scholarly journals On the recursive sequence xn+1 = xn-7/1+xn-3

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1381-1386
Author(s):  
D. Simsek ◽  
Kyzy Esengul ◽  
Imash Kyzy
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Recursive Sequence

In this paper the solutions of the following difference equation is examined: xn+1 = xn-7/1 + xn-3, n = 0, 1, 2, 3,... where the initial conditions are positive real numbers.

On the global attractivity and the solution of recursive sequence

2010 ◽  
Vol 47 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence ◽  
The Difference

In this paper we study the behavior of the difference equation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x_{n + 1} = ax_{n - 2} + \frac{{bx_n x_{n - 2} }}{{cx_n + dx_{n - 3} }},n = 0,1,...$$ \end{document} where the initial conditions x−3 , x−2 , x−1 , x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.

scholarly journals Solution and Attractivity for a Rational Recursive Sequence

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
E. M. Elsayed
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Specific Form ◽  
Nonlinear Difference Equation ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence

This paper is concerned with the behavior of solution of the nonlinear difference equation , where the initial conditions , , are arbitrary positive real numbers and are positive constants. Also, we give specific form of the solution of four special cases of this equation.

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 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.

On the solutions of a higher order difference equation

2020 ◽  
Vol 27 (2) ◽  
pp. 165-175 ◽  
Author(s):  
Raafat Abo-Zeid
Keyword(s):  
Difference Equation ◽  
Explicit Formula ◽  
Initial Conditions ◽  
Global Behavior ◽  
Real Numbers ◽  
Positive Real ◽  
Order Difference ◽  
Forbidden Set ◽  
Order Difference Equation ◽  
The Difference

AbstractIn this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equationx_{n+1}=\frac{ax_{n}x_{n-k}}{bx_{n}-cx_{n-k-1}},\quad n=0,1,\ldots,where{a,b,c}are positive real numbers and the initial conditions{x_{-k-1},x_{-k},\ldots,x_{-1},x_{0}}are real numbers. We show that when{a=b=c}, the behavior of the solutions depends on whetherkis even or odd.

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).

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

On the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }} $

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
İlhan Öztürk ◽  
Saime Zengin
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Equilibrium Point ◽  
Global Attractor ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference

AbstractIn this paper, we investigate the global stability and the periodic nature of solutions of the difference equation $y_{n + 1} = \frac{{\alpha + y_n^p }} {{\beta y_{n - 1}^p }} - \frac{{\gamma + y_{n - 1}^p }} {{\beta y_n^p }},n = 0,1,2,... $ where α, β, γ ∈ (0,∞), α(1 − p) − γ > 0, 0 < p < 1, every y n ≠ 0 for n = −1, 0, 1, 2, … and the initial conditions y−1, y0 are arbitrary positive real numbers. We show that the equilibrium point of the difference equation is a global attractor with a basin that depends on the conditions of the coefficients.

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