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In this paper, we considered some systems of rational difference equations of higher order as follows$ \begin{eqnarray*} u_{n+1} & = &\frac{v_{n-6}}{1\pm v_{n}u_{n-1}v_{n-2}u_{n-3}v_{n-4}u_{n-5}v_{n-6}}, \\ v_{n+1} & = &\frac{u_{n-6}}{1\pm u_{n}v_{n-1}u_{n-2}v_{n-3}u_{n-4}v_{n-5}u_{n-6}}, \end{eqnarray*} $where the initial conditions $ u_{0, } $ $ u_{-1}, $ $ u_{-2}, $ $ u_{-3}, $ $ u_{-4}, $ $ u_{-5}, $ $ u_{-6}, $ $ v_{0, } $ $ v_{-1}, $ $ v_{-2}, $ $ v_{-3}, $ $ v_{-4}, $ $ v_{-5} $ and $ v_{-6} $ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.

In this paper, we considered some systems of rational difference equations of higher order as follows u n + 1 = v n − 6 1 ± v n u n − 1 v n − 2 u n − 3 v n − 4 u n − 5 v n − 6 , v n + 1 = u n − 6 1 ± u n v n − 1 u n − 2 v n − 3 u n − 4 v n − 5 u n − 6 , where the initial conditions u 0 , u − 1 , u − 2 , u − 3 , u − 4 , u − 5 , u − 6 , v 0 , v − 1 , v − 2 , v − 3 , v − 4 , v − 5 and v − 6 were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to discuss a nonlinear rational difference equation system with three variables. Firstly, by utilizing Jacobian theory, the stability of the system is obtained. Then, we explore the saddle point, nonhyperbolic point, and prime period‐two solution and its boundedness. Finally, we verify the results obtained in the paper via computer numerical calculations. [ABSTRACT FROM AUTHOR]

In this paper we show that the following four-dimensional system of difference equations x n + 1 = y n α z n − 1 β , y n + 1 = z n γ t n − 1 δ , z n + 1 = t n ϵ x n − 1 μ , t n + 1 = x n ξ y n − 1 ρ , n ∈ N 0 , where the parameters α, β, γ, δ, ϵ, μ, ξ, ρ ∈ ℤ and the initial values x–i, y–i, z–i, t–i, i ∈ {0, 1}, are real numbers, can be solved in closed forms, extending further some results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of two-dimensional nonlinear difference equations system of second order, which is a considerably extension of some recent results in the literature. Our main results show that class of system of difference equations is solvable in closed form theoretically. It is noteworthy that the solutions of aforementioned system are associated with generalized Mersenne numbers. The asymptotic behavior of solution to aforementioned system of difference equations when a = b and p = 0 are also given. Finally, numerical examples are given to support the theoretical results presented in this paper. [ABSTRACT FROM AUTHOR]

We propose some theoretical explanations pertaining to the representation for the solution of the system of the higher-order difference equations x n + 1 = 1 A − y n − k , y n + 1 = 1 A − x n − k , n , k ∈ N 0 , where ℕ0 = ℕ ∪ {0}, A ∈ ℕ1 and the initial conditions x–k, x–k+1, ..., x0, y–k, y–k+1, ..., y0 are non zero real numbers such that their solution is related to a generalized Balancing numbers. We also study the stability character and asymptotic behavior of this system. [ABSTRACT FROM AUTHOR]

This study is involved with a class of three-dimensional system of difference equations incorporating quadratic term, which naturally extends and improve several results in the literature. Firstly, we demonstrate the existence of fixed points, the boundedness, persistence and invariance of positive solution of the mentioned system. Later, for this system, we give the global asymptotic stability at fixed point and the rate of convergence result which play an important role in the discrete dynamical systems. And lastly, some numerical examples are given to validate the effectiveness and feasibility of the theoretical findings. [ABSTRACT FROM AUTHOR]

