EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6583 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Natural Extension of the Banach fixed-point Theorem in a b-Metric Space with an Orthogonal Direct Sum Structure Ghadah ALbeladi1,∗, Saleh Omran2 1 Department of Mathematics, College of Sciences & Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia 2 Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt Abstract. This study aims to develop new versions of the Banach fixed-point theorem in general- ized metric spaces endowed with a direct sum structure. Specifically, we assume a diagonal matrix A in Rd×d and establish more appropriate contraction conditions to improve the applicability of fixed-point results within this framework. Since the condition that the matrix A must converge to zero is unnecessary, our approach yields stronger results than the Perov one. As an application of our findings, we examine the existence and uniqueness of solutions for a system of matrix equa- tions. This version is more powerful than the Perov version. We introduced some examples and applications to illustrate our result. 2020 Mathematics Subject Classifications: 47H10, 54H25, 46A19, 15A24 Key Words and Phrases: Fixed point, Banach contraction principle, generalized b-metric space, direct sum 1. Introduction A fixed-point of F on X is an element q such that F (q) = q. Under certain cir- cumstances on the operator F and the space X , the operator F admits one or more fixed points, according to the fixed-point theorem. For various applications of the fixed- point theorem, there are numerous outcomes Brouwer’s fixed-point theorem, Schauder’s fixed-point theorem, and the Banach contraction principle are the foundational ideas of fixed-point theory. Perov generalized the notion of metric spaces in [1] by replacing the set of real num- bers with vector-valued space Rd. Thus, the classical contraction operator principle was extended to contraction operators in a vector-valued metric space, known as a generalized ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6583 Email addresses: galbeladi@kau.edu.sa (G. ALbeladi), salehomran@yahoo.com (S. Omran) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 2 of 27 metric space. In this context, several interesting results have been introduced by various authors (see [2, 3]). Omran presented versions of Banach fixed-point theorems in a generalized metric space endowed with the Hadamard product. Omran [4] applied the Hadamard product to solve a system of symmetric positive definite matrix equations, demonstrating that results based on the Hadamard product of a vector-valued metric are stronger than those using Perov’s condition. Specifically, it eliminates the need to assume that A is a matrix converging to zero. In Banach’s 1922 thesis [5], the fixed-point theorem—also known as the Banach contrac- tion principle—was first used in its final form in 1922 to prove that an integral equation had a solution. Since then, it has been found to be a useful tool for solving many practical problems in various branches of mathematical analysis. In this principle, b-metric space X is complete if F : X → X is a contraction operator; that is, δR(Fx,Fy) ≤ λδR(x, y) for all x, y ∈ X , where λ ∈ (0, 1s ) is a constant, then F has a fixed point. In 1989, Bakhtin [6] introduced the concept of b-metric spaces, which Czerwik further developed in 1993 [7]. This advancement sparked significant research into generalizing the Banach fixed-point theorem for b-metric spaces. Notable contributions from Mehmet Kir [8], Boriceanu [9], Bota [10], Pacurar [11], and Czerwik [7] have extended fixed-point theorems in this context. Additionally, Czerwik explored the convergence of measurable functions with respect to measure and generalized the Banach contraction principle . In the same year, Czerwik [7] proposed a new axiom for semimetric spaces, relaxing the classical triangle inequality to broaden the scope of the Banach contraction principle. This idea aligns with the nonlinear elastic matching (NEM) distance discussed by Fagin et al. [12], which has found applications in diverse fields, such as trademark shape analysis [13] and ice floe measurements [14]. Building on this concept, Q. Xia [15] applied semi- metric distances to study optimal transport paths between probability measures, coining the term quasi-metric spaces, a designation also used in Heinonen’s book [16]. Our goal is to further extend well-known fixed-point theorems within the framework of b-metric spaces. Some articles examine various generalizations of classical metric spaces to establish new fixed-point theorems. Building on the framework of b-metric spaces presented by Jleli and Samet [17]. Albargi and Ahmad [18] developed standard F-fuzzy fixed-point theorems under rational (β-ϕ)-contractive conditions, with applications to fuzzy integrodifferential equations via the generalized Hukuhara derivative. Furthermore, Huang and Samet [19] introduced two novel self-mapping classes on complete metric spaces: p-contractions con- cerning families of mappings, and (ψ,Γ, α)-contractions, which involve the Euler Gamma function. Also, [20, 21] developed fixed-point results to fuzzy normed spaces using a newly introduced triangle property and to algebra fuzzy metric spaces. These results generalized, unified, and improved many classical