EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6218 ISSN 1307-5543 – ejpam.com Published by New York Business Global Behavior and Solution Representations of Fourth-Order Rational Systems of Difference Equations Hanan S. Gafel1,∗, Haya A. Altamimi1 1 Department of Mathematics and Statistics, College of Science, Taif University, Taif 21944, Saudi Arabia Abstract. Our aim in this paper is to obtain formulas expressions for solutions of the difference equations. The purpose of this article is to determine the expressions of solutions for the following rational difference systems Θn+1 = Θn−2Ωn α ± Θn−2 ± Ωn−3 , Ωn+1 = ΘnΩn−2 β + Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , where the real numbers α and β are arbitrary. Additionally, the qualitative behavior of the solutions is analyzed, including their boundedness as well as their local and global stability. We will illustrate our findings with numerical examples. 2020 Mathematics Subject Classifications: 39A10, 39A23 Key Words and Phrases: Difference equations, Fourth-Order, Rational systems, Local and global stability 1. Introduction The study of rational difference equations and their systems has garnered signif- icant attention in recent years due to their essential role in analyzing dynamic systems. These equations serve as mathematical models for simulating various real-world phenom- ena and act as effective tools for approximating solutions to continuous problems. As the demand for a deeper understanding of these equations grows, an increasing number of researchers are investigating their qualitative properties. This interest arises from their capacity to accurately represent a wide range of physical, biological, and economic phe- nomena. Moreover, research in this field has led to notable advancements by providing numerical solutions and analytical techniques that establish links between difference and differential equations, ultimately contributing to the development of more precise and adaptable models for understanding complex systems. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6218 Email addresses: h.gafal@tu.edu.sa (H. S. Gafel), s44580205@students.tu.edu.sa (H. A. Altamimi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 2 of 28 Extensive research has been carried out regarding the method of determining the gen- eral form of the solution for certain special cases of the problem. The systems and behavior of rational difference equations have been extensively studied (see, for instance, [1]-[2]). The solutions of the following three-dimensional system of difference equations were examined by Tollu and Yalçınkaya [3]. sn+1 = a + tnun tn + un , tn+1 = a + snun un + sn , un+1 = a + sntn sn + tn . Khaliq et al. [4]. conducted a dynamical analysis of the following discrete-time Lotka– Volterra system with two predators and one prey sn+1 = αsn − βsntn − γsnun 1 + δsn , tn+1 = ζyn + ηsntn − µtnun 1 + εtn , un+1 = vzn + ρsnun − σtnun 1 + ωun . Tollu et al. [2]. examined the general solutions of the following system of second-order difference equations. sn+1 = a +1 + tnsn−1 , tn+1 = b +1 + tn−1sn . In [5]. Khaliq and Shoaib analyzed the global and local stability of the equilibrium point for the three-dimensional system of difference equations sn+1 = sn a + sn−1tn−1un−1 , tn+1 = tn b + tn−1un−1sn−1 , un+1 = un −c + un−1sn−1tn−1 . Elsayed and Alharbi [6]. derived a closed-form expression for the solutions of the following systems. sn+1 = sntn−1 sn + tn , tn+1 = tnsn−1 sn + tn . The formula for the general solution of the system has been determined by Halim et al. [7]. sn+1 = tn−1sn−2 tn (α + βtn−1sn−2) , tn+1 = sn−1tn−2 sn (α + βsn−1tn−2) . [8]. Elsayed et al. discuss the structure of the system’s solutions: H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 3 of 28 sn+1 = α1un−1tn−1 sn−1 + tn−1 + un−1 , tn+1 = α2un−1sn−1 sn−1 + tn−1 + un−1 , un+1 = α3sn−1tn−1 sn−1 + tn−1 + un−1 . Finding the solution expressions for the following rational difference systems is the primary objective of this article: Θn+1 = Θn−2Ωn α ± Θn−2 ± Ωn−3 , Ωn+1 = ΘnΩn−2 β + Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , (1) where α and β are arbitrary real values, and the initial conditions Θ−3, Θ−2, Θ−1, Θ0, Ω−3, Ω−2, Ω−1 and Ω0 are nonzero real integers. Additionally, we examine the behavior of the solutions, including their boundedness and their local and global stability. 