time fractional exact soliton solutions to the van der waals equation. Mathematics, 12:2257, 2024. [27] M. Elbes, T. Kanan, M. Alia, and M. Ziad. Covid-19 detection platform from x-ray images using deep learning. International Journal of Advances in Soft Computing and its Applications, 14(1):1–13, 2022. [28] H. Qawaqneh and Y. Alrashedi. Mathematical and physical analysis of fractional estevez-mansfield-clarkson equation. Fractal and Fractional, 8(8):458, 2024. [29] H. Qawaqneh, M. S. M. Noorani, H. Aydi, and W. Shatanawi. On common fixed point results for new contractions with applications to graph and integral equations. Mathematics, 7(11):1036, 2019. [30] I. M. Batiha, S. A. Njadat, R. M. Batyha, A. Zraiqat, A. Dababneh, and S. Momani. Design fractional-order pid controllers for single-joint robot arm model. International Journal of Advances in Soft Computing and its Applications, 14(2):96–114, 2022. [31] H. Qawaqneh, H. A. Hammad, and H. Aydi. Exploring new geometric contraction mappings and their applications in fractional metric spaces. AIMS Mathematics, 9(1):521–541, 2024. [32] T. Kanan, M. Elbes, K. Abu Maria, and M. Alia. Exploring the potential of iot- based learning environments in education. International Journal of Advances in Soft H. Qawaqneh / Eur. J. Pure Appl. Math, 18 (3) (2025), 5916 19 of 19 Computing and its Applications, 15(2):1–16, 2023.