scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

The exact difference equation of the first order

1933 ◽  
Vol 29 (3) ◽  
pp. 373-381
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Linear Equation ◽  
Sufficient Conditions ◽  
First Order ◽  
Non Linear Equation ◽  
Exact Difference ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Limiting Case

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.

scholarly journals Some representations of the general solution to a difference equation of additive type

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Stevo Stević
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Representation Formula ◽  
Double Root ◽  
General Representation ◽  
Additive Type ◽  
The Difference

Abstract The general solution to the difference equation $$x_{n+1}=\frac {ax_{n}x_{n-1}x_{n-2}+bx_{n-1}x_{n-2}+cx_{n-2}+d}{x_{n}x_{n-1}x_{n-2}},\quad n\in\mathbb{N}_{0}, $$ x n + 1 = a x n x n − 1 x n − 2 + b x n − 1 x n − 2 + c x n − 2 + d x n x n − 1 x n − 2 , n ∈ N 0 , where $a, b, c\in\mathbb{C}$ a , b , c ∈ C , $d\in\mathbb{C}\setminus\{0\}$ d ∈ C ∖ { 0 } , is presented by using the coefficients, the initial values $x_{-j}$ x − j , $j=\overline{0,2}$ j = 0 , 2 ‾ , and the solution to the difference equation $$y_{n+1}=ay_{n}+by_{n-1}+cy_{n-2}+dy_{n-3}, \quad n\in\mathbb{N}_{0}, $$ y n + 1 = a y n + b y n − 1 + c y n − 2 + d y n − 3 , n ∈ N 0 , satisfying the initial conditions $y_{-3}=y_{-2}=y_{-1}=0$ y − 3 = y − 2 = y − 1 = 0 , $y_{0}=1$ y 0 = 1 . The representation complements known ones of the general solutions to the corresponding difference equations of the first and second order. Besides, the general representation formula is investigated in detail and refined by using the roots of the characteristic polynomial $$P_{4}(\lambda )=\lambda ^{4}-a\lambda ^{3}-b\lambda ^{2}-c\lambda -d $$ P 4 ( λ ) = λ 4 − a λ 3 − b λ 2 − c λ − d of the linear equation. The following cases are considered separately: (1) all the roots of the polynomial are distinct; (2) there is a unique double root of the polynomial; (3) there is a triple root of the polynomial and one simple; (4) there is a quadruple root of the polynomial; (5) there are two distinct double roots of the polynomial.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

Well-defined solutions of the difference equation xn = xn−3kxn−4kxn−5k xn−kxn−2k(±1±xn−3kxn−4kxn−5k)

2019 ◽  
Vol 12 (06) ◽  
pp. 2040016
Author(s):  
Güven Çi̇nar ◽  
Ali̇ Geli̇şken ◽  
Ozan Özkan
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Numerical Results ◽  
Real Numbers ◽  
The Difference

We investigate the behavior of well-defined solutions of the difference equation [Formula: see text] where the initial conditions [Formula: see text], [Formula: see text] are arbitrary nonzero real numbers. Also, we give some special results and numerical results.

Dynamics of a Higher Order Rational Difference Equation

2011 ◽  
Vol 216 ◽  
pp. 50-55 ◽  
Author(s):  
Yi Yang ◽  
Fei Bao Lv
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we address the difference equation xn=pxn-s+xn-t/q+xn-t n=0,1,... with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0. Our results not only include some previously known results, but apply to some difference equations that have not been investigated so far.

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.

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 Some Discussions on the Difference Equationxn+1=α+(xn-1m/xnk)

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Awad A. Bakery
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give in this work the sufficient conditions on the positive solutions of the difference equationxn+1=α+(xn-1m/xnk),  n=0,1,…, whereα,k, andm∈(0,∞)under positive initial conditionsx-1,  x0to be bounded,α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the casesm=k=1of Amleh et al. (1999) andm=1of Hamza and Morsy (2009). We offer improving conditions in the case ofm=1of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.

scholarly journals Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Zhi-Qiang Zhu
Keyword(s):  
Time Scales ◽  
Difference Equations ◽  
Nonlinear Dynamic ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations ◽  
Necessary And Sufficient Criteria ◽  
The Difference ◽  
Necessary And Sufficient

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the -difference equations.

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).

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