theorems. G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 3 of 27 The aim of this work is to present new versions of Banach fixed-point theorems in a generalized b-metric space endowed with the orthogonal direct sum, where A is a diago- nal matrix in Rd. This approach, constructed via the direct sum, ensures more suitable contraction conditions. Now, we can summarize the content of this manuscript as follows: Section 1 provides a brief historical overview to put the fixed-point theorems established in this paper into context. Section 2 introduces the definitions and basic concepts that lay the foundation for understanding generalized b-metric spaces endowed with the orthogo- nal direct sum. Section 3 presents several theorems that generalize the theorems given in [1, 4], which prove the existence and uniqueness of fixed points. 2. Preliminaries The real vector space sum of the real vector spaces R1, . . . ,Rd in a nonempty finite ordered list is the real vector space whose underlying set is the Cartesian product R1 × · · · × Rd = ∏d i=1Ri, with vector space operations defined by the following formulas: (x1, x2, · · · , xd) + (y1, y2, · · · , yd) = (x1 + y1, x2 + y2, · · · , xd + yd). α(x1, x2, · · · , xd) = (αx1, αx2, · · · , αxd). Definition 1. [22] The direct sum, or external direct sum, of a family (Ri)i∈I is the following sub-module of ∏ i∈I Ri: R⊕I = ⊕ i∈I Ri = {(xi)i∈I ∈ ∏ i∈I Ri | xi = 0 for almost all i ∈ I}. Immediately, this defines a sub-module. If I = ∅, then ⊕ i∈I Ri = 0. If I = {1}, then ⊕ i∈I Ri = R1. If I = {1, 2, · · · , d}, then ⊕ i∈I Ri is also denoted by R1 ⊕ R2 ⊕ R3 ⊕ · · · ⊕ Rd = ⊕∑d i=1Ri, and coincides with R1 × R2 × · · · × Rd. The direct sum ⊕ i∈I Ri comes with an injection ιj : Rj → ⊕ i∈I Ri for every j ∈ I, defined by its components: for all xj ∈ Rj , ιj(x)j = x ∈ Rj , ιj(x)i = 0 ∈ Ri for all i ̸= j, every ιj is an injective homomorphism. If R1, . . . ,Rd are normed spaces, their vector space sum can be equipped with a norm inspired by the norm of the Euclidean n space. Specifically, a norm on the vector space sum can be defined in a way that resembles the standard Euclidean norm. Definition 2. Let R1, · · · ,Rd be a normed space with respective norms | · |R1 , · · · , | · |Rd is the normed space whose underlying vector space is the vector space sum of R1, · · · ,Rd G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 4 of 27 and whose norm is the direct sum norm given by the formula ∥(x1, · · · , xn)∥ =  n∑ j=1 | xj |2Rj  1 2 . This normed space is dented by R1 ⊕ · · · ⊕ Rd. Corolary 1. For any two spaces A and B, the direct sum can be defined as a homomor- phism f : A⊕B →Ms, where the matrix Ms of size s, given by: F (A,B) = A⊕B = ( A 0 0 B ) , where A and B are embedded as diagonal blocks in the matrix form, and 0 represents appropriately sized zero matrices. Theorem 1. [23] Let R1,R2, · · · ,Rd be a family of sets of all real numbers. Then the following are equivalent: 1. R⊕d = ⊕∑d i=1Ri is a direct sum. 2. If 0 = ∑d i=1 xi, xi ∈ Ri, then xi = 0, i = 1, 2, · · · , d. 3. Ri ∩ ⊕∑d j=1, j ̸=iRj = 0̃, i = 1, 2, · · · , d. Theorem 2. [23] Let R1,R2, · · · ,Rd be a family of sets of all real numbers, and let R1×· · ·×Rd = ∏d i=1Ri be their direct product. Let R∗ i = {(, · · · , 0, xi, 0, · · · , 0)| xi ∈ Ri}. Then R⊕d = ⊕∑d i=1R∗ i is a direct sum of R∗ i . We recall some reminders and auxiliary results that will be used throughout this paper. Let Rd denotes the orthogonal direct sum R ⊕ R ⊕ · · · ⊕ R of d-copies of R. Let R⊕d be the direct sum set, and x, y ∈ R⊕d , x = (x1, x2, · · · , xd), y = (y1, y2, · · · , yd) by x ≾ y (respectively, xi < yi) for all i = 1, 2, · · · d. In addition, R⊕d + = {x ∈ R⊕d : xi ≥ 0, ∀i = 1, 2, · · · , d} is the set of positive elements in R⊕d , and we denote x ≿ 0̃ if xi ≥ 0 for all i = 1, 2, · · · , d, where 0̃ is the d× d zero matrix in R⊕d . By taking the product of two square diagonal matrices, we can define x · y as follows: x · y =  x1y1 0 · · · 0 0 x2y2 · · · 0 ... ... . . . ... 0 0 · · · xdyd  = diag ( x1y1, x2y2, · · · , xdyd ) ∈ R⊕d , and the direct sum unit over R⊕d will be denoted by IR ⊕d + =  1 0 · · · 0 0 1 · · · 0 ... ... . . . ... 0 0 · · · 1  = diag ( 1, 1, · · · , 1 ) . G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 5 of 27 An element x = (x1, x2, · · · , xd) ∈ R⊕d has an invers if xi ̸= 0, for all i = 1, 2, · · · , d and is denoted by x−1 = diag ( x−1 1 , x−1 2 , · · · x−1 d ) = diag ( 1 x1 , 1 x2 , · · · 1 xd ) , and x is called invertible if it has an inverse. Definition 3. Let X be a nonempty set, s ≥ 1 be a constant. Let Rd denote the orthogonal direct sum R ⊕ R ⊕ · · · ⊕ R of d-copies of R. An operator δb R⊕d : X × X → R⊕d is called a generalized b-metric endowed with the orthogonal direct sum if the following properties are satisfied: (B1) δ b R⊕d (x, y) ≿ 0̃, for all x, y ∈ X and δb R⊕d (x, y) = 0̃ if and only if x = y; (B2) δ b R⊕d (x, y) = δb R⊕d (y, x), for all x, y ∈ X ; (B3) δ b R⊕d (x, z) ≾ s [ δb R⊕d (x, y) + δb R⊕d (y, z) ] , for all x, y, z ∈ X . Then, we call (X ,R⊕d , δb R⊕d ) a generalized b-metric space on X . Example 1. Let (X , δR) be a metric space, and δb R⊕d : X × X → R⊕d be defined by δb R⊕d (x, y) = diag ( [d(x, y)]p, [d(x, y)]p, · · · , [d(x, y)]p ) ∈ R⊕d , where p > 1 is a fixed real number. Then δb R⊕d is the generalized b-metric space endowed with the orthogonal direct sum, where