2. Main Results Let IΘ and IΩ be any real number intervals, and let f : I3 Θ ×I3 Ω → IΘ, g : I3 Θ ×I3 Ω → IΩ be continuously differentiable functions. Then for each initial condition (Θi, Ωi) ∈ IΘ × IΩ for i ∈ {−3, −2, −1, 0}, the system of difference equations: Θn+1 = f (Θn, Θn−2, Θn−3, Ωn, Ωn−2, Ωn−3) , Ωn+1 = g (Θn, Θn−2, Θn−3, Ωn, Ωn−2, Ωn−3) , n = 0, 1, 2, . . . , (2) has a unique solution {Θn, Ωn}∞ n=−3. Definition 1. A point ( Θ̄, Ω̄ ) is said to be an equilibrium point of (2) if Θ̄ = f(Θ̄, Θ̄, Θ̄, Ω̄, Ω̄, Ω̄), Ω̄ = g(Θ̄, Θ̄, Θ̄, Ω̄, Ω̄, Ω̄) are satisfied. Definition 2. Assume that ( Θ̄, Ω̄ ) is a fixed point of (2). (i): (Θ̄, Ω̄) is said to be stable if, for every ε > 0, there exists δ > 0 such that, for every initial condition (Θi, Ωi) ∈ IΘ × IΩ for i ∈ {−3, −2, −1, 0} if∥∥∥∥∥∥ 0∑ i=−3 (Θi, Ωi) − (Θ̄, Ω̄) ∥∥∥∥∥∥ < δ ⇒ ∥∥∥(Θn, Ωn) − (Θ̄, Ω̄) ∥∥∥ < ε for all n > 0. (ii): (Θ̄, Ω̄) is said to be unstable if it is not stable. (iii): (Θ̄, Ω̄) is called asymptotically stable if there exists γ > 0 such that H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 4 of 28 ∥∥∥∥∥∥ 0∑ i=−3 (Θi, Ωi) − (Θ̄, Ω̄) ∥∥∥∥∥∥ < γ, (Θn, Ωn) → (Θ̄, Ω̄) as n → ∞. (iv): (Θ̄, Ω̄) is called global attractor if (Θn, Ωn) → (Θ̄, Ω̄) as n → ∞. (v): (Θ̄, Ω̄) is called globally asymptotically stable if it is a global attractor and stable. Theorem 1. [9]. Assume that (Θn+1, Ωn+1) = F (Θn, Ωn) , n = 0, 1, . . ., is a system of difference equations where F is continuously differentiable on open neighborhood H ⊆ Rn+1 and (Θ̄, Ω̄) is a fixed point of F then (1) If all eigenvalues of the Jacobian matrix JF at equilibrium point (Θ̄, Ω̄) lie inside the unit disk i.e. |λi| < 1 then (Θ̄, Ω̄) is locally asymptotically stable. (2) If at least one eigenvalues at equilibrium point (Θ̄, Ω̄) outside the unit disk, then ( Θ̄, Ω̄ ) is unstable. Theorem 2. [10]. Let [a1, a2] and [b1, b2] be an interval of real numbers Moreover, suppose that f : [a1, a2] × [b1, b2] → [a1, a2] and g : [a1, a2] × [b1, b2] → [b1, b2] are continuous functions. Let Θn+1 = f (Θn, Ωn) and Ωn+1 = g (Θn, Ωn) , and assume ( m1, m2, M1, M2 ) be a solution of the system m1 = f (m1, m2) , M1 = f (M1, M2) , m2 = g (m1, m2) , M2 = g (M1, M2) . where mi = { m if f or g nondecreasing in Θ or Ω, M if f or g nonincreasing in Θ or Ω, and Mi = { M if f or g nondecreasing in Θ or Ω, m if f or g nonincreasing in Θ or Ω. Thus, m1 = M1 and m2 = M2. Then, the system of difference equations has a unique equilibrium point and it is a global attractor. 3. On the System: Θn+1 = Θn−2Ωn α+Θn−2+Ωn−3 , Ωn+1 = ΘnΩn−2 β+Θn−3+Ωn−2 The behavior of the solutions of the following system of difference equations is analyzed in this section Θn+1 = Θn−2Ωn α + Θn−2 + Ωn−3 , Ωn+1 = ΘnΩn−2 β + Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , (3) H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 5 of 28 Then, we derive a specific expression for the solutions of the following system of difference equations in (3) by considering the special case α = β = 0 Θn+1 = Θn−2Ωn Θn−2 + Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , (4) where Θ−3, Θ−2, Θ−1, Θ0, Ω−3, Ω−2, Ω−1 and Ω0 are nonzero real numbers that serve as initial conditions. 3.1. Local Stability of Equilibrium Point The stability of the critical point O = (0, 0) of system (3) is examined in this subsection. Theorem 3. The equilibrium point O is locally asymptotically stable. Proof. To investigate the stability of the critical point O, we assume xn = Θn−3, yn = Θn−2, zn = Θn−1, un = Ωn−3, vn = Ωn−2, wn = Ωn−1, Consequently, system (3) can be expressed as follows: xn+1 yn+1 zn+1 Θn+1 un+1 vn+1 wn+1 Ωn+1  =  yn zn Θn ynΩn α+yn+un vn wn Ωn Θnvn β+xn+vn  . (5) The Jacobian matrix of (5) is given by: JF =  0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 αΩn+unΩn (α+yn+un)2 0 0 −ynΩn (α+yn+un)2 0 0 yn α+yn+un 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 −Θnvn (β+xn+vn)2 0 0 vn β+xn+vn 0 βΘn+xnΘn (β+xn+vn)2 0 0  . If we evaluate the Jacobian matrix about the equilibrium point, we get all eigenvalues |λi| = 0, i = 1, 2, . . . , 8. So all eigenvalues are inside the unit disk. Therefore Theorem 1 ensures that the origin is locally asymptotically stable. H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 6 of 28 3.2. Global Attractor We will show in this subsection that the global attractor is the critical point. Theorem 4. The equilibrium point O of the system (3) is the global attractor.Proof. To prove this, we will assume that f(p, q) = pq α + p + q , and g(p, q) = pq β + p + q , from here, ∂f ∂p = αq + q2 (α + p + q)2 , ∂f ∂q = αp + p2 (α + p + q)2 , ∂g ∂p = βq + q2 (β + p + q)2 , ∂g ∂q = βq + q2 (β + p + q)2 . We observe that f(p, q) and g(p, q) are nondecreasing in p and q. Let (m1, m2, M1, M2) be a solution of system (3) such that: m1 = f (m1, m2) , M1 = f (M1, M2) , m2 = g (m1, m2) , M2 = g (M1, M2) . So we have m1 = m1m2 α + m1 + m2 ⇒ αm1 + m2 1 + m1m2 = m1m2, (6) M1 = M1M2 α + M1 + M2 ⇒ αM1 + M2 1 + M1M2 = M1M2, (7) m2 = m1m2 β + m1 + m2 ⇒ βm2 + m1m2 + m2 2 = m1m2, (8) M2 = M1M2 β + M1 + M2 ⇒ βM2 + M1M2 + M2 2 = M1M2. (9) By subtracting (6) from (7), we obtain: α (m1 − M1) + ( m2 1 − M2 1 ) = 0, α (m1 − M1) + (m1 − M1) (m1 + M1) = 0, (m1 − M1) [α + m1 + M1] = 0, since α + m1 + M1 ̸= 0, hence, we find m1 − M1 = 0 tends to m1 = M1. Likewise, we obtain m2 = M2. Thus, system (3) has a unique equilibrium point, which is a global attractor according to Theorem 2. H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 7 of 28 Theorem 5. system (3) has a globally asymptotically stable equilibrium point O. Proof. The equilibrium point O is locally stable according to Theorem 3. Furthermore, it is a universal attractor according to Theorem 4. Therefore, the equilibrium point O is globally asymptotically stable according to Definition 2. 