scholarly journals On the global behavior of the nonlinear difference equationxn+1=f(pn,xn−m,xn−t(k+1)+1)

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Positive Sequence ◽  
Nonlinear Difference Equation ◽  
Global Behavior ◽  
Initial Values

We consider the following nonlinear difference equation:xn+1=f(pn,xn−m,xn−t(k+1)+1),n=0,1,2,…, wherem∈{0,1,2,…}andk,t∈{1,2,…}with0≤m<t(k+1)−1, the initial valuesx−t(k+1)+1,x−t(k+1)+2,…,x0∈(0,+∞), and{pn}n=0∞is a positive sequence of the periodk+1. We give sufficient conditions under which every positive solution of this equation tends to the periodk+1solution.

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 On the recursive sequencexn+1=−1/xn+A/xn−1

2001 ◽  
Vol 27 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation

We investigate the periodic character of solutions of the nonlinear difference equationxn+1=−1/xn+A/xn−1. We give sufficient conditions under which every positive solution of this equation converges to a period two solution. This confirms a conjecture in the work of DeVault et al. (2000).

scholarly journals Existence of Monotone Solutions of a Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Weizhen Quan
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Initial Values ◽  
Monotone Positive Solutions ◽  
Monotone Solutions

We consider the nonlinear difference equationxn+1=f(xn−k,xn−k+1,…,xn),n=0,1,…,wherek∈{1,2,…}and the initial valuesx−k,x−k+1,…,x0∈(0,+∞). We give sufficient conditions under which this equation has monotone positive solutions which converge to the equilibrium, extending and including in this way some results of the literature.

Nonexistence of positive solution of a class of nonlinear difference equation

2001 ◽  
Vol 119 (2-3) ◽  
pp. 187-195
Author(s):  
Jun Yang ◽  
Xinping Guan ◽  
Shutang Liu
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Nonlinear Difference Equation

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 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 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 Qualitative Inspection Fourth order Non Linear Difference Equations

2019 ◽  
Vol 8 (11) ◽  
pp. 3467-3469
Keyword(s):  
Difference Equation ◽  
Fourth Order ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Linear Difference Equations ◽  
All Solutions ◽  
Non Linear

In this paper, the authors obtained some new sufficient conditions for the oscillation of all solutions of the fourth order nonlinear difference equation of the form ( ) ( 1 ) 0 3  anxn  pnxn  qn f xn  n = 0,1,2, … ., where an, pn, qn positive sequences. The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the main result.

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