scholarly journals On boundedness of the solutions of the difference equationxn+1=xn−1/(p+xn)

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Hui Wu
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Open Problem ◽  
Initial Values ◽  
The Difference

We study the difference equationxn+1=xn−1/(p+xn),n=0,1,…,where initial valuesx−1,x0∈(0,+∞)and0<p<1, and obtain the set of all initial values(x−1,x0)∈(0,+∞)×(0,+∞)such that the positive solution{xn}n=−1∞is bounded. This answers the Open Problem 2 proposed by Kulenović and Ladas.

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 Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

The long-term behavior of solutions of the following difference equation: , , where the initial values , , are real numbers, is investigated in the paper.

scholarly journals On the Difference Equationxn=anxn-k/(bn+cnxn-1⋯xn-k)

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Stevo Stević ◽  
Josef Diblík ◽  
Bratislav Iričanin ◽  
Zdeněk Šmarda
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference

The behavior of well-defined solutions of the difference equationxn=anxn-k/(bn+cnxn-1⋯xn-k), n∈ℕ0, wherek∈ℕis fixed, the sequencesan,bnandcnare real,(bn,cn)≠(0,0),n∈ℕ0, and the initial valuesx-k,…,x-1are real numbers, is described.

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 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 Long-Term Behavior of Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

We investigate the long-term behavior of solutions of the following difference equation: , , where the initial values , , and are real numbers. Numerous fascinating properties of the solutions of the equation are presented.

scholarly journals On the Asymptotic Behavior of a Difference Equation with Maximum

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Fangkuan Sun
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
The Difference

We study the asymptotic behavior of positive solutions to the difference equationxn=max{A/xn-1α,B/xn−2β},n=0,1,…,where0<α, β<1, A,B>0. We prove that every positive solution to this equation converges tox∗=max{A1/(α+1),B1/(β+1)}.

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 On the Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
İbrahim Yalçinkaya
Keyword(s):  
Difference Equation ◽  
Higher Order ◽  
Real Numbers ◽  
Positive Real ◽  
Global Behaviour ◽  
Initial Values ◽  
The Difference

We investigate the global behaviour of the difference equation of higher order , where the parameters and the initial values and are arbitrary positive real numbers.

Sign in / Sign up

Export Citation Format

Share Document