s = 2p−1. Indeed, conditions (B1) and (B2) in Definition 3 are satisfied and thus we only need to show that condition (B3) holds for δb R⊕d . It is easy to see that if 1 < p < +∞, then the convexity of the function F (x) = xp, where x ≥ 0, implies ( a+ c 2 )p ≤ 1 2 (ap + cp), and hence (a+ c)p ≤ 2p−1(ap + cp). Therefore, for each x, y, z ∈ X, we get δb R⊕d (x, y) = diag ( [d(x, y)]p, [d(x, y)]p, · · · , [d(x, y)]p ) ≤ diag ( [d(x, z) + d(z, y)]p, [d(x, z) + d(z, y)]p, · · · , [d(x, z) + d(z, y)]p ) ≤ 2p−1 diag ( d(x, z)p + d(z, y)p, d(x, z)p + d(z, y)p, · · · , d(x, z)p + d(z, y)p ) = 2p−1 [ diag ( d(x, z)p, d(x, z)p, · · · , d(x, z)p ) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 6 of 27 +diag ( d(z, y)p, d(z, y)p, · · · , d(z, y)p ) ] = 2p−1 [ δb R⊕d (x, z) + δb R⊕d (z, y) ] . So condition (B3) in Definition 3 holds and then δb R⊕d is a b-metric coefficient s = 2p−1 > 1. Definition 4. Suppose that (X ,R⊕d , δb R⊕d ) is a generalized b-metric space endowed with the orthogonal direct sum. Then a sequence {xn} in X is called: 1. b-convergent sequence with respect to R⊕d . If, for every ϵ̃ ∈ R⊕d , with ϵ̃ ≻ 0̃, if there exists x ∈ X and there is N ∈ N such that for all n > N , δb R⊕d (xn, x) ≺ ϵ̃. 2. b-Cauchy sequence with respect to R⊕d . If, for any ϵ̃ ≻ 0̃ there exists N ∈ N such that for all n,m > N , δb R⊕d (xn, xm) ≺ ϵ̃. Example 2. Assume that X = lp(R) with p ∈ (0, 1), where lp(R) := { {xn} ⊆ R | +∞∑ n=1 | xn |p< +∞ } , together with the operator δb R⊕d : lp(R)× lp(R) → R⊕d + defined by δb R⊕d (x, y) = diag ((∑+∞ n=1 | xn − yn |p ) 1 p , · · · , (∑+∞ n=1 | xn − yn |p ) 1 p ) ∈ R⊕d , where x = {xn}, y = {yn} ∈ lp(R), is a generalized b-metric space endowed with the orthogonal direct sum, where coefficient s = 2 1 p > 1. Then (lp,Rd, d) is a complete gener- alized b-metric space endowed with the orthogonal direct sum. Example 3. Assume that X = Lp(R) with p ∈ (0, 1), where Lp(R) := x(t) ⊆ C([0, 1],R)| 1∫ 0 | x(t) |p< +∞  , together with the operator d : Lp(R)× Lp(R) → R⊕d + defined by δb R⊕d (x, y) = diag (( 1∫ 0 | x(t)− y(t) |p dt ) 1 p , · · · , ( 1∫ 0 | x(t)− y(t) |p dt ) 1 p ) ∈ R⊕d , where x = {xn}, y = {yn} ∈ lp(R), is a b-metric space with coefficient s = 2 1 p > 1. Then (lp,R⊕d , δb R⊕d ) is a complete generalized b-metric endowed with the orthogonal direct sum space. G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 7 of 27 Example 4. Let X = {0, 1, 2} and the operator δb : X × X → R+ defined by δb(0, 0) = δb(1, 1) = δb(2, 2) = 0, δb(0, 1) = δb(1, 0) = δb(1, 2) = δb(2, 1) = 1, and δb(2, 0) = δb(0, 2) = m, where m is given real number such that m ≥ 2. δb R⊕d (x, y) = diag ( δb(x, y), δb(x, y), · · · , δb(x, y) ) ∈ R⊕d + . It is easy to see that δb R⊕d (x, y) ≤ m 2 [ δb R⊕d (x, z) + δb R⊕d (z, y) ] , for all x, y, z ∈ X . Therefore, (X ,R⊕d + , δb R⊕d ) is a generalized b-metric space with coefficient s = m 2 . We obtain that the ordinary triangle inequality does not hold if m > 2, and then (X , d) is not a metric space. Corolary 2. Let (X ,R⊕d + , δb R⊕d ) be a generalized b-metric space. Then, the following assertions hold: 1. A b-convergent sequence with respect to R⊕d has a unique limit. 2. Each b-convergent sequence with respect to R⊕d is a b-Cauchy sequence. Definition 5. Suppose that (X ,R⊕d , δb R⊕d ) and (Y,R⊕d , δb R⊕d ) are two generalized b-metric spaces endowed with the orthogonal direct sum. 1. The space (X ,R⊕d , δb R⊕d ) is a complete if every Cauchy sequence with respect to Rd is converges. 2. A function F : X → Y is said to be continuous at a point x ∈ X if, for every sequence {xn} ⊆ X that converges to x, the sequence F (xn) converges to F (x). 3. Main Results In this section, we will present some Banach fixed-point theorems combining the direct sum. Definition 6. Suppose that (X ,R⊕d , δb R⊕d ) is a complete generalized b-metric space en- dowed with the orthogonal direct sum. An operator F : X → X is said to be a con- traction and endowed with the orthogonal direct sum if there exists a diagonal matrix Λ = diag(λ1, λ2, · · · , λd) ∈ Rd + with 0̃ ≾ λi ≺ 1 s such that δb R⊕d (F (x), F (y)) ≾ Λ δb R⊕d (x, y), for all x, y ∈ X G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 8 of 27 Example 5. Let X = R and δb R⊕d : X ×X → R⊕d be a generalized b-metric space endowed with the orthogonal direct sum defined by δb R⊕d (x, y) = diag ( | x− y |2, | x− y |2, · · · , | x− y |2 ) ∈ R⊕d , and let Λ = diag ( 1 4 , 1 4 , · · · , 1 4 ) and F : X → X such that F (x) = x 2 . Then, δb R⊕d (F (x),F (y)) = diag ( 1 4 · | x− y |2, 1 4 · | x− y |2, · · · , 1 4 · | x− y |2 ) ∈ R⊕d . it is clear that δb R⊕d (F (x),F (y)) ≾ Λ · δb R⊕d (x, y), where Λ ≺ IR ⊕d + . Therefore, F is a contraction and endowed with the orthogonal direct sum on a gen- eralized b-metric space endowed with direct sum (X ,R⊕d , δb R⊕d ). Example 6. (Numerical example) Consider all hypotheses of Example 4 are true. Let F : X → X such that F (x) = x 4 . Then, δb R⊕d (F (x),F (y)) = 1 4 · IR⊕d + diag ( δb(x, y), δb(x, y), · · · , δb(x, y) ) = diag ( 1 4 , 1 4 , · · · , 1 4 ) · diag ( δb(x, y), δb(x, y), · · · , δb(x, y) ) . It is clear that δb R⊕d (F (x),F (y)) ≾ Λ · δb R⊕d (x, y), where Λ ≺ IR ⊕d + . Therefore, F is a contraction endowed the direct sum for the a generalized b-metric space endowed with the orthogonal direct sum (X ,R⊕d , δb R⊕d ). Theorem 3 (Banach contraction Type). Suppose that (X ,R⊕d , δb R⊕d ) is a complete gen- eralized b-metric space endowed with the orthogonal direct sum, and let F : X → X be a contraction operator endowed with the direct sum. That is, δb R⊕d (F (x),F (y)) ≾ diag(λ1, λ2, . . . , λd) δ b R⊕d (x, y), for all x, y ∈ X , where λi ∈ (0, 1) for i = 1, 2, . . . , d. Then F has a unique fixed-point in X . G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 9 of 27 Proof. Let x0 be any point in X , that is x0 ∈ X . Let us define a sequence {xn} in X as given below. xn+1 = F (xn) = Fn+1(x0) ∀ n ≥ 0. By the contraction condition, we get δb R⊕d (xn−1, xn) = δb R⊕d (F (xn−2),F (xn−1)) ≾ diag ( λ1, λ2, · · · , λd ) [δb R⊕d (xn−2, xn−1)] ... ≾ diag ( λn−1 1 , λn−1 2 , · · · , λn−1 d ) δb R⊕d (x0, x1). Let us prove that {xn} is a Cauchy sequence. Suppose that n > m from the contraction and triangle inequality property, we can write as: δb R⊕d (xm, xn) ≾ sδb R⊕d (xm, xm+1) + s2 δb R⊕d (xm+1, xm+2) + s3 [ δb R⊕d (xm+2, xm+3) + δb R⊕d (xm+3, xn) ] ... ≾ s δb R⊕d (xm, xm+1) + s2 δb R⊕d (xm+1, xm+2) + s3 δb R⊕d (xm+2, xm+3) + · · ·+ sn−m δb R⊕d (xn−1, xn) ≾ s diag(λm1 , λ m 2 , · · · , λmd )δb R⊕d (x0, x1) + s2 diag(λm+1 1 , λm+1 2 , · · · , λm+1 d )δb R⊕d (x0, x1) + · · ·+ sn−m diag(λn−1 1 , λn−1 2 , · · · , λn−1 d )δb R⊕d (x0, x1) = ( (s λm1 + s2 λm+1 1 + · · ·+ sn−m λn−1 1 )δb R⊕d (x0, x1), (s λ m 2 + s2 λm+1 2 + · · · +sn−m λn−1 2 )δb R⊕d (x0, x1), · · · , (sλmd + s2 λm+1 d + · · ·+ sn−m λn−1 d )δb R⊕d (x0, x1) ) = diag ( n−m∑ k=1 n−1∑ j=m skλj1δ b R⊕d (x0, x1), n−m∑ k=1 n−1∑ j=m skλj2δ b R⊕d (x0, x1), · · · , n−m∑ k=1 n−1∑ j=m sk λjdδ b R⊕d (x0, x1) ) = diag ( n−m∑ k=1 n−1∑ j=m skλj1, n−m∑ k=1 n−1∑ j=m skλj2, · · · , n−m∑ k=1 n−1∑ j=m skλjd ) δb R⊕d (x0, x1) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 10 of 27 = diag ( sλm1 n−m−1∑ j=0 (sλ1) j , sλm2 n−m−1∑ j=0 (sλ2) j , · · · , sλmd n−m−1∑ j=0 (sλd) j ) δb R⊕d (x0, x1) ≾ diag ( sλm1 +∞∑ j=0 (sλ1) j + sλm2 +∞∑ j=0 (sλ2) j + · · ·+ sλmd +∞∑ j=0 (sλd) j ) δb R⊕d (x0, x1) = diag ( sλm1 [ 1 1− sλ1 ] , sλm2 [ 1 1− sλ2 ] , · · · , sλmd [ 1 1− sλd ]) δb R⊕d (x0, x1) ≾ diag(sλm1 , · · · , sλmd ) diag ([ 1 1− sλ1 ] , · · · , [ 1 1− sλd ]) δb R⊕d (x0, x1) → 0, as n,m → +∞. Since 0̃ ≺ diag(sλj1, sλ j 2, · · · , sλ j d) ≺ IR ⊕d + and δb R⊕d (x0, x1) are fixed, it is evident that by selectingm sufficiently large (with n > m), we can make δb R⊕d (xn, xm) arbi- trarily small. This shows that {xn} is a Cauchy sequence. Finally, because (X ,R⊕d , δb R⊕d ) is complete, there exists some z ∈ X such that xn → z. To show that X has a fixed point, we consider the distance δb R⊕d (z,F (z)). From the triangle inequality and contraction condition, we get δb R⊕d (z,F (z)) ≾ s [ δb R⊕d (z, xn) + δb R⊕d (xn,F (z)) ] = s [ δb R⊕d (z, xn) + δb R⊕d (F (xn−1),F (z)) ] ≾ s [ δb R⊕d (z, xn) + diag ( λ1, λ2, · · · , λd ) δb R⊕d (xn−1, z) ] , and since xn → z, it is clear that we can make this distance as small as we please by choosing n sufficiently large. We conclude that δb R⊕d (z,F (z)) = 0 =⇒ F (z) = z, so x ∈ X is a fixed-point of F . To prove uniqueness, suppose there are two fixed points x = F (x) and y = F (x). Then, from the contraction condition, we have δb R⊕d (x, y) = δb R⊕d (F (x),F (y)) ≾ diag ( λ1, λ2, · · · , λd ) δb R⊕d (x, y) this implies δb R⊕d (x, y) = 0 since 0̃ ≺ Λ ≺ IR ⊕d + . Hence x = y, and the fixed-point X of F is unique. Example 7 (A Numerical Iterative Example). Let X = R with the generalized metric δb R⊕d defined as: δb R⊕d (x, y) = diag(|x− y|2, |x− y|2, . . . , |x− y|2). G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 11 of 27 Consider the function: F (x) = x 3 . We choose the diagonal contraction matrix: Λ = diag ( 1 9 , 1 9 , . . . , 1 9 ) , which satisfies: δb R⊕d (F (x),F (y)) = 1 9 · δb R⊕d (x, y). Since 0 < 1 9 < 1, the operator F satisfies the contraction condition required by Theorem 3, we can find the fixed-point by using the iterative sequence: xn+1 = F (xn), starting from an arbitrary initial point x0 = 6, we get: x1 = 6 3 = 2, x2 = 2 3 ≈ 0.67, x3 = 0.67 3 ≈ 0.22, ... xn = 6 3n , ... Taking the limit as n→ +∞: lim n→+∞ xn = lim n→+∞ 6 3n = 0. Thus, the iterative process confirms that the fixed-point of F is indeed z = 0. Example 8 (A Fredholm Integral Equation). Let X = C([0, 1],R), the space of continuous functions on [0, 1], equipped with the norm: ∥f∥ = sup x∈[0,1] |f(x)|. Define the generalized metric: δb R⊕d (f, g) = diag(∥f − g∥2, ∥f − g∥2, . . . , ∥f − g∥2). G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 12 of 27 Consider the Fredholm integral equation: f(x) = 1∫ 0 k(x, t)f(t)dt, where k(x, t) is a continuous kernel. Define the operator F : C([0, 1],R) → C([0, 1],R) by: (Ff)(x) = 1∫ 0 k(x, t)f(t)dt. Suppose there exists λ ∈ (0, 1s ) such that: sup x∈[0,1] 1∫ 0 |k(x, t)|dt ≤ λ. For any two functions f, g ∈ C([0, 1],R): ∥Ff − Fg∥2 = sup x∈[0,1] ∣∣∣∣∣∣ 1∫ 0 k(x, t)(f(t)− g(t))dt ∣∣∣∣∣∣ 2 ≤ λ sup t∈[0,1] |f(t)− g(t)|2 = λ∥f − g∥2. Thus, δb R⊕d (f, g) is a contraction with Λ = diag(λ, λ, . . . , λ), where 0 < λ < 1 s and δb R⊕d (Ff,Fg) ≾ Λδb R⊕d (f, g). By Theorem 3, F has a unique fixed point, meaning that the integral equation has a unique solution. If d = 1 and s = 1, then the theorem 3 can be reduced to the standard Banach contraction principle in a metric space. Corolary 3 (Banach contraction principle, [5]). Suppose that (X , d) is a complete metric space and F : X → X is a contraction