3.3. On Solution of System (4) In this subsection, we find the solution to system (4). Theorem 6. Assume that system (4) has a solution {Θn, Ωn}∞ n=−3. For n = 0, 1, 2, . . ., Θ6n−3 = ηnλnσnζµnτnκn[ ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i)κ) ] , Θ6n−2 = ηnλnσn+1µnτnκn[ ∏n−1 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ] , Θ6n−1 = ηnλn+1σnµnτnκn[ ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ] , Θ6n = ηn+1λnσnµnτnκn[ ∏n−1 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ] , Θ6n+1 = ηnλnσn+1µn+1τnκn[ (σ + δ) ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 1)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ] , Θ6n+2 = ηn+1λn+1σnµnτnκn+1[ (λ + κ)(ζ + κ) ∏n−1 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 3)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ] , Ω6n−3 = ηnλnσnµnτnκnδ[ ∏n−1 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i)τ) ((2i)σ + δ)(ζ + (2i + 1)κ) ] , Ω6n−2 = ηnλnσnµnτnκn+1[ ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ] , Ω6n−1 = ηnλnσnµnτn+1κn[ ∏n−1 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ] , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 8 of 28 Ω6n = ηnλnσnµn+1τnκn[ ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ] , Ω6n+1 = ηn+1λnσnµnτnκn+1[ (ζ + κ) ∏n−1 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ] , Ω6n+2 = ηnλnσn+1µn+1τn+1κn[ (σ + τ)(σ + δ) ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ] , where Θ−3 = ζ, Θ−2 = σ, Θ−1 = λ, Θ0 = η, Ω−3 = δ, Ω−2 = κ, Ω−1 = τ and Ω0 = µ. Proof. The outcome holds true for n = 0. Assuming that n > 0 and that our hypothesis is correct for n − 1, we have: Θ6n−9 = ηn−1λn−1σn−1ζµn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i)κ) ] , Θ6n−8 = ηn−1λn−1σnµn−1τn−1κn−1[ ∏n−2 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ] , Θ6n−7 = ηn−1λnσn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ] , Θ6n−6 = ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ] , Θ6n−5 = ηn−1λn−1σnµnτn−1κn−1[ (σ + δ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 1)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ] , Θ6n−4 = ηnλnσn−1µn−1τn−1κn[ (λ + κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 3)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ] , Ω6n−9 = ηn−1λn−1σn−1µn−1τn−1κn−1δ[ ∏n−2 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i)τ) ((2i)σ + δ)(ζ + (2i + 1)κ) ] , Ω6n−8 = ηn−1λn−1σn−1µn−1τn−1κn[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ] , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 9 of 28 Ω6n−7 = ηn−1λn−1σn−1µn−1τnκn−1[ ∏n−2 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ] , Ω6n−6 = ηn−1λn−1σn−1µnτn−1κn−1[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ] , Ω6n−5 = ηnλn−1σn−1µn−1τn−1κn[ (ζ + κ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ] , Ω6n−4 = ηn−1λn−1σnµnτnκn−1[ (σ + τ)(σ + δ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ] . Now, it can be inferred from system (4) that Θ6n−3 = Θ6n−6 Ω6n−4 Θ6n−6 + Ω6n−7 =  ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ]  ηn−1λn−1σnµnτnκn−1[ (σ + τ)(σ + δ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ]   ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ]  +  ηn−1λn−1σn−1µn−1τnκn−1[ ∏n−2 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i + 2)τ) ((2i + 2)σ + δ)(ζ + (2i + 1)κ) ]  H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 10 of 28 =  ηnλn−1σnµnτnκn−1[ (σ + τ)(σ + δ) ∏n−2 i=0 ((2i + 2)η + τ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ) ((2i + 2)λ + κ)(σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ]  [ η∏n−2 i=0 ((2i+2)η+τ) ] + [ τ∏n−2 i=0 ((2i)η+τ) ] =  ηnλn−1σnµnτnκn−1[ (σ + τ)(σ + δ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ]  η + [ τ ∏n−2 i=0 ((2i+2)η+τ)∏n−2 i=0 ((2i)η+τ) ] =  ηnλn−1σnµnτnκn−1[ (σ + τ)(σ + δ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ) (σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ]  η + ((2n − 2)η + τ) = ηnλn−1σnµnτnκn−1[ (σ + τ)(σ + δ)((2n − 1)η + τ) ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ) ((2i + 2)λ + κ)(σ + (2i + 3)τ)((2i + 3)σ + δ)(ζ + (2i + 2)κ) ] . As a result, we obtain Θ6n−3 = ηnλnσnζµnτnκn[ ∏n−1 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i)κ) ] . Similarly, we