We present six classes of nonlinear two-dimensional systems of difference equations which are solvable. Some methods for finding their general solutions are described in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the global behavior of positive solutions of the following system of difference equations ..., where the parameter A>0, the initial values x-i,y-i, z-i, i ∈ {0,1,2, . . ., k}, are arbitrary positive real number and k ∈ N. Moreover, we provide semi-cycle analysis of positive solutions to the above system of difference equations. Finally, we also give some numerical examples which support our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we derive the well-defined solutions to a θ -dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we focus on obtaining the closed-form solution for the following bidimensional system of higher-order rational difference equations: u(1) n+1 = 1 3 - 2u(2) n-m, u(2) n+1 = 1 3 - 2u(1) n-m, n,m N0, where the initial values u(1) -j and u(2) -j, j {0, 1, ...,m} are real numbers not equal to 3/2. We show that the solutions of this system are associated with Mersenne numbers and/or Mersenne-Lucas numbers. It is shown that the global stability of positive solutions of this system holds. Finally, we provide numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper.

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order $ m+1 $, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research.

In this study, we investigate the boundedness, persistence of positive solutions, local and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the following difference equations systems of exponential forms: $ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Psi_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Omega_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Upsilon_{n}}, \end{equation*} $ $ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Upsilon_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Psi_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Omega_{n}}, \end{equation*} $ for $ n\in \mathbb{N}_{0} $, where the initial conditions $ \Upsilon_{-j} $, $ \Psi_{-j} $, $ \Omega_{-j} $, for $ j\in\{0, 1\} $ and the parameters $ \Gamma_{i} $, $ \delta_{i} $, $ \Theta_{i} $ for $ i\in\{1, 2, 3\} $ are positive constants.

This paper investigates both the local and global stability of a system of rational difference equations and its connection to co-balancing numbers. The study delves into the intricate dynamics of mathematical models and their stability properties, emphasizing the broader implications of global stability. Additionally, the investigation extends to the role of co-balancing numbers, elucidating their significance in achieving equilibrium within the solutions of the rational difference equations. The interplay between global stability and co-balancing numbers forms a foundational aspect of the analysis. The findings contribute to a deeper understanding of the mathematical structures underlying dynamic systems and offer insights into the factors influencing their stability and equilibrium. This article serves as a valuable resource for mathematicians, researchers, and scholars interested in the intersection of global stability and co-balancing sequences in the realm of rational difference equations. Moreover, the presented examples and figures consistently demonstrate the global asymptotic stability of the equilibrium point throughout the paper. [ABSTRACT FROM AUTHOR]

We presented a comprehensive theory for deriving closed-form expressions and representations of the general solutions for a specific case of systems involving Riccati difference equations of order m + 1, as discussed in the literature. However, our focus was on coefficients dependent on the Jacobsthal sequence. Importantly, this system of difference equations represents a natural extension of the corresponding one-dimensional difference equation, uniquely characterized by its theoretical solvability in a closed form. Our primary objective was to demonstrate a direct linkage between the solutions of this system and Jacobsthal and Lucas-Jacobsthal numbers. The system's capacity for theoretical solvability in a closed form enhances its distinctiveness and potential applications. To accomplish this, we detailed offer theoretical explanations and proofs, establishing the relationship between the solutions and the Jacobsthal sequence. Subsequently, our exploration addressed key aspects of the Jacobsthal system, placing particular emphasis on the local stability of positive solutions. Additionally, we employed mathematical software to validate the theoretical results of this novel system in our research. [ABSTRACT FROM AUTHOR]

In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: ...xn+1 = Lm+2+Lm+1yn􀀀1 Lm+3+Lm+2yn􀀀1; yn+1 = Lm+2+Lm+1zn􀀀1 Lm+3+Lm+2zn􀀀1; zn+1 = Lm+2+Lm+1wn􀀀1 Lm+3+Lm+2wn􀀀1; wn+1 = Lm+2 +Lm+1xn􀀀1 Lm+3 +Lm+2xn􀀀1: where n 2 N0, fLmg+¥ m=􀀀¥ is Lucas sequence and the initial conditions x􀀀1, x0, y􀀀1, y0, z􀀀1, z0, w􀀀1, w0 are arbitrary real numbers such that v􀀀i 6= 􀀀 Lm+3 Lm+2, where v􀀀i = x􀀀i;y􀀀i; z􀀀i;w􀀀i, i = 0;1 and m 2 Z. [ABSTRACT FROM AUTHOR]