operator with Lipschitz constant λ < 1. Then F has a unique fixed-point z ∈ X . Theorem 4. (Kannan Type) Suppose that (X ,R⊕d , δb R⊕d ) is a complete generalized b- metric space endowed with the orthogonal direct sum and F : X → X is an operator satisfying the following condition δb R⊕d (F (x),F (y)) ≾ diag ( α1, α2, · · · , αd ) [δb R⊕d (x,F (x)) + δb R⊕d (y,F (y))], (1) where αi ∈ (0, 12) for all i = 1, 2, · · · , d. Then, F has a unique fixed-point in X . G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 13 of 27 Proof. Let x0 be any point in X , that is x0 ∈ X . Let us define a sequence {xn} in X as given below. xn+1 = F (xn) = Fn+1(x0) ∀ n ≥ 0. We have δb R⊕d (xn, xn+1) = δb R⊕d (F (xn−1),F (xn)) ≾ diag ( α1, α2, · · · , αd ) [ δb R⊕d (xn−1,F (xn−1)) + δb R⊕d (xn,F (xn)) ] = diag ( α1, α2, · · · , αd ) δb R⊕d (xn−1, xn) +diag ( α1, α2, · · · , αd ) δb R⊕d (xn, xn+1), and diag ( (1− α1), (1− α2), · · · , (1− αd) ) δb R⊕d (xn, xn+1) ≾ diag ( α1, α2, · · · , · · · αd ) δb R⊕d (xn−1, xn), since the inverse of the diagonal matrix is diag ( (1− α1) −1, (1− α2) −1, · · · , (1− αd) −1 ) . Thus, δb R⊕d (xn, xn+1) ≾ diag ( α1 (1− α1) , · · · · · · , αd (1− αd) ) δb R⊕d (xn−1, xn) ≾ diag (( α1 (1− α1) )2 , · · · · · · , ( αd (1− αd) )2 ) δb R⊕d (xn−2, xn−1) ≾ diag (( α1 (1− α1) )3 , · · · , ( αd (1− αd) )3 ) δb R⊕d (xn−3, xn−2) ... ≾ diag (( α1 (1− α1) )n , · · · · · · , ( αd (1− αd) )n) δb R⊕d (x0, x1), If we let βi = αi 1− αi , then δb R⊕d (xn, xn+1) ≾ diag ( βn1 , βn2 , · · · , βnd ) δb R⊕d (x0, x1). Let us prove that {xn} is a Cauchy sequence. Suppose that n > m, then from (1) and the triangle inequality property, we can write as: δb R⊕d (xm, xn) ≾ s [ δb R⊕d (xm, xm+1) + δb R⊕d (xm+1, xn) ] ≾ sδb R⊕d (xm, xm+1) + s2 [ δb R⊕d (xm+1, xm+2) + δb R⊕d (xm+2, xn) ] G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 14 of 27 ≾ s δb R⊕d (xm, xm+1) + s2 δb R⊕d (xm+1, xm+2) +s3 [ δb R⊕d (xm+2, xm+3) + δb R⊕d (xm+3, xn) ] ... ≾ s δb R⊕d (xm, xm+1) + s2 δb R⊕d (xm+1, xm+2) +s3 δb R⊕d (xm+2, xm+3) + · · ·+ sn−mδb R⊕d (xn−1, xn) = diag (sβm1 , · · · , sβmd ) δb R⊕d (x0, x1) + diag(s2βm+1 1 , · · · , s2βm+1 d )δb R⊕d (x0, x1) + · · ·+ diag ( sn−mβn−1 1 , · · · , sn−mβn−1 d ) δb R⊕d (x0, x1) = diag ([ sβm1 + s2βm+1 1 + · · ·+ sn−mβn−1 1 ] , · · · , [ sβmd + s2βm+1 d + · · ·+ sn−mβn−1 d ]) δb R⊕d (x0, x1) = diag n−m∑ i=1 n−1∑ j=m siβj1 , · · · , n−m∑ i=1 n−1∑ j=m siβjd  δb R⊕d (x0, x1) = diag ( sβm1 n−m−1∑ j=0 (sβ1) j , · · · , sβmd n−m−1∑ j=0 (sβd) j ) δb R⊕d (x0, x1) ≾ diag ( sβm1 +∞∑ j=0 (sβ1) j , · · · , sβmd +∞∑ j=0 (sβd) j ) δb R⊕d (x0, x1) = diag ( sβm1 , · · · , sβmd ) diag ( +∞∑ j=0 (sβ1) j , · · · +∞∑ j=0 (sβd) j ) δb R⊕d (x0, x1) = diag ( sβm1 , · · · , sβmd ) diag ([ 1 1− sβ1 ] , · · · , [ 1 1− sβd ]) δb R⊕d (x0, x1). Since sβi ∈ (0, 1) for all i = 1, 2, · · · , d and δb R⊕d (x0, x1) are fixed, it is evident that by selecting m sufficiently large (with n > m), we can make δb R⊕d (xn, xm) arbitrarily small. This shows that {xn} is a Cauchy sequence. Finally, because (X ,R⊕d , δb R⊕d ) is complete, there exists some z ∈ X such that xn → z. To show that X has a fixed point, we consider the distance δb R⊕d (z,F (z)). From the triangle inequality and contraction condition, we get δb R⊕d (z,F (z)) ≾ s [ δb R⊕d (z, xn) + δb R⊕d (xn,F (z)) ] G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 15 of 27 = s [ δb R⊕d (z, xn) + δb R⊕d (F (xn−1),F (z)) ] ≾ s [ δb R⊕d (z, xn) + diag ( α1, · · · , αd ) δb R⊕d (xn−1,F (xn−1)) + δb R⊕d (z,F (z)) ] ≾ diag ( sα1, · · · , sαd ) δb R⊕d (z,F (z)), since xn → z and sαi ∈ (0, 1), ∀i = {1, 2, · · · , d} it is clear that we can make this distance as small as we please by choosing n sufficiently large. We conclude that δb R⊕d (z,F (z)) = 0 =⇒ F (z) = z, so z ∈ X is a fixed-point of F . To prove uniqueness, suppose there are two fixed points x = F (x) and y = F (x). Then, from the contraction condition, we have δb R⊕d (x, y) = δb R⊕d (F (x),F (y)) ≾ diag ( sα1, · · · , sαd ) [δb R⊕d (x,F (x))+δb R⊕d (y,F (y))], this implies δb R⊕d (x, y) = 0. Hence x = y, and the fixed-point X of F is unique. If d = 1 and s = 1, then the theorem 4 can be reduced to the standard Kannan in the metric space. Corolary 4 (Kannan, [24] ). Suppose that (X , d) is a complete metric space and F : X → X is an operator such that δ(Fx,Fy) ≤ α [δ(Fx, x) + δ(Fy, y)] , for constant α ∈ (0, 12) and for every x, y ∈ X . Then F has a unique fixed-point z ∈ X . Theorem 5. ( Chatterjee Type) Suppose that (X ,R⊕d , δb R⊕d ) is a complete generalized b-metric space endowed with the orthogonal direct sum and F : X → X is an operator satisfying the following condition δb R⊕d (F (x),F (y)) ≾ diag ( α1, · · · , αd ) [δb R⊕d (F (x), y) + δb R⊕d (F (y), x)], (2) for sαi ∈ (0, 12) for all i = 1, 2, · · · , d, and for each x, y ∈ X . Then, F has a unique fixed-point in X . Proof. Let x0 be any point in X , that is x0 ∈ X . Let us define a sequence {xn} in X as given below. xn+1 = F (xn) = Fn+1(x0) ∀ n ≥ 0. We have δb R⊕d (xn, xn+1) = δb R⊕d (F (xn−1),F (xn)) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 16 of 27 ≾ diag ( α1, · · · , αd ) [ δb R⊕d (F (xn−1), xn) + δb R⊕d (F (xn), xn−1) ] = diag ( α1, · · · , αd ) [δb R⊕d (xn, xn) + δb R⊕d (xn+1, xn−1)] = diag ( α1, · · · , αd ) δb R⊕d (F (xn),F (xn−2)) ≾ diag ( sα1, · · · , sαd ) [ δb R⊕d (F (xn),F (xn−1)) + δb R⊕d (F (xn−1),F (xn−2)) ] = diag ( sα1, · · · , sαd ) [ δb R⊕d (xn+1, xn) + δb R⊕d (xn, xn−1) ] . Thus, δb R⊕d (xn, xn+1) ≾ diag ( sα1 (1−sα1) , · · · , sαd (1−sαd) ) δb R⊕d (xn−1, xn), put ( sαi (1− sαi) = λi ) = diag(λ1, · · · , λd) δbR⊕d (xn−1, xn), where (λ1, , · · · , λd) ≺ IR ⊕d + ≾ diag(λ21, · · · , λ2d) δbR⊕d (xn−2, xn−1) ≾ diag(λ31, · · · , λ3d) δbR⊕d (xn−3, xn−2) ... ≾ diag(λn1 , · · · , λnd )δbR⊕d (x0, x1). Let us prove that {xn} is a Cauchy sequence. Suppose that n > m, then from (2) and the triangle inequality property, we can write as: δb R⊕d (xm, xn) ≾ s [ δb R⊕d (xm, xm+1) + δb R⊕d (xm+1, xn) ] ≾ sδb R⊕d (xm, xm+1) + s2 [ δb R⊕d (xm+1, xm+2) + δb R⊕d (xm+2, xn) ] ≾ sδb R⊕d (xm, xm+1) + s2δb R⊕d (xm+1, xm+2) + s3 [ δb R⊕d (xm+2, xm+3) + δb R⊕d (xm+3, xn) ] ... ≾ sδb R⊕d (xm, xm+1) + s2δb R⊕d (xm+1, xm+2) + · · ·+ sn−mδb R⊕d (xn−1, xn) = diag(sλm1 , sλ m 2 , · · · , sλmd )δb R⊕d (x0, x1) + diag(s2λm+1 1 , s2λm+1 2 , · · · , s2λm+1 d )δb R⊕d (x0, x1) + · · ·+ diag(sn−mλn−1 1 , sn−mλn−1 2 , · · · , sn−mλn−1 d )δb R⊕d (x0, x1) = ( (sλm1 + s2λm+1 1 + · · ·+ sn−mλn−1 1 )δb R⊕d (x0, x1), (sλ m 2 + s2λm+1 2 + · · ·+ sn−mλn−1 2 )δb R⊕d (x0, x1) , · · · , (sλmd + s2λm+1 d + · · ·+ sn−mλn−1 d )δb R⊕d (x0, x1) ) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 17 of 27 = diag ( n−m∑ i=1 n−1∑ j=m siλj1δ b R⊕d (x0, x1), n−m∑ i=1 n−1∑ j=m siλj2δ b R⊕d (x0, x1), · · · , n−m∑ i=1 n−1∑ j=m siλjdδ b R⊕d (x0, x1) ) = diag ( n−m∑ i=1 n−1∑ j=m siλj1, n−m∑ i=1 n−1∑ j=m siλj2, · · · , n−m∑ i=1 n−1∑ j=m siλjd ) δb R⊕d (x0, x1) = diag ( sλm1 n−m−1∑ j=0 (sλ1) j , sλm2 n−m−1∑ j=0 (sλ2) j , · · · , sλmd n−m−1∑ j=0 (sλd) j ) δb R⊕d (x0, x1) ≾ diag ( sλm1 +∞∑ j=0 (sλ1) j , sλm2 +∞∑ j=0 (sλ2) j , · · · , sλm1 +∞∑ j=0 (sλd) j ) δb R⊕d (x0, x1) = diag ( sλm1 [ 1 1− sλ1 ] , sλm2 [ 1 1− sλ2 ] , · · · , sλm1 [ 1 1− sλd ]) δb R⊕d (x0, x1) ≾ diag(sλm1 , sλ m 2 , · · · , sλmd ) diag ([ 1 1− sλ1 ] , [ 1 1− sλ2 ] , · · · , [ 1 1− sλd ]) δb R⊕d (x0, x1) → 0, as n,m → +∞. Since 0̃ ≺ diag(sλj1, sλ j 2, · · · , sλ j d) ≺ IR ⊕d + and δb R⊕d (x0, x1) are fixed, it is evident that by selectingm sufficiently large (with n > m), we can make δb R⊕d (xn, xm) arbi- trarily small. This shows that {xn} is a Cauchy sequence. Finally, because (X ,R⊕d , δb R⊕d ) is complete, there exists some z ∈ X such that xn → z. To show that X has a fixed point, we consider the distance δb R⊕d (z,F (z)). From the triangle inequality and contraction condition, we get δb R⊕d (z,F (z)) ≾ s [ δb R⊕d (z, xn) + δb R⊕d (xn,F (z)) ] = s [ δb R⊕d (z, xn) + δb R⊕d (F (xn−1),F (z)) ] ≾ s [ δb R⊕d (z, xn) + diag ( α1, α2, · · · , αd ) (δb R⊕d (F (xn−1), z) + δb R⊕d (F (z), xn−1)) ] = diag ( s(1 + α1), s(1 + α2), · · · , s(1 + αd) ) δb R⊕d (xn, z) +diag ( sα1, sα2, · · · , sαd ) δb R⊕d (F (z), xn−1), since xn → z it is clear that δb R⊕d (z,F (z)) = 0 =⇒ F (z) = z, so z ∈ X is a fixed-point of F . To prove uniqueness, suppose there are two fixed points x = F (x) and y = F (y). Then, from the contraction condition, we have δb R⊕d (x, y) = δb R⊕d (F (x),F (y)) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 18 of 27 ≾ diag ( sα1, · · · , sαd ) [δb R⊕d (F (x), x) + δb R⊕d (x, y) + δb R⊕d (F (y), y) + δb R⊕d (y, x)] = 2 diag ( sα1, · · · , sαd ) δb R⊕d (x, y) ≺ δb R⊕d (x, y), this implies δb R⊕d (x, y) = 0. Hence x = y, and the fixed-point X of F is unique. If d = 1 and s = 1, then the theorem 5 can be reduced to the standard Chatterjee in the metric space. Corolary 5 (Chatterjee, [25]). Suppose that (X , d) is a complete metric space and F : X → X is an operator such that δ(Fx,Fy) ≤ α [δ(Fx, y) + δ(Fy, x)] , for constant α ∈ (0, 12) and for every x, y ∈ X . Then F has a unique fixed-point z ∈ X . Theorem 6. Suppose that (X ,R⊕d , δb R⊕d ) is a complete generalized b-metric space endowed with the orthogonal direct sum and F : X → X is an operator satisfying the following condition δb R⊕d (Fx,Fy) ≾ diag ( α1, α2, · · · , αd ) δb R⊕d (x, y) +diag ( β1, β2, · · · , βd ) [δb R⊕d (x,F (x)) + δb R⊕d (y,F (y))], for αi, βi ∈ Rd + and αi + 2βi ∈ [0, 1), i = 1, 2, · · · , d for each x, y ∈ X . Then, F has a unique fixed-point in X . Proof. Let x0 be any point in X , that is x0 ∈ X . Let us define a sequence {xn} in X as given below. xn+1 = F (xn) = Fn+1(x0) ∀ n ≥ 0. We have δb R⊕d (xn, xn+1) = δb R⊕d (F (xn−1),F (xn)) ≾ diag ( α1, α2, · · · , αd ) δb R⊕d (xn−1, xn) +diag ( β1, β2, · · · , βd ) [δb R⊕d (xn−1,F (xn−1)) + δb R⊕d (xn,F (xn))] = diag ( α1, α2, · · · , αd ) δb R⊕d (xn−1, xn) +diag ( β1, β2, · · · , βd ) [δb R⊕d (xn−1, xn) + δb R⊕d (xn, xn+1)] = diag ( (α1 + β1), (α2 + β2), · · · , (αd + βd) ) δb R⊕d (F (xn−2),F (xn−1)) +diag ( β1 β2, · · · , βd ) δb R⊕d (xn, xn+1). Thus, diag ( (1− β1), (1− β2), · · · , (1− βd) ) δb R⊕d (xn, xn+1) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 19 of 27 ≾ diag ( (α1 + β1), (α2 + β2), · · · , (αd + βd) ) δb R⊕d (F (xn−2),F (xn−1)). This implies that δb R⊕d (xn, xn+1) ≾ diag ( (α1 + β1) (1− β1) , · · · , (αd + βd) (1− βd) ) δb R⊕d (xn−1, xn), we use λi = αi + βi 1− βi ∈ (0, 1), i = 1, 2, · · · , d we obtain δb R⊕d (xn, xn+1) ≾ diag ( λ1, λ2, · · · , λd ) δb R⊕d (xn, xn−1). So δb R⊕d (xn, xn+1) ≾ diag ( λ1, λ2, · · · , λd ) δb R⊕d (xn−1, xn) ≾ diag ( λ21, λ22, · · · , λ2d ) δb R⊕d (xn−2, xn−1) ≾ diag ( λ31, λ32, · · · , λ3d ) δb R⊕d (xn−3, xn−2) ≾ ... ≾ diag ( λn1 , λn2 , · · · , λnd ) δb R⊕d (x0, x1). Let us prove that {xn} is a Cauchy sequence. Suppose that m < n, then