can infer from system (4) that Ω6n−3 = Θ6n−4Ω6n−6 Θ6n−7 + Ω6n−6 H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 11 of 28 =  ηnλnσn−1µn−1τn−1κn[ (λ + κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 3)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ]  ηn−1λn−1σn−1µnτn−1κn−1[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ]   ηn−1λnσn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ]  +  ηn−1λn−1σn−1µnτn−1κn−1[ ∏n−2 i=0 ((2i + 1)η + τ)(λ + (2i + 2)µ)((2i + 2)λ + κ)(σ + (2i + 1)τ) ((2i + 1)σ + δ)(ζ + (2i + 2)κ) ]  =  ηnλnσn−1µnτn−1κn[ (λ + κ)(ζ + κ) ∏n−2 i=0 (λ + (2i + 2)µ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ) ((2i + 3)λ + κ)(σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ]  [ λ∏n−2 i=0 (λ+(2i)µ) ] + [ µ∏n−2 i=0 (λ+(2i+2)µ) ] =  ηnλnσn−1µnτn−1κn[ (λ + κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 3)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ]  [ λ ∏n−2 i=0 (λ+(2i+2)µ)∏n−2 i=0 (λ+(2i)µ) ] + µ =  ηnλnσn−1µnτn−1κn[ (λ + κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ)((2i + 3)λ + κ) (σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ]  (λ + (2n − 2)µ) + µ H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 12 of 28 = ηnλnσn−1µnτn−1κn[ (λ + κ)(ζ + κ)(λ + (2n − 1)µ) ∏n−2 i=0 ((2i + 2)η + τ)(λ + (2i + 1)µ) ((2i + 3)λ + κ)(σ + (2i + 2)τ)((2i + 2)σ + δ)(ζ + (2i + 3)κ) ] . Hence, we obtain Ω6n−3 = ηnλnσnµnτnκnδ[ ∏n−1 i=0 ((2i)η + τ)(λ + (2i + 1)µ)((2i + 1)λ + κ)(σ + (2i)τ) ((2i)σ + δ)(ζ + (2i + 1)κ) ] . A similar approach can be used to study other expressions. To confirm the results presented in this section, a numerical example is provided to demonstrate a representative type of solution to system (4). Example 1. The initial conditions are taken as Θ−3 = 0.15, Θ−2 = −0.02, Θ−1 = 0.08, Θ0 = −0.05, Ω−3 = 0.1, Ω−2 = −0.06, Ω−1 = 0.05 and Ω0 = −0.02. The result- ing solution and represented graphically and its behavior is illustrated in Figure 1. Figure 1 From Figure 1, we conclude that the solutions are bounded and that O is globally asymp- totically stable and bounded. It is observed that the solutions has oscillatory harmonically in the all range. The results confirm Theorem 5 and Lemma 1. This result is in good agreement with the results obtained by Elsayed and Alshabi (2023) 3.4. Boundedness of The Solution In this subsection, we show that the positive solutions of system (4) are bounded. Lemma 1. All positive solutions of system (4) are bounded and converge to zero. H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 13 of 28 Proof. system (4) demonstrates that Θn+1 = Θn−2Ωn Θn−2 + Ωn−3 ≤ Θn−2Ωn Θn−2 ≤ Ωn, Ωn+1 = ΘnΩn−2 Θn−3 + Ωn−2 ≤ ΘnΩn−2 Ωn−2 ≤ Θn. We conclude that Θn+1 ≤ Ωn and Ωn+1 ≤ Θn. Setting n = n + 1, n + 2, . . ., yields Θn+2 ≤ Ωn+1 ≤ Θn and Ωn+2 ≤ Θn+1 ≤ Ωn Θn+3 ≤ Ωn+2 ≤ Θn+1 and Ωn+3 ≤ Θn+2 ≤ Ωn+1, . . . , and so on. This suggests that the subsequences {Θ6n−3}∞ n=0, {Θ6n−2}∞ n=0, {Θ6n−1}∞ n=0, {Θ6n}∞ n=0, {Θ6n+1}∞ n=0, {Θ6n+2}∞ n=0 and {Ω6n−3}∞ n=0, {Ω6n−2}∞ n=0, {Ω6n−1}∞ n=0, {Ω6n}∞ n=0, {Ω6n+1}∞ n=0, {Ω6n+2}∞ n=0 are decreasing and are therefore bounded above by Θmax = max {Θ−3, Θ−2, Θ−1, Θ0} , and Ωmax = max {Ω−3, Ω−2, Ω−1, Ω0} . Setting α = β = 0 in (1) yields the solution expression in the following cases. 4. On the System: Θn+1 = Θn−2Ωn Θn−2−Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3+Ωn−2 The solutions of the following system of difference equations will be explicitly ex- pressed in this section Θn+1 = Θn−2Ωn Θn−2 − Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , (10) where Θ−3, Θ−2, Θ−1, Θ0, Ω−3, Ω−2, Ω−1 and Ω0 are nonzero real numbers that serve as initial conditions. Theorem 7. Assume that system (10) has a solution {Θn, Ωn}∞ n=−3. For n = 0, 1, 2, . . ., Θ6n−3 = ηnλnσnζµnτn (η − τ)n(σ − δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i)κ) , Θ6n−2 = ηnλnσn+1µnκn δn(λ − κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Θ6n−1 = ηnλn+1σnµnτn (η − τ)n(σ − δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 14 of 28 Θ6n = ηn+1λnσnµnκn δn(λ − κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Θ6n+1 = ηnλnσn+1µn+1τn (η − τ)n(σ − δ)n+1 ∏n−1 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n+2 = ηn+1λn+1σnµnκn+1 (ζ + κ)δn(λ − κ)n+1 ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n−3 = ηnλnσnµnκn δn−1(λ − κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Ω6n−2 = ηnλnσnµnτnκ (η − τ)n(σ − δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Ω6n−1 = ηnλnσnµnτκn δn(λ − κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Ω6n = ηnλnσnµn+1τn (η − τ)n(σ − δ)n ∏n−1 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Ω6n+1 = ηn+1λnσnµnκn+1 (ζ + κ)δn(λ − κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n+2 = ηnλnσn+1µn+1τn+1 (σ + τ)(η − τ)n(σ − δ)n+1 ∏n−1 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) , where