In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations: $ \begin{equation} S_{n+1} = \dfrac{T_{n} S_{n-2}}{S_{n-2} + T_{n-1}},\quad T_{n+1} = \dfrac{S_{n} T_{n-2}}{\pm T_{n-2} \pm S_{n-1}},\quad \;\;\;\; n = 0,1,2,...., \quad\quad\quad(0.1)\end{equation} $ where the initial conditions $ s_{-2}, s_{-1}, s_0, t_{-2}, t_{-1}, t_0 $ are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper.

In this paper, we study the following two-dimesional system of difference equations ... where the parameters a;b; c;d and the initial values x i; y i, i 2 f1;2;3;4;5;6g, are real numbers. We show that some subclasses of nonlinear two-dimensional system of difference equations are solvable in closed form. We also describe the forbidden set of solutions of the system of differ- ence equations. Some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We provide some theoretical justifications pertaining to the representation for the solution of the system of the higher-order rational difference equations xn+1 = 1/6−yn−k, yn+1 = 1/6−xn−k, n, k ∈ N0. where N0 = N ∪ {0}, and the initial conditions x−k, x−k+1, . . ., x0, y−k, y−k+1, . . ., y0 are non zero real numbers such that their solution is related to Balancing numbers. We also study the stability character and asymptotic behavior of this system. [ABSTRACT FROM AUTHOR]

In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations: S n + 1 = T n S n − 2 S n − 2 + T n − 1 , T n + 1 = S n T n − 2 ± T n − 2 ± S n − 1 , n = 0 , 1 , 2 ,.... , (0.1) where the initial conditions s − 2 , s − 1 , s 0 , t − 2 , t − 1 , t 0 are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper. [ABSTRACT FROM AUTHOR]

This paper aims to derive analytical expressions for solutions of fractional bidimensional systems of difference equations with higher-order terms under specific parametric conditions. Additionally, formulations of solutions for one-dimensional equations derived from these systems are explored. Furthermore, rigorous proof is provided for the local stability of the unique positive equilibrium point of the proposed systems. The theoretical findings are validated through numerical examples using MATLAB, facilitating graphical illustrations of the results.

In this paper, we represent that the following three-dimensional system of difference equations x n + 1 = α y n + a y n y n − β z n − 1 , y n + 1 = β z n + b z n z n − γ x n − 1 , z n + 1 = γ x n + c x n x n − α y n − 1 , n ∈ ℕ 0 , where the parameters a, b, c, α, β, γ and the initial values x−i, y−i, z−i, i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we derive the well-defined solutions to a θ-dimensional system of difference equations. We show that, the well-defined solutions to that system are represented in terms of Fibonacci and Lucas sequences. Moreover, we study the global stability of the solutions to that system. Finally, we give some numerical examples which confirm our theoretical results.

In this paper, we obtain the form of the solutions of the following rational systems of difference equations $ \begin{equation*} x_{n+1} = \dfrac{y_{n-1}z_{n}}{z_{n}\pm x_{n-2}}, \;y_{n+1} = \dfrac{z_{n-1}x_{n} }{x_{n}\pm y_{n-2}}, \ z_{n+1} = \dfrac{x_{n-1}y_{n}}{y_{n}\pm z_{n-2}}, \end{equation*} $ with initial values are non-zero real numbers.

We present thirty-six classes of three-dimensional systems of difference equations of the hyperbolic-cotangent type which are solvable in closed form.

