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 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 Existence and Global Behavior of Positive Solutions for Some Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ramzi S. Alsaedi
Keyword(s):  
Continuous Function ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Global Behavior ◽  
Nonnegative Continuous Function

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem:u(4)(x)=a(x)uσ(x),x∈(0,1)with the boundary conditionsu(0)=u(1)=u'(0)=u'(1)=0, whereσ∈(-1,1)andais a nonnegative continuous function on (0, 1) that may be singular atx=0orx=1. We also give the global behavior of such a solution.

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 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 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).

scholarly journals On the Dynamics of the Recursive Sequence

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Mehmet Gümüş ◽  
Özkan Öcalan ◽  
Nilüfer B. Felah
Keyword(s):  
Periodic Solution ◽  
Difference Equation ◽  
Periodic Solutions ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Recursive Sequence ◽  
Boundedness Character ◽  
The Difference

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation , where , , and the initial conditions are arbitrary positive numbers. We investigate the boundedness character for . Also, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no prime two periodic solutions of the equation above.

scholarly journals On a difference equation with min-max response

2004 ◽  
Vol 2004 (55) ◽  
pp. 2915-2926 ◽  
Author(s):  
George L. Karakostas ◽  
Stevo Stevic
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Global Behavior ◽  
The Difference

We investigate the global behavior of the (positive) solutions of the difference equationxn+1=αn+F(xn,…,xn−k),n=0,1,…,where(αn)is a sequence of positive reals andFis a min-max function in the sense introduced here. Our results extend several results obtained in the literature.

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 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.

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