from (3) and the triangle inequality property, we can write as: δb R⊕d (xm, xn) ≾ sδb R⊕d (xm, xm+1) + sδb R⊕d (xm+1, xn) ≾ sδb R⊕d (xm, xm+1) + s2 [ δb R⊕d (xm+1, xm+2) + δb R⊕d (xm+2, xn) ] ≾ sδb R⊕d (xm, xm+1) + s2δb R⊕d (xm+1, xm+2) + s3 [ δb R⊕d (xm+2, xm+3) + δb R⊕d (xm+3, xn) ] ... ≾ sδb R⊕d (xm, xm+1) + s2δb R⊕d (xm+1, xm+2) + · · ·+ sn−mδb R⊕d (xn−1, xn) = diag ( sλm1 , sλm2 , · · · , sλmd ) δb R⊕d (x0, x1) +diag ( s2λm+1 1 , s2λm+1 2 , · · · , s2λm+1 d ) δb R⊕d (x0, x1) +diag ( s3λm+2 1 , s3λm+2 2 , · · · , s3λm+2 d ) δb R⊕d (x0, x1) + · · · +diag ( sn−mλn−1 1 , sn−mλn−1 2 , · · · , sn−mλn−1 d ) δb R⊕d (x0, x1) = ( (sλm1 + s2λm+1 1 + · · ·+ sn−mλn−1 1 )δb R⊕d (x0, x1), G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 20 of 27 (sλm2 + s2λm+1 2 + · · ·+ sn−mλn−1 2 )δb R⊕d (x0, x1) , · · · , (sλmd + s2λm+1 d + · · ·+ sn−mλn−1 d )δb R⊕d (x0, x1) ) = diag (∑n−m i=1 ∑n−1 j=m s iλj1, · · · , ∑n−m i=1 ∑n−1 j=m s iλjd ) δb R⊕d (x0, x1) = diag ( sλm1 ∑n−m−1 j=0 (sλ1) j , · · · , sλm1 ∑n−m−1 j=0 (sλd) j ) δb R⊕d (x0, x1) ≾ diag ( sλm1 ∑+∞ j=0 (sλ1) j , · · · , sλm1 ∑+∞ j=0 (sλd) j ) δb R⊕d (x0, x1) = diag ( sλm1 [ 1 1− sλ1 ] , · · · , sλmd [ 1 1− sλd ]) δb R⊕d (x0, x1) ≾ diag ( sλm1 , · · · , sλmd ) diag ([ 1 1− sλ1 ] , · · · , [ 1 1− sλd ]) δb R⊕d (x0, x1) → 0, as n,m → +∞. Since sλi ∈ (0, 1) for all i = 1, 2, · · · , d and δb R⊕d (x0, x1) are fixed, it is evident that by selectingm sufficiently large (with n > m), we can make δb R⊕d (xn, xm) arbi- trarily small. This shows that {xn} is a Cauchy sequence. Finally, because (X ,R⊕d , δb R⊕d ) is complete, there exists some z ∈ X such that xn → z. To show that z is a fixed point, we consider the distance δb R⊕d (z,F (z)). From the triangle inequality and contraction condition, we get δb R⊕d (z,F (z)) ≾ s[δb R⊕d (z, xn) + δb R⊕d (xn,F (z))] = s[δb R⊕d (z, xn) + δb R⊕d (F (xn−1),F (z))] ≾ s [ δb R⊕d (z, xn) + diag ( α1, · · · , αd ) δb R⊕d (xn−1, z) + diag ( β1, · · · , βd ) [ δb R⊕d (xn−1,F (xn−1)) + δb R⊕d (z,F (z)) ]] , therefore ( 1− sβ1, · · · , 1− sβd ) δb R⊕d (z,F (z)) ≾ sδb R⊕d (z, xn) + diag ( sα1, · · · , sαd ) δb R⊕d (xn−1, z) + diag ( sβ1, · · · , sβd ) [ δb R⊕d (xn−1,F (xn−1)) ] → 0, and since xn → z it is clear that we can make this distance as small as we please by choosing n sufficiently large. We conclude that δb R⊕d (z,F (z)) = 0 =⇒ F (z) = z, so z ∈ X is a fixed-point of F . To prove uniqueness, suppose there are two fixed points x = F (x) and y = F (y). Then, from the contraction condition, we have δb R⊕d (x, y) = δb R⊕d (F (x),F (y)) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 21 of 27 ≾ diag ( α1, α2, · · · , αd ) δb R⊕d (x, y) +diag ( β1, β2, · · · , βd ) [δb R⊕d (x,F (x)) + δb R⊕d (y,F (y))] = diag ( α1, α2, · · · , αd ) δb R⊕d (x, y) ≺ δb R⊕d (x, y), this implies δb R⊕d (x, y) = 0. Hence x = y, and the fixed-point x of F is unique. Theorem 7. Suppose that (X ,R⊕d , δb R⊕d ) is a complete generalized b-metric space endowed with the orthogonal direct sum and F : X → X is a continuous operator satisfying the following condition δb R⊕d (Fx,Fy) ≾ diag ( α1, α2, · · · , αd ) δb R⊕d (x, y) (3) +diag ( β1, β2, · · · , βd ) [δb R⊕d (x,F (x)) + δb R⊕d (y,F (y))] + diag ( γ1, γ2, · · · , γd ) [δb R⊕d (x,F (y)) + δb R⊕d (y,F (x))],(4) for αi, βi, γi ∈ R+ ∀ i = 1, 2, 3, · · · , d and αi + 2(βi + sγi) ∈ [0, 1), for each x, y ∈ X . Then, F has a unique fixed-point in X . Proof. Let x0 be any point in X , that is x0 ∈ X . Let us define a sequence {xn} in X as given below. xn+1 = F (xn) = Fn+1(x0) ∀ n ≥ 0. We have δb R⊕d (xn, xn+1) = δb R⊕d (F (xn−1),F (xn)) ≾ ( α1, α2, · · · , αd ) δb R⊕d (xn−1, xn) +diag ( β1, β2, · · · , βd ) [δb R⊕d (xn−1,F (xn−1)) + δb R⊕d (xn,F (xn))] + diag ( γ1, γ2, · · · , γd ) [δb R⊕d (xn−1,F (xn)) + δb R⊕d (xn,F (xn−1))] = diag ( α1, α2, · · · , αd ) δb R⊕d (xn−1, xn) +diag ( β1, β2, · · · , βd ) [δb R⊕d (xn−1, xn) + δb R⊕d (xn, xn+1)] + diag ( γ1, γ2, · · · , γd ) [δb R⊕d (xn−1, xn+1))] ≾ diag ( α1 + β1, α2 + β2, · · · , αd + βd ) δb R⊕d (xn−1, xn) +diag ( β1, β2, · · · , βd ) δb R⊕d (xn, xn+1) +diag ( sγ1, sγ2, · · · , sγd ) [δb R⊕d (xn−1, xn) + δb R⊕d (xn, xn+1))] = diag ( α1 + β1 + sγ1, α2 + β2 + sγ2, · · · , αd + βd + sγd ) δb R⊕d (xn−1, xn) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 22 of 27 +diag ( β1 + sγ1, β2 + sγ2, · · · , βd + sγd ) δb R⊕d (xn, xn+1). Thus, diag ( (1− (β1 + sγ1)), (1− (β2 + sγ2)), · · · , (1− (βd + sγd)) ) δb R⊕d (xn, xn+1) ≾ diag ( α1 + β1 + sγ1, α2 + β2 + sγ2, · · · , αd + βd + sγd ) δb R⊕d (xn−1, xn). This implies that δb R⊕d (xn, xn+1) ≾ diag ( (α1 + β1 + sγ1) (1− (β1 + sγ1)) , (α2 + β2 + sγ2) (1− (β2 + sγ2)) , · · · , (αd + βd + sγd) (1− (βd + sγd)) ) δb R⊕d (xn−1, xn). we use λi = αi + βi + sγi 1− (βi + sγi) ∈ (0, 1) for all i = 1, 2, · · · , d, we obtain δb R⊕d (xn, xn+1) ≾ diag ( λ1, λ2, · · · , λd ) δb R⊕d (xn−1, xn). So δb R⊕d (xn, xn+1) ≾ diag ( λ1, λ2, · · · , λd ) δb R⊕d (xn−1, xn) ≾ diag ( λ21, λ22, · · · , λ2d ) δb R⊕d (xn−2, xn−1) ≾ diag ( λ31, λ32, · · · , λ3d ) δb R⊕d (xn−3, xn−2) ≾ · · · ≾ diag ( λn1 , λn2 , · · · , λnd ) δb R⊕d (x0, x1). (5) Let us prove that {xn} is a Cauchy sequence. Suppose that m < n, then from (5) and the triangle inequality property, we can write as: δb R⊕d (xm, xn) ≾ s [ δb R⊕d (xm, xm+1) + δb R⊕d (xm+1, xn) ] ≾ sδb R⊕d (xm, xm+1) + s2 [ δb R⊕d (xm+1, xm+2) + δb R⊕d (xm+2, xn) ] ≾ sδb R⊕d (xm, xm+1) + s2δb R⊕d (xm+1, xm+2) + s3 [ δb R⊕d (xm+2, xm+3) + δb R⊕d (xm+3, xn) ] ... ≾ sδb R⊕d (xm, xm+1) + s2δb R⊕d (xm+1, xm+2) + s3δb R⊕d (xm+2, xm+3) + · · ·+ sn−mδb R⊕d (xn−1, xn) = diag ( sλm1 , sλm2 , · · · , sλmd ) δb R⊕d (x0, x1) +diag ( s2λm+1 1 , s2λm+1 2 , · · · , s2λm+1 d ) δb R⊕d (x0, x1) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 23 of 27 +diag ( s3λm+2 1 , s3λm+2 2 , · · · , s3λm+2 d ) δb R⊕d (x0, x1) + · · ·+ diag ( sn−mλn−1 1 , sn−mλn−1 2 , · · · , sn−mλn−1 d ) δb R⊕d (x0, x1) = diag ( (sλm1 + s2λm+1 1 + · · ·+ sn−mλn−1 1 ), (sλm2 + s2λm+1 2 + · · ·+ sn−mλn−1 2 ), · · · , (sλmd + s2λm+1 d + · · ·+ sn−mλn−1 d ) ) δb R⊕d (x0, x1) = diag ( n−m∑ i=1 n−1∑ j=m siλj1, n−m∑ i=1 n−1∑ j=m siλj2, · · · , n−m∑ i=1 n−1∑ j=m siλjd ) δb R⊕d (x0, x1) = diag ( sλm1 n−m−1∑ j=0 (sλ1) j , sλm2 n−m−1∑ j=0 (sλ2) j , · · · , sλm1 n−m−1∑ j=0 (sλd) j ) δb R⊕d (x0, x1) ≾ diag ( sλm1 +∞∑ j=0 (sλ1) j , sλm2 +∞∑ j=0 (sλ2) j , · · · , sλm1 +∞∑ j=0 (sλd) j ) δb R⊕d (x0, x1) = diag ( sλm1 [ 1 1− sλ1 ] , sλm2 [ 1 1− sλ2 ] , · · · , sλmd [ 1 1− sλd ]) δb R⊕d (x0, x1) ≾ diag(sλm1 , · · · , sλmd ) diag ([ 1 1− sλ1 ] , · · · , [ 1 1− sλd ]) δb R⊕d (x0, x1) → 0, as n,m → +∞. Since λi ∈ (0, 1) ∀i and δb R⊕d (x0, x1) are fixed, it is evident that by selecting m sufficiently large (with n > m), we can make δb R⊕d (xn, xm) arbitrarily small. This shows that {xn} is a Cauchy sequence. Finally, because (X ,R⊕d , δb R⊕d ) is complete, there exists some z ∈ X such that xn → z. To show that x is a fixed point, we consider the distance δb R⊕d (z,F (z)). From the triangle G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 24 of 27 inequality and contraction condition, we get δb R⊕d (z,F (z)) ≾ sδb R⊕d (z, xn) + sδb R⊕d (xn,F (z)) = sδb R⊕d (z, xn) + sδb R⊕d (F (xn−1),F (z)) ≾ sδb R⊕d (z, xn) + sdiag ( α1, α2, · · · , αd ) δb R⊕d (xn−1, z) + sdiag ( β1, β2, · · · , βd ) [δb R⊕d (xn−1,F (xn−1)) + δb R⊕d (z,F (z))] + sdiag ( γ1, γ2, · · · , γd ) [δb R⊕d (xn−1,F (z)) + δb R⊕d (z,F (xn−1))] = sdiag ( 1 + γ1, 1 + γ2, · · · , 1 + γd ) δb R⊕d (z, xn) + sdiag ( α1, α2, · · · , αd ) δb R⊕d (xn−1, z) + sdiag ( β1, β2, · · · , βd ) [δb R⊕d (xn−1, xn) + δb R⊕d (z,F (z))] + sdiag ( γ1, γ2, · · · , γd ) δb R⊕d (xn−1,F (z)), hence diag ( 1 + sβ1, · · · , 1 + sβd ) δb R⊕d (z,F (z)) ≾ sdiag ( 1 + γ1, · · · , 1 + γd ) δb R⊕d (z, xn) + sdiag ( α1, · · · , αd ) δb R⊕d (xn−1, z) + sdiag ( β1, · · · , βd ) δb R⊕d (xn−1, xn) + sdiag ( γ1, · · · , γd ) δb R⊕d (F (xn−2),F (z)), and since xn → z it is clear that we can make this distance as small as we please by choosing n sufficiently large. We conclude that δb R⊕d (z,F (z)) = 0 =⇒ F (z) = z, so z ∈ X is a fixed-point of F . To prove uniqueness, suppose there are two fixed points x = F (x) and y = F (y). Then, from the contraction condition, we have δb R⊕d (x, y) = δb R⊕d (F (x),F (y)) ≾ diag ( α1, α2, · · · , αd ) δb R⊕d (x, y) +diag ( β1, β2, · · · , βd ) [δb R⊕d (x,F (x)) + δb R⊕d (y,F (y))] + diag ( γ1, γ2, · · · , γd ) [δb R⊕d (x,F (y)) + δb R⊕d (y,F (x))] = diag ( α1, α2, · · · , αd ) δb R⊕d (x, y) +diag ( 2γ1, 2γ2, · · · , 2γd ) δb R⊕d (x, y) = diag ( α1 + 2γ1, α2 + 2γ2, · · · , αd + 2γd ) δb R⊕d (x, y) G. ALbeladi, S. Omran / Eur. J. Pure Appl. Math, 18 (3) (2025), 6583 25 of 27 ≺ δb R⊕d (x, y), this implies δb R⊕d (x, y) = 0. Hence x = y, and the fixed-point x of F is unique. 4. Conclusions and Future Work This research presents novel extensions of the Banach fixed-point theorem within gen- eralized metric spaces that include a direct sum structure. By introducing a diagonal matrix A in Rd, we establish evolved contraction conditions that enhance the applica- bility of fixed-point results in this framework. Our method provides a more structured and flexible methodology for analyzing contraction operators, thereby contributing to the promotion of fixed-point theory in vector-valued metric spaces. The results obtained ex- tend classical fixed-point theorems and offer new understandings of the interplay between metric structures and direct sum operations. Future studies may expand these results to more comprehensive classes of generalised metric spaces, such as G-metric or partial metric spaces. Studying non-diagonal or operator-valued matrix systems within the direct sum framework could yield more profound insights. Applications to vector-valued integral and differential equations, as well as iterative schemes for approximating fixed points under the new conditions, also certify further study. Acknowledgements We would like to extend my sincere thanks and appreciation to Dr. Mohamed Gamal for reviewing the paper and making valuable modifications. Declarations • Conflicts of interest The authors declare that they have no conflicts of interest. • Author’s contributions Conceptualization, Writing original draft (G. Albeladi); Methodology, Data curation (S. Omran); Formal analysis, Revisions (S. Omran); Project administration (G. Albeladi); Supervision (S. Omran). References [1] AI Perov. On the cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn, 2(1964):115–134, 1964. [2] Muhammad Usman Ali and Jong Kyu Kim. An extension of vector-valued met- ric spaces and perov’s fixed point theorem (nonlinear analysis and convex analysis). Research Institute for Mathematical Sciences Kokyuroku, 2114:12–20, 2019. [3] Shaoyuan Xu, Yan Han, Suzana Aleksić, and Stojan Radenovic. 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