Our aim in this paper is to obtain formulas for solutions of rational difference equations such as x n + 1 = 1 ± x n − 1 y n / 1 − y n , y n + 1 = 1 ± y n − 1 x n / 1 − x n , and x n + 1 = 1 ± x n − 1 y n − 2 / 1 − y n , y n + 1 = 1 ± y n − 1 x n − 2 / 1 − x n , where the initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 are non-zero real numbers. In addition, we show that the some of these systems are periodic with different periods. We also verify our theoretical outcomes at the end with some numerical applications and draw it by using some mathematical programs to illustrate the results. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the boundedness of the positive solutions as well as local and global asymptotic stable of the equilibrium points of difference equations system in the second order Un+1=b1Vn+c1VnUn−1r+Vn+Un−1,Vn+1=b2Un+c2UnVn−1r+Un±Vn−1,$$ {\displaystyle \begin{array}{ll}{U}_{n+1}& ={b}_1{V}_n+\frac{c_1{V}_n{U}_{n-1}}{r+{V}_n+{U}_{n-1}},\\ {}{V}_{n+1}& ={b}_2{U}_n+\frac{c2{U}_n{V}_{n-1}}{r+{U}_n\pm {V}_{n-1}},\end{array}} $$ Furthermore, we study the solutions for three special cases of system. Numerical examples are provided to illustrate results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the dynamics of the system of two rational difference equations: xn+1 = αn + yn−k/ yn, yn+1 = αn + xn−k / xn, n = 0, 1, . . . where {αn}n≥0 is a two periodic sequence of nonnegative real numbers and the initial conditions xi, yi are arbitrary positive numbers for i = −k, −k + 1, −k + 2, . . ., 0 and k ∈ N. We investigate the boundedness character of positive solutions. In addition, we establish some sufficient conditions under which the local asymptotic stability and the global asymptotic stability are assured. Furthermore, we determine the rate of the convergence of the solutions. Some numerical are considered in order to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed study of the following system of difference equations $ \begin{equation*} x_{n+1} = \frac{a}{1+y_{n}x_{n-1}}, \ y_{n+1} = \frac{b}{1+x_{n}y_{n-1}}, \ n\in\mathbb{N}_{0}, \end{equation*} $ where the parameters $ a $, $ b $, and the initial values $ x_{-1}, \; x_{0}, \ y_{-1}, \; y_{0} $ are arbitrary real numbers such that $ x_{n} $ and $ y_{n} $ are defined. We mainly show by using a practical method that the general solution of the above system can be represented by characteristic zeros of the associated third-order linear equation. Also, we characterized the well-defined solutions of the system. Finally, we study long-term behavior of the well-defined solutions by using the obtained representation forms.

We show practical solvability of the following two-dimensional systems of difference equations $$x_{n+1}=\frac{u_{n-2}v_{n-3}+a}{u_{n-2}+v_{n-3}},\quad y_{n+1}=\frac{w_{n-2}s_{n-3}+a}{w_{n-2}+s_{n-3}},\quad n\in\mathbb{N}_0,$$ where $u_n$, $v_n,$ $w_n$ and $s_n$ are $x_n$ or $y_n$, by presenting closed-form formulas for their solutions in terms of parameter $a$, initial values, and some sequences for which there are closed-form formulas in terms of index $n.$ This shows that a recently introduced class of systems of difference equations, contains a subclass such that one of the delays in the systems is equal to four, and that they all are practically solvable, which is a bit unexpected fact.

This paper is concerned with the dynamics of positive solutions for a system of rational difference equations of the following form un+1 = au2 n-1 b + gvn-2 , vn+1 = a1v 2 n-1 b1 + g1un-2 , n = 0, 1, . . . , where the parameters a, b, g, a1, b1, g1 and the initial values u-i, v-i ∈ (0, ∞), i = 0, 1, 2. Moreover, the rate of convergence of a solution that converges to the zero equilibrium of the system is discussed. Finally, some numerical examples are given to demonstrate the effectiveness of the results obtained.

