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.

Non Linear Difference Equation of Orlicz Type

2017 ◽  
Vol 14 (1) ◽  
pp. 306-313
Author(s):  
Awad. A Bakery ◽  
Afaf. R. Abou Elmatty
Keyword(s):  
Difference Equation ◽  
Equilibrium Point ◽  
Positive Solution ◽  
Orlicz Function ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Numerical Examples ◽  
The Difference

We give here the sufficient conditions on the positive solutions of the difference equation xn+1 = α+M((xn−1)/xn), n = 0, 1, …, where M is an Orlicz function, α∈ (0, ∞) with arbitrary positive initials x−1, x0 to be bounded, α-convergent and the equilibrium point to be globally asymptotically stable. Finally we present the condition for which every positive solution converges to a prime two periodic solution. Our results coincide with that known for the cases M(x) = x in Ref. [3] and M(x) = xk, where k ∈ (0, ∞) in Ref. [7]. We have given the solution of open problem proposed in Ref. [7] about the existence of the positive solution which eventually alternates above and below equilibrium and converges to the equilibrium point. Some numerical examples with figures will be given to show our results.

scholarly journals Global Behavior of a Discrete Survival Model with Several Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Meirong Xu ◽  
Yuzhen Wang
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Survival Model ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
All Solutions ◽  
The Difference ◽  
Several Delays

The difference equationyn+1−yn=−αyn+∑j=1mβje−γjyn−kjis studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.

scholarly journals Global Behavior ofxn+1=(α+βxn-k)/(γ+xn)

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chenquan Gan ◽  
Xiaofan Yang ◽  
Wanping Liu
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Global Stability ◽  
Initial Conditions ◽  
Global Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Difference

This paper aims to investigate the global stability of negative solutions of the difference equationxn+1=(α+βxn-k)/(γ+xn),n=0,1,2,…, where the initial conditionsx-k,…,x0∈-∞,0,kis a positive integer, and the parametersβ,  γ<0,  α>0. By utilizing the invariant interval and periodic character of solutions, it is found that the unique negative equilibrium is globally asymptotically stable under some parameter conditions. Additionally, two examples are given to illustrate the main results in the end.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

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.

scholarly journals Global Behavior of the Difference Equationxn+1=(p+xn-1)/(qxn+xn-1)

2010 ◽  
Vol 2010 ◽  
pp. 1-6
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Hui Wu ◽  
Caihong Han
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Initial Conditions ◽  
Global Behavior ◽  
Finite Limit ◽  
The Difference

We study the following difference equationxn+1=(p+xn-1)/(qxn+xn-1),n=0,1,…,wherep,q∈(0,+∞)and the initial conditionsx-1,x0∈(0,+∞). We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true.

scholarly journals Global asymptotic stability in a periodic Lotka-Volterra system

1985 ◽  
Vol 27 (1) ◽  
pp. 66-72 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Volterra System ◽  
Periodic Coefficients ◽  
Globally Asymptotically Stable ◽  
Stable Periodic

AbstractA set of easily verifiable sufficient conditions are obtained for the existence of a globally asymptotically stable periodic solution in a Lotka-Volterra system with periodic coefficients.

Analysis of nonautonomous two species system in a polluted environment

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
G. Samanta
Keyword(s):  
Periodic Solution ◽  
Population Growth ◽  
Asymptotic Behaviour ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Volterra Model ◽  
Asymptotically Stable ◽  
Polluted Environment ◽  
Globally Asymptotically Stable ◽  
Bounded Functions

AbstractIn this paper, a two-species nonautonomous Lotka-Volterra model of population growth in a polluted environment is proposed. Global asymptotic behaviour of this model by constructing suitable bounded functions has been investigated. It is proved that each population for competition, predation and cooperation systems respectively is uniformly persistent (permanent) under appropriate conditions. Sufficient conditions are derived to confirm that if each of competition, predation and cooperation systems respectively admits a positive periodic solution, then it is globally asymptotically stable.

scholarly journals Dynamics of a Family of Nonlinear Delay Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Qiuli He ◽  
Taixiang Sun ◽  
Hongjian Xi
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We study the global asymptotic stability of the following difference equation:xn+1=f(xn-k1,xn-k2,…,xn-ks;xn-m1,xn-m2,…,xn-mt),n=0,1,…,where0≤k1<k2<⋯<ksand0≤m1<m2<⋯<mtwith{k1,k2,…,ks}⋂‍{m1,m2,…,mt}=∅,the initial values are positive, andf∈C(Es+t,(0,+∞))withE∈{(0,+∞),[0,+∞)}. We give sufficient conditions under which the unique positive equilibriumx-of that equation is globally asymptotically stable.

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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