scholarly journals On the Solutions of Three-Dimensional Rational Difference Equation Systems

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
H. S. Alayachi ◽  
A. Q. Khan ◽  
M. S. M. Noorani
Keyword(s):  
Difference Equation ◽  
Mathematical Program ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Boundedness Of Solutions ◽  
Real Numbers ◽  
Third Order ◽  
Numerical Examples ◽  
Rational Difference Equation ◽  
Special Case

In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system: x n + 1 = y n z n − 1 / y n ± x n − 2 , y n + 1 = z n x n − 1 / z n ± y n − 2 ,  and  z n + 1 = x n y n − 1 / x n ± z n − 2 with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 , z − 2 , z − 1 ,  and  z 0 are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.

scholarly journals Boundedness of solutions and stability of certain second-order difference equation with quadratic term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Emin Bešo ◽  
Senada Kalabušić ◽  
Naida Mujić ◽  
Esmir Pilav
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Second Order ◽  
Quadratic Term ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractWe consider the second-order rational difference equation $$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$xn+1=γ+δxnxn−12, where γ, δ are positive real numbers and the initial conditions $x_{-1}$x−1 and $x_{0}$x0 are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.

scholarly journals Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-22
Author(s):  
M. R. S. Kulenović ◽  
S. Moranjkić ◽  
M. Nurkanović ◽  
Z. Nurkanović
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Local Asymptotic Stability ◽  
Stable Periodic ◽  
Rational Difference Equation ◽  
Unknown Period

We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12,  n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

scholarly journals Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

This is a continuation part of our investigation in which the second order nonlinear rational difference equation xn+1=(α+βxn+γxn-1)/(A+Bxn+Cxn-1), n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0, is considered. The first part handled the global asymptotic stability of the hyperbolic equilibrium solution of the equation. Our concentration in this part is on the global asymptotic stability of the nonhyperbolic equilibrium solution of the equation.

scholarly journals Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
S. Atawna ◽  
R. Abu-Saris ◽  
E. S. Ismail ◽  
I. Hashim
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Global Asymptotic Stability ◽  
Equilibrium Solution ◽  
Initial Conditions ◽  
Second Order ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
Hyperbolic Equilibrium

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation xn+1=α+βxn+γxn-1/A+Bxn+Cxn-1, n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).

On the solutions of a second-order difference equation in terms of generalized Padovan sequences

2018 ◽  
Vol 68 (3) ◽  
pp. 625-638 ◽  
Author(s):  
Yacine Halim ◽  
Julius Fergy T. Rabago
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Initial Conditions ◽  
Solution Stability ◽  
Two Dimensional ◽  
Real Numbers ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Rational Difference Equation

AbstractThis paper deals with the solution, stability character and asymptotic behavior of the rational difference equation$$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$where ℕ0= ℕ ∪ {0},α,β,γ∈ ℝ+, and the initial conditionsx–1andx0are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by$$\begin{array}{} \displaystyle x_{n+1} = \frac{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0. \end{array}$$

scholarly journals Qualitative Behavior of Rational Difference Equation of Big Order

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
M. M. El-Dessoky
Keyword(s):  
Global Convergence ◽  
Difference Equation ◽  
Initial Conditions ◽  
Qualitative Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Recursive Sequence ◽  
Rational Difference Equation

We investigate the global convergence, boundedness, and periodicity of solutions of the recursive sequencexn+1=axn-l+bxn-x/c+dxn-lxn-k,n=0,1,…,where the parametersa,  b,  c,anddare positive real numbers, and the initial conditionsx-t,x-t+1,…,x-1andx0are positive real numbers wheret=maxk,l.

scholarly journals On the Solutions of a System of Third-Order Rational Difference Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
A. M. Alotaibi ◽  
M. S. M. Noorani ◽  
M. A. El-Moneam
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Theoretical Discussion ◽  
Real Numbers ◽  
Nonlinear Difference Equations ◽  
Positive Real ◽  
Third Order ◽  
Numerical Examples ◽  
Rational Difference Equations

The structure of the solutions for the system nonlinear difference equations xn+1=ynyn-2/(xn-1+yn-2), yn+1=xnxn-2/(±yn-1±xn-2), n=0,1,…, is clarified in which the initial conditions x-2, x-1, x0, y-2, y-1, y0 are considered as arbitrary positive real numbers. To exemplify the theoretical discussion, some numerical examples are presented.

scholarly journals Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Michael A. Radin ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Boundedness Of Solutions ◽  
Real Numbers ◽  
Unbounded Solutions ◽  
The Difference ◽  
Long Term Behavior

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: , where the initial conditions and are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.

scholarly journals Global behavior of a higher order rational difference equation

Filomat ◽  
2016 ◽  
Vol 30 (12) ◽  
pp. 3265-3276 ◽  
Author(s):  
R. Abo-Zeida
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Higher Order ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
All Solutions ◽  
Forbidden Set ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we derive the forbidden set and discuss the global behavior of all solutions of the difference equation xn+1=Axn-k/B-C ?k,i=0 xn-i, n = 0,1,... where A,B,C are positive real numbers and the initial conditions x-k,..., x-1, x0 are real numbers.

scholarly journals Dynamics and Solutions of a Fifth-Order Nonlinear Difference Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-21 ◽  
Author(s):  
M. M. El-Dessoky ◽  
E. M. Elabbasy ◽  
Asim Asiri
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Rational Difference Equation ◽  
Fifth Order

The main objective of this paper is to study the behavior of the rational difference equation of the fifth-order yn+1=αyn+βynyn-3/(Ayn-4+Byn-3), n=0,1,…, where α,β,A, and B are real numbers and the initial conditions y-4,y-3,y-2,y-1 and y0 are positive real numbers such that Ay-4+By-3≠0. Also, we obtain the solution of some special cases of this equation.

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