We present the following new class of three‐dimensional systems of difference equations xn+3=1a2+a1yn+2+a0yn+2zn+1,yn+3=1b2+b1zn+2+b0zn+2xn+1,zn+3=1c2+c1xn+2+c0xn+2yn+1,n∈ℕ0,where aj,bj,cj,x1,x2,y1,y2,z1,z2∈ℝ,j=0,2‾, and show that the systems are solvable in closed form, considerably extending some results in the literature. [ABSTRACT FROM AUTHOR]

Abstract Recently there has been some interest in difference equations and systems whose forms resemble some trigonometric formulas. One of the classes of such systems is the so-called hyperbolic-cotangent class of systems of difference equations. The corresponding two-dimensional class has two delays denoted by k and l. So far the class has been studied for the case k ≠ l $k\ne l$ , and it was shown that it is practically solvable when max { k , l } ≤ 2 $\max \{k,l\}\le 2$ . In this note we show practical solvability of the system in the case k = l $k=l$ , not only for small values of k and l, but for all k = l ∈ N $k=l\in {\mathbb {N}}$ , which is the first result of such generality.

The systems of difference equations $$x_{n+1}=\frac{u_nv_{n-2}+a}{u_n+v_{n-2}},\quad y_{n+1}=\frac{w_ns_{n-2}+a}{w_n+s_{n-2}},\quad n\in\mathbb{N}_0,$$ where $a, u_0, w_0, v_j, s_j$ $j=-2,-1,0,$ are complex numbers, and the sequences $u_n$, $v_n,$ $w_n$, $s_n$ are $x_n$ or $y_n$, are studied. It is shown that each of these sixteen systems is practically solvable, complementing some recent results on solvability of related systems of difference equations.

The advances in the mathematical explanation of the dynamics underlying treated cancer has opened the door to the mathematical design of optimal therapies. In parallel, the improvements and cost reductions in experimentation and data analysis techniques have made the formulation of personalized therapies possible. However, the design of cancer therapies making use of optimal control theory has not fully considered this possibility in detail. In this paper we contribute to the existing literature by analyzing the diverse alternatives that optimal therapy models offer to design personalized treatments. Taking as the starting point the Chronic Myeloid Leukemia (CML) optimal therapy model in [25], we design personalized optimal therapy models for patients with: CML; CML with intrinsic and/or induced resistance to the administered drug; CML and suffering high drug toxicity and/or allergy to the administered drug; and CML with presence of adverse factors. Along the paper we show that the clinical and medical applicability -the ultimate objective of this biomathematical research- of our proposed personalized models relies on the joint and proper use of the implemented calibration, simulation, and mathematical approaches and techniques. All the theoretical results generated by our personalized optimal therapy models are corroborated by clinical evidence.

Abstract This paper deals with a typical system of Caputo fractional difference equations. Using the Guo–Krasnosel’skii fixed point theorem, we find a parameter interval for which at least one positive solution of the system exists. We give two examples to illustrate the results.

In this paper, we investigate the form of the solutions of the following systems of difference equations of second order xn+1xn+1==xnyn−1xn+yn,yn+1=xn−1ynxn+yn,xnyn−1xn−yn,yn+1=xn−1ynxn−yn, n=0,1,...,xn+1=xnyn−1xn+yn,yn+1=xn−1ynxn+yn,xn+1=xnyn−1xn−yn,yn+1=xn−1ynxn−yn, n=0,1,..., where the initial conditions x−1, x0, y−1 x−1, x0, y−1 and y0 y0 are arbitrary nonzero real numbers.

In this paper, we present that the following system of difference equations ... where n ∈ ℕ0, k, l ∈ ℕ0, the initial values x-i,y-i, z-i are real numbers, for i ∈ 1,k +1, and sequences (an)... (bn)..., (cn)..., (dn)..., (en)... and (fn)... are non-zero real numbers, for all n e No, which can be solved in closed form. We describe the forbidden set of the initial values using the obtained formulas and also determine the asymptotic behavior of solutions for the case k = 3, l = l, and the sequences (an)..., (bn)..., (cn)..., (dn)..., (en)... and (fn)... are constant. Our results considerably extend and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations x n + 1 = α 1 + β 1 e − x n − 1 γ 1 + y n , y n + 1 = α 2 + β 2 e − y n − 1 γ 2 + x n , x n + 1 = α 1 + β 1 e − y n − 1 γ 1 + x n , y n + 1 = α 2 + β 2 e − x n − 1 γ 2 + y n , where the parameters α i , β i , γ i for i ∈ { 1 , 2 } and the initial conditions x − 1 , x 0 , y − 1 , y 0 are positive real numbers. Some numerical example are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The global asymptotic stability of the unique positive equilibrium point and the rate of convergence of positive solutions of the system of two recursive sequences has been studied recently. Here we generalize this study to the system of recursive sequences , , where , are arbitrary positive numbers for and We also give some numerical examples to demonstrate the effectiveness of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we show that the following three-dimensional system of difference equations xn+1 = ynxn−2/axn−2 + bzn−1, yn+1 = znyn−2/cyn−2 + dxn−1, zn+1 = xnzn−2/ezn−2 + ƒyn−1, n ∈ N0, where the parameters a, b, c, d, e, f and the initial values x−i, y−i, z−i, i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given. [ABSTRACT FROM AUTHOR]

Here we solve the following system of difference equations x n + 1 (j) = F m + 2 + F m + 1 x n - k ((j + 1) m o d (p)) F m + 3 + F m + 2 x n - k ((j + 1) m o d (p)) , n , m , p , k ∈ N 0 , j = 1 , p ¯ , where F n n = 0 + ∞ is the Fibonacci sequence. We give a representation of its general solution in terms of Fibonacci numbers and the initial values. Some theoretical justifications related to the representation for the general solution are also given. [ABSTRACT FROM AUTHOR]

In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations $$ x_{n+1} = \frac{5 y_{n-k}-5}{y_{n-k}}, \qquad y_{n+1} = \frac{5 x_{n-k}-5}{x_{n-k}} ,\qquad n, k\in \mathbb{N}_0, $$ where $\mathbb{N}_{0}=\mathbb{N}\cup \left\{0\right\}$, and the initial conditions $x_{-k}$, $x_{-k+1},\ldots$, $x_{0}$, $y_{-k}$, $y_{-k+1},\ldots$, $y_{0}$ are non zero real numbers such that their solutions are associated to Lucas numbers. We also study the stability character and asymptotic behavior of this system.

Abstract Closed-form formulas for general solutions to sixteen hyperbolic-cotangent-type systems of difference equations of interest are obtained, showing their practical solvability and completely solving a solvability problem for some concrete values of delays.

Abstract We show that the following class of two-dimensional hyperbolic-cotangent-type systems of difference equations xn+1=un−kvn−l+aun−k+vn−l,yn+1=wn−ksn−l+awn−k+sn−l,n∈N0, $$ x_{n+1}=\frac{u_{n-k}v_{n-l}+a}{u_{n-k}+v_{n-l}},\quad \quad y_{n+1}= \frac{w _{n-k}s_{n-l}+a}{w_{n-k}+s_{n-l}},\quad n\in {\mathbb {N}} _{0}, $$ where k,l∈N0 $k,l\in {\mathbb {N}} _{0}$, a∈C $a\in {\mathbb {C}} $, u−j,w−j∈C $u_{-j}, w_{-j}\in {\mathbb {C}} $, j=1,k‾ $j=\overline{1,k}$, v−j′ $v_{-j'}$, s−j′ $s_{-j'}$, j′=1,l‾ $j'=\overline{1,l}$, and each of the sequences un $u_{n}$, vn $v_{n}$, wn $w_{n}$, sn $s_{n}$ is equal to xn $x_{n}$ or yn $y_{n}$, is theoretically solvable. When k=0 $k=0$ and l=1 $l=1$, we show that the systems are practically solvable by presenting closed-form formulas for their solutions. To do this, we employ a constructive method, which is possible to use on the complex domain, presenting in this way a new and elegant solution to the problem in this case, and giving a hint how such type of systems can be solved.