Θ−3 = ζ, Θ−2 = σ, Θ−1 = λ, Θ0 = η, Ω−3 = δ, Ω−2 = κ, Ω−1 = τ and Ω0 = µ. Proof. The outcome holds true for n = 0. Assuming that n > 0 and that our hypothesis is correct for n − 1, we have: Θ6n−9 = ηn−1λn−1σn−1ζµn−1τn−1 (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i)κ) , Θ6n−8 = ηn−1λn−1σnµn−1κn−1 δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Θ6n−7 = ηn−1λnσn−1µn−1τn−1 (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n−6 = ηnλn−1σn−1µn−1κn−1 δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Θ6n−5 = ηn−1λn−1σnµnτn−1 (η − τ)n−1(σ − δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n−4 = ηnλnσn−1µn−1κn (ζ + κ)δn−1(λ − κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n−9 = ηn−1λn−1σn−1µn−1κn−1 δn−2(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Ω6n−8 = ηn−1λn−1σn−1µn−1τn−1κ (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 15 of 28 Ω6n−7 = ηn−1λn−1σn−1µn−1τκn−1 δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Ω6n−6 = ηn−1λn−1σn−1µnτn−1 (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Ω6n−5 = ηnλn−1σn−1µn−1κn (ζ + κ)δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n−4 = ηn−1λn−1σnµnτn (σ + τ)(η − τ)n−1(σ − δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) . We now derive from system (10) that Θ6n−3 = Θ6n−6Ω6n−4 Θ6n−6 − Ω6n−7 = [ ηnλn−1σn−1µn−1κn−1 (δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) ] [ ηn−1λn−1σnµnτn (σ + τ)(η − τ)n−1(σ − δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) ] [ ηnλn−1σn−1µn−1κn−1 (δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) ] − [ ηn−1λn−1σn−1µn−1τκn−1 δn−1(λ − κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) ] = [ ηnλn−1σnµnτn (σ + τ)(η − τ)n−1(σ − δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) ] (η − τ) = ηnλn−1σnµnτn (σ + τ)(η − τ)n(σ − δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) . Thus, we obtain Θ6n−3 = ηnλnσnµnτnζ (η − τ)n(σ − δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i)κ) . Likewise, we can analyze the relationships using the same approach Ω6n−3 = Θ6n−4Ω6n−6 Θ6n−7 + Ω6n−6 H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 16 of 28 = [ ηnλnσn−1µn−1κn (ζ + κ)δn−1(λ − κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) ] [ ηn−1λn−1σn−1µnτn−1 (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) ] [ ηn−1λnσn−1µn−1τn−1 (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) ] + [ ηn−1λn−1σn−1µnτn−1 (η − τ)n−1(σ − δ)n−1 ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) ] =  ηnλnσn−1µnκn[ (ζ + κ)δn−1(λ − κ)n ∏n−2 i=0 (λ + (2i + 2)µ) ∏n−2 i=0 (λ + (2i + 1)µ) (σ + (2i + 2)τ)(ζ + (2i + 3)κ) ]  [ λ∏n−2 i=0 (λ+(2i)µ) ] + [ µ∏n−2 i=0 (λ+(2i+2)µ) ] =  ηnλnσn−1µnκn[ (ζ + κ)δn−1(λ − κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) ] [ λ ∏n−2 i=0 (λ+(2i+2)µ)∏n−2 i=0 (λ+(2i)µ) ] + µ =  ηnλnσn−1µnκn[ (ζ + κ)δn−1(λ − κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) ] (λ + (2n − 2)µ) + µ = ηnλnσn−1µnκn[ (ζ + κ)δn−1(λ − κ)n(λ + (2n − 1)µ) ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ) (ζ + (2i + 3)κ) ] . Then Ω6n−3 = ηnλnσnµnκn δn−1(λ − κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) . The same method can be used to prove other relations. Example 2. Let the initial values be Θ−3 = −0.1, Θ−2 = 0.09, Θ−1 = −0.03, Θ0 = 0.05, Ω−3 = 0.03, Ω−2 = −0.04, Ω−1 = 0.1 and Ω0 = −0.02. See Figure 2. H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 17 of 28 Figure 2 Figure 2, explained that the solutions are bounded and that O is globally asymptotically stable and bounded. It is observed that the solutions has oscillatory harmonically in the all range and its converges to zero for n > 10. The results confirm Theorem 5 and Lemma 1. This result is in good agreement with the results obtained by Khan et al. (2019). 5. On the System: Θn+1 = Θn−2Ωn −Θn−2+Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3+Ωn−2 The solutions of the following system of difference equations will be explicitly ex- pressed in this section Θn+1 = Θn−2Ωn −Θn−2 + Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , (11) where Θ−3, Θ−2, Θ−1, Θ0, Ω−3, Ω−2, Ω−1 and Ω0 are nonzero real numbers that serve as initial conditions. Theorem 8. Assume that system (11) has a solution {Θn, Ωn}∞ n=−3. For n = 0, 1, 2, . . ., Θ6n−3 = (−1)nηnλnσnζµnτnκn[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i)κ) ] , Θ6n−2 = (−1)nηnλnσn+1µnτnκn[ ∏n−1 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Θ6n−1 = (−1)nηnλn+1σnµnτnκn[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 18 of 28 Θ6n = (−1)nηn+1λnσnµnτnκn[ ∏n−1 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Θ6n+1 = (−1)n+1ηnλnσn+1µn+1τnκn[ (σ − δ) ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 1)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] , Θ6n+2 = (−1)n+1ηn+1λn+1σnµnτnκn+1[ (λ − κ)(ζ + κ) ∏n−1 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 3)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] , Ω6n−3 = (−1)nηnλnσnµnτnκnδ[ ∏n−1 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i)τ) ((2i)σ − δ)(ζ + (2i + 1)κ) ] , Ω6n−2 = (−1)nηnλnσnµnτnκn+1[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Ω6n−1 = (−1)nηnλnσnµnτn+1κn[ ∏n−1 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Ω6n = (−1)nηnλnσnµn+1τnκn[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Ω6n+1 = (−1)nηn+1λnσnµnτnκn+1[ (ζ + κ) ∏n−1 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] , Ω6n+2 = (−1)n+1ηnλnσn+1µn+1τn+1κn[ (σ + τ)(σ − δ) ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] , where Θ−3 = ζ, Θ−2 = σ, Θ−1 = λ, Θ0 = η, Ω−3 = δ, Ω−2 = κ, Ω−1 = τ and Ω0 = µ. Proof. The outcome holds true for n = 0. Assuming that n > 0 and that our hypothesis is correct for n − 1, we have: Θ6n−9 = (−1)n−1ηn−1λn−1σn−1ζµn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i)κ) ] , Θ6n−8 = (−1)n−1ηn−1λn−1σnµn−1τn−1κn−1[ ∏n−2 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 19 of 28 Θ6n−7 = (−1)n−1ηn−1λnσn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Θ6n−6 = (−1)n−1ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Θ6n−5 = (−1)nηn−1λn−1σnµnτn−1κn−1[ (σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 1)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] , Θ6n−4 = (−1)nηnλnσn−1µn−1τn−1κn[ (λ − κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 3)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] , Ω6n−9 = (−1)n−1ηn−1λn−1σn−1µn−1τn−1κn−1δ[ ∏n−2 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i)τ) ((2i)σ − δ)(ζ + (2i + 1)κ) ] , Ω6n−8 = (−1)n−1ηn−1λn−1σn−1µn−1τn−1κn[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Ω6n−7 = (−1)n−1ηn−1λn−1σn−1µn−1τnκn−1[ ∏n−2 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ] , Ω6n−6 = (−1)n−1ηn−1λn−1σn−1µnτn−1κn−1[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ] , Ω6n−5 = (−1)n−1ηnλn−1σn−1µn−1τn−1κn[ (ζ + κ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] , Ω6n−4 = (−1)nηn−1λn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] . Now, it can be inferred from system (11) that Θ6n−3 = Θ6n−6Ω6n−4 −Θ6n−6 + Ω6n−7 H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 20 of 28 =  (−1)n−1ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ]  (−1)nηn−1λn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  −  (−1)n−1ηnλn−1σn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ]  +  (−1)n−1ηn−1λn−1σn−1µn−1τnκn−1[ ∏n−2 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i + 2)τ) ((2i + 2)σ − δ)(ζ + (2i + 1)κ) ]  =  (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 2)η − τ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ) ((2i + 2)λ − κ)(σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  [ η∏n−2 i=0 ((2i+2)η−τ) ] − [ τ∏n−2 i=0 ((2i)η−τ) ] =  (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  η − [ τ ∏n−2 i=0 ((2i+2)η−τ)∏n−2 i=0 ((2i)η−τ) ] =  (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ) (σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ]  η + ((2n − 2)η − τ) H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 21 of 28 = (−1)n−1ηnλn−1σnµnτnκn−1[ (σ + τ)(σ − δ)((2n − 1)η − τ) ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ) ((2i + 2)λ − κ)((σ + (2i + 3)τ)((2i + 3)σ − δ)(ζ + (2i + 2)κ) ] . Therefore, we get Θ6n−3 = (−1)nηnλnσnζµnτnκn[ ∏n−1 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i)κ) ] . Similarly, we can infer from system (11) that Ω6n−3 = Θ6n−4Ω6n−6 Θ6n−7 + Ω6n−6 =  (−1)nηnλnσn−1µn−1τn−1κn[ (λ − κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 3)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ]  (−1)n−1ηn−1λn−1σn−1µnτn−1κn−1[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ]   (−1)n−1ηn−1λnσn−1µn−1τn−1κn−1[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ]  +  (−1)n−1ηn−1λn−1σn−1µnτn−1κn−1[ ∏n−2 i=0 ((2i + 1)η − τ)(λ + (2i + 2)µ)((2i + 2)λ − κ)(σ + (2i + 1)τ) ((2i + 1)σ − δ)(ζ + (2i + 2)κ) ]  =  (−1)nηnλnσn−1µnτn−1κn[ (λ − κ)(ζ + κ) ∏n−2 i=0 (λ + (2i + 2)µ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ) ((2i + 3)λ − κ)(σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ]  [ λ∏n−2 i=0 (λ+(2i)µ) ] + [ µ∏n−2 i=0 (λ+(2i+2)µ) ] H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 22 of 28 =  (−1)nηnλnσn−1µnτn−1κn[ (λ − κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 3)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ]  [ λ ∏n−2 i=0 (λ+(2i+2)µ)∏n−2 i=0 (λ+(2i)µ) ] + µ =  (−1)nηnλnσn−1µnτn−1κn[ (λ − κ)(ζ + κ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ)((2i + 3)λ − κ) (σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ]  (λ + (2n − 2)µ) + µ = (−1)nηnλnσn−1µnτn−1κn[ (λ − κ)(ζ + κ)(λ + (2n − 1)µ) ∏n−2 i=0 ((2i + 2)η − τ)(λ + (2i + 1)µ) ((2i + 3)λ − κ)(σ + (2i + 2)τ)((2i + 2)σ − δ)(ζ + (2i + 3)κ) ] . Hence, we obtain Ω6n−3 = (−1)nηnλnσnµnτnκnδ[ ∏n−1 i=0 ((2i)η − τ)(λ + (2i + 1)µ)((2i + 1)λ − κ)(σ + (2i)τ) ((2i)σ − δ)(ζ + (2i + 1)κ) ] . A similar method can be applied to prove the following cases. Example 3. Consider the initial values Θ−3 = 0.07, Θ−2 = −0.03, Θ−1 = 0.1, Θ0 = −0.09, Ω−3 = −0.02, Ω−2 = 0.05, Ω−1 = −0.08 and Ω0 = 0.04. See Figure 3. Figure 3 H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 23 of 28 Figure 3, demonstrated that the solutions are bounded and that O is globally asymptot- ically stable and bounded. It is observed that the solutions has oscillatory harmonically in the all range and its converges to zero for n > 12. The results confirm Theorem 5 and Lemma 1. This result is in good agreement with the results obtained by Khaliq et al. (2022). 6. On the System: Θn+1 = Θn−2Ωn −Θn−2−Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3+Ωn−2 The solutions of the following system of difference equations will be explicitly expressed in this section Θn+1 = Θn−2Ωn −Θn−2 − Ωn−3 , Ωn+1 = ΘnΩn−2 Θn−3 + Ωn−2 , n = 0, 1, 2, . . . , (12) where Θ−3, Θ−2, Θ−1, Θ0, Ω−3, Ω−2, Ω−1 and Ω0 are nonzero real numbers that serve as initial conditions. Theorem 9. Assume that system (12) has a solution {Θn, Ωn}∞ n=−3. For n = 0, 1, 2, . . ., Θ6n−3 = ηnλnσnζµnτn (η + τ)n(σ + δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i)κ) , Θ6n−2 = (−1)nηnλnσn+1µnκn δn(λ + κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Θ6n−1 = ηnλn+1σnµnτn (η + τ)n(σ + δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n = (−1)nηn+1λnσnµnκn δn(λ + κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Θ6n+1 = −ηnλnσn+1µn+1τn (η + τ)n(σ + δ)n+1 ∏n−1 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n+2 = (−1)n+1ηn+1λn+1σnµnκn+1 (ζ + κ)δn(λ + κ)n+1 ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n−3 = (−1)nηnλnσnµnκn δn−1(λ + κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Ω6n−2 = ηnλnσnµnτnκ (η + τ)n(σ + δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Ω6n−1 = (−1)nηnλnσnµnτκn δn(λ + κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Ω6n = ηnλnσnµn+1τn (η + τ)n(σ + δ)n ∏n−1 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 24 of 28 Ω6n+1 = (−1)nηn+1λnσnµnκn+1 (ζ + κ)δn(λ + κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n+2 = −ηnλnσn+1µn+1τn+1 (σ + τ)(η + τ)n(σ + δ)n+1 ∏n−1 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) , where Θ−3 = ζ, Θ−2 = σ, Θ−1 = λ, Θ0 = η, Ω−3 = δ, Ω−2 = κ, Ω−1 = τ and Ω0 = µ. Proof. The outcome holds true for n = 0. Assuming that n > 0 and that our hypothesis is correct for n − 1, we have: Θ6n−9 = ηn−1λn−1σn−1ζµn−1τn−1 (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i)κ) , Θ6n−8 = (−1)n−1ηn−1λn−1σnµn−1κn−1 δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Θ6n−7 = ηn−1λnσn−1µn−1τn−1 (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n−6 = (−1)n−1ηnλn−1σn−1µn−1κn−1 δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Θ6n−5 = −ηn−1λn−1σnµnτn−1 (η + τ)n−1(σ + δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Θ6n−4 = (−1)nηnλnσn−1µn−1κn (ζ + κ)δn−1(λ + κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n−9 = (−1)n−1ηn−1λn−1σn−1µn−1κn−1 δn−2(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) , Ω6n−8 = ηn−1λn−1σn−1µn−1τn−1κ (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Ω6n−7 = (−1)n−1ηn−1λn−1σn−1µn−1τκn−1 δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) , Ω6n−6 = ηn−1λn−1σn−1µnτn−1 (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) , Ω6n−5 = (−1)n−1ηnλn−1σn−1µn−1κn (ζ + κ)δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) , Ω6n−4 = −ηn−1λn−1σnµnτn (σ + τ)(η + τ)n−1(σ + δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) . We now demonstrate that the findings hold true for n. It can be inferred from system (12) H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 25 of 28 that Θ6n−3 = Θ6n−6Ω6n−4 −Θ6n−6 − Ω6n−7 = [ (−1)n−1ηnλn−1σn−1µn−1κn−1 δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) ] [ −ηn−1λn−1σnµnτn (σ + τ)(η + τ)n−1(σ + δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) ] − [ (−1)n−1ηn−1λn−1σn−1µn−1τκn−1 δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) ] − [ (−1)n−1ηn−1λn−1σn−1µn−1τκn−1 δn−1(λ + κ)n−1 ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 1)κ) ] = [ ηnλn−1σnµnτn (σ + τ)(η + τ)n−1(σ + δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) ] (η + τ) = ηnλn−1σnµnτn (σ + τ)(η + τ)n(σ + δ)n ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 3)τ)(ζ + (2i + 2)κ) . So, we have Θ6n−3 = ηnλnσnζµnτn (η + τ)n(σ + δ)n ∏n−1 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i)κ) . Additionally, we can see from system (12) that Ω6n−3 = Θ6n−4Ω6n−6 Θ6n−7 + Ω6n−6 = [ (−1)nηnλnσn−1µn−1κn (ζ + κ)δn−1(λ + κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) ] [ ηn−1λn−1σn−1µnτn−1 (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) ] [ ηn−1λnσn−1µn−1τn−1 (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) ] + [ ηn−1λn−1σn−1µnτn−1 (η + τ)n−1(σ + δ)n−1 ∏n−2 i=0 (λ + (2i + 2)µ)(σ + (2i + 1)τ)(ζ + (2i + 2)κ) ] H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 26 of 28 =  (−1)nηnλnσn−1µnκn[ (ζ + κ)δn−1(λ + κ)n ∏n−2 i=0 (λ + (2i + 2)µ) ∏n−2 i=0 (λ + (2i + 1)µ) (σ + (2i + 2)τ)(ζ + (2i + 3)κ) ]  [ λ∏n−2 i=0 (λ+(2i)µ) ] + [ µ∏n−2 i=0 (λ+(2i+2)µ) ] =  (−1)nηnλnσn−1µnκn[ (ζ + κ)δn−1(λ + κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) ] [ λ ∏n−2 i=0 (λ+(2i+2)µ)∏n−2 i=0 (λ+(2i)µ) ] + µ =  (−1)nηnλnσn−1µnκn[ (ζ + κ)δn−1(λ + κ)n ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ)(ζ + (2i + 3)κ) ] (λ + (2n − 2)µ) + µ = (−1)nηnλnσn−1µnκn[ (ζ + κ)δn−1(λ + κ)n(λ + (2n − 1)µ) ∏n−2 i=0 (λ + (2i + 1)µ)(σ + (2i + 2)τ) (ζ + (2i + 3)κ) ] . Thus, we obtain Ω6n−3 = (−1)nηnλnσnµnκn δn−1(λ + κ)n ∏n−1 i=0 (λ + (2i + 1)µ)(σ + (2i)τ)(ζ + (2i + 1)κ) . Likewise, we can investigate additional relationships by applying the same methodology. Example 4. Take the initial values Θ−3 = 0.1, Θ−2 = −0.04, Θ−1 = 0.06, Θ0 = −0.02, Ω−3 = 0.05, Ω−2 = −0.03, Ω−1 = 0.09 and Ω0 = −0.1. See Figure 4. H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 27 of 28 Figure 4 Figure 4, shows that the solutions are bounded and that O is globally asymptotically stable and bounded. It is observed that the solutions has oscillatory harmonically in the all range. The results confirm Theorem 5 and Lemma 1. This result is in good agreement with the results obtained by Elsayed and Ibrahim (2015). 7. Conclusion The majority of studies on nonlinear rational difference equations analyze the be- havior of solutions by presenting a general solution form. The initial conditions of the systems considered in this study are nonzero real integers. Since obtaining explicit solu- tion formulations can be challenging, researchers investigate the stability characteristics of the equilibrium point. This article presents the solution expressions for a few specific examples that serve as applications of fourth-order rational difference systems. In Section 3, we examined the qualitative behavior of the solutions, including their boundedness, as well as local and global stability. Additionally, we derived the general solution form of system 4. Moreover, we obtained solutions for three specific cases of the studied systems, namely systems 10, 11, and 12, which are discussed in Sections 4, 5, and 6 respectively. Illustra- tive and numerical examples are presented within Sections 3, 4, 5, and 6 to support the theoretical discussions and verify the analytical results. The expressions and results reported in this research might be beneficial for applica- tions in seismology and materials characterization of difference equations. H. S. Gafel, H. A. Altamimi / Eur. J. Pure Appl. Math, 18 (3) (2025), 6218 28 of 28 Acknowledgements We would like to thank the readers and editorial team of the European Journal of Pure and Applied Mathematics for their continued interest and contributions to the advancement of mathematical research. References [1] R Abo-Zeid and H Kamal. Global behavior of two rational third order difference equations. Univ. J. Math. Appl., 2:212–217, 2019. [2] D T Tollu, I Yalçınkaya, H Ahmad, and S W Yao. A detailed study on a solv- able system related to the linear fractional difference equation. Math. Biosci. Eng., 18:5392–5408, 2021. [3] D Tollu and I Yalçınkaya. On solvability of a three-dimensional system of nonlinear difference equations. Dyn. Contin. Discr. Impul. Syst. Ser. B: Appl. Algorithms, 29:35–47, 2022. [4] A Khaliq, T F Ibrahim, A M Alotaibi, M Shoaib, and M ElMoneam. Dynamical analysis of discrete-time two-predators one-prey lotka-volterra model. Mathematics, 10(4015), 2022. [5] A Khaliq and M Shoaib. Dynamics of three-dimensional system of second order rational difference equations. Electron. J. Math. Anal. Appl., 9:308–319, 2021. [6] K N Alharbi and E M Elsayed. The expressions and behavior of solutions for nonlinear systems of rational difference equations. J. Innov. Appl. Math. Comput. Sci., 2:78–91, 2022. [7] Y Halim, A Khelifa, and M Berkal. Representation of solutions of a two-dimensional system of difference equations. Misk. Math. Notes, 21:203–218, 2020. [8] E M Elsayed, A Alshareef, and F Alzahrani. Qualitative behavior and solution of a system of three-dimensional rational difference equations. Math. Methods Appl. Sci., 45:5456–5470, 2022. [9] E M Elsayed, Q Din, F A Al-Rakhami, and N M Seyam. Dynamics and expressions of solutions of fourth-order rational systems of difference equations. Int. J. Anal. Appl., 22(163), 2024. [10] S Elaydi. An Introduction to Difference Equations. Springer-Verlag, New York, 2005.