Periodic Solutions of a System of Nonlinear Difference Equations with Periodic Coefficients

Positive Real ◽

This paper is dealt with the following system of difference equations x n + 1 = a n / x n + b n / y n , y n + 1 = c n / x n + d n / y n , where n ∈ ℕ 0 = ℕ ∪ 0 , the initial values x 0 and y 0 are the positive real numbers, and the sequences a n n ≥ 0 , b n n ≥ 0 , c n n ≥ 0 , and d n n ≥ 0 are two-periodic and positive. The system is an extension of a system where every positive solution is two-periodic or converges to a two-periodic solution. Here, the long-term behavior of positive solutions of the system is examined by using a new method to solve the system.

On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

10.1155/2009/325296 ◽

2009 ◽

Vol 2009 ◽

pp. 1-11 ◽

Author(s):

Dağistan Simsek ◽

Bilal Demir ◽

Cengiz Cinar

Keyword(s):

Difference Equations ◽

Initial Conditions ◽

Real Numbers ◽

Positive Real

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.

Long-term behavior of positive solutions of an exponentially self-regulating system of difference equations

International Journal of Biomathematics ◽

10.1142/s1793524517500450 ◽

2017 ◽

Vol 10 (03) ◽

pp. 1750045 ◽

Cited By ~ 1

Author(s):

N. Psarros ◽

G. Papaschinopoulos ◽

K. B. Papadopoulos

Keyword(s):

Asymptotic Behavior ◽

Positive Solutions ◽

Difference Equations ◽

Initial Conditions ◽

In this paper, we study the asymptotic behavior of the positive solutions of a system of the following difference equations: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants and the initial conditions [Formula: see text] and [Formula: see text] are positive numbers.

Dynamics of a Second-Order System of Nonlinear Difference Equations

10.20944/preprints201910.0357.v1 ◽

2019 ◽

Author(s):

Erkan Taşdemir

Keyword(s):

Equilibrium Point ◽

Periodic Solutions ◽

Difference Equations ◽

Equilibrium Points ◽

Order System ◽

Real Numbers ◽

Numerical Examples ◽

Unbounded Solutions ◽

Theoretical Results

In this paper, we investigate the equilibrium points, stability of two equilibrium points, convergences of negative equilibrium point, periodic solutions, and existence of bounded or unbounded solutions of a system of nonlinear difference equations xn+1 =xn-1yn - 1, yn+1 = yn-1xn - 1 n = 0,1,..., where the initial values are real numbers. Additionally we present some numerical examples to verify our theoretical results.

Dynamics of a Higher-Order System of Difference Equations

10.1155/2017/4312640 ◽

2017 ◽

Vol 2017 ◽

pp. 1-9

Author(s):

Qi Wang ◽

Qinqin Zhang ◽

Qirui Li

Keyword(s):

Positive Integer ◽

Positive Solutions ◽

Difference Equations ◽

Initial Conditions ◽

Higher Order ◽

Order System ◽

Real Numbers ◽

Precise Description ◽

Positive Real

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

Global Character of a Six-Dimensional Nonlinear System of Difference Equations

10.1155/2016/6842521 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 4

Author(s):

Mehmet Gümüş ◽

Yüksel Soykan

Keyword(s):

Nonlinear System ◽

Positive Solutions ◽

Difference Equations ◽

Dynamical Behavior ◽

Real Numbers ◽

Positive Real ◽

Rational Difference Equations ◽

Initial Values ◽

Global Character

The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form:un+1=αun-1/β+γvn-2p,vn+1=α1vn-1/β1+γ1un-2p,n=0,1,…, where the parametersα,β,γ,α1,β1,γ1,pand the initial valuesu-i,v-ifori=0,1,2are positive real numbers.

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

10.1155/2018/1743540 ◽

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 ◽

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.

On a class of difference equations with interlacing indices

Advances in Difference Equations ◽

10.1186/s13662-021-03452-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Stevo Stević ◽

A. El-Sayed Ahmed ◽

Witold Kosmala ◽

Zdeněk Šmarda

Keyword(s):

Difference Equations ◽

Natural Connection ◽

AbstractSome results on the long-term behavior of solutions to a class of difference equations, which includes numerous nonlinear difference equations of various orders that attracted some attention in the last 15 years, are presented. We also present a natural connection among these difference equations, compare some results on the equations with some other ones in the literature, and give a list of a considerable number of difference equations which can be treated in a similar way.

Asymptotic Stability for a Class of Nonlinear Difference Equations

10.1155/2010/791610 ◽

2010 ◽

Vol 2010 ◽

pp. 1-10 ◽

Author(s):

Chang-you Wang ◽

Shu Wang ◽

Zhi-wei Wang ◽

Fei Gong ◽

Rui-fang Wang

Keyword(s):

Asymptotic Stability ◽

Difference Equation ◽

Difference Equations ◽

Global Asymptotic Stability ◽

Initial Conditions ◽

Real Numbers ◽

Positive Real ◽

Fractional Difference ◽

Fractional Difference Equation

We study the global asymptotic stability of the equilibrium point for the fractional difference equationxn+1=(axn-lxn-k)/(α+bxn-s+cxn-t),n=0,1,…, where the initial conditionsx-r,x-r+1,…,x1,x0are arbitrary positive real numbers of the interval(0,α/2a),l,k,s,tare nonnegative integers,r=max⁡⁡{l,k,s,t}andα,a,b,care positive constants. Moreover, some numerical simulations are given to illustrate our results.

On a class of solvable higher-order difference equations

Filomat ◽

10.2298/fil1702461s ◽

2017 ◽

Vol 31 (2) ◽

pp. 461-477 ◽

Author(s):

Stevo Stevic ◽

Mohammed Alghamdi ◽

Abdullah Alotaibi ◽

Elsayed Elsayed

Keyword(s):

Difference Equation ◽

Closed Form ◽

Difference Equations ◽

Higher Order ◽

Real Numbers ◽

Order Difference ◽

Initial Values ◽

Closed form formulas for well-defined solutions of the next difference equation xn = xn-2xn-k-2/xn-k(an + bnxn-2xn-k-2), n ? N0, where k ? N, (an)n?N0, (bn)n?N0, and initial values x-i, i = 1,k+2 are real numbers, are given. Long-term behavior of well-defined solutions of the equation when (an)n?N0 and (bn)n?N0 are constant sequences is described in detail by using the formulas. We also describe the domain of undefinable solutions of the equation. Our results explain and considerably improve some recent results in the literature.

On the solutions of a class of max-min-type difference equation system

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm180108008m ◽

2019 ◽

Vol 13 (1) ◽

pp. 165-177

Author(s):

Huili Ma ◽

Haixia Wang

Keyword(s):

Difference Equation ◽

Periodic Solutions ◽

Difference Equations ◽

Equation System ◽

Real Numbers ◽

Positive Real ◽

General Solutions ◽

Initial Values

We mainly investigate the general solutions and periodic solutions to the following system of max-type difference equations xn+1 = max{y2n-1, An/yn-1}, yn+1 = min{x2n-1,Bn/xn-1}, where n ? N, (An)n?N and (Bn)n?N are positive real sequences, and the initial values x-1 = ?, x0= ?; y-1= ?, y0 = ? are real numbers.