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.

scholarly journals Monotonicity of Eventually Positive Solutions for a Second Order Nonlinear Difference Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Huiqin Chen ◽  
Zhen Jin ◽  
Shugui Kang
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Monotone Solutions

We derive several sufficient conditions for monotonicity of eventually positive solutions on a class of second order perturbed nonlinear difference equation. Furthermore, we obtain a few nonexistence criteria for eventually positive monotone solutions of this equation. Examples are provided to illustrate our main results.

scholarly journals ASYMPTOTICALLY MONOTONE SOLUTIONS OF A NONLINEAR DIFFERENCE EQUATION

1993 ◽  
Vol 24 (3) ◽  
pp. 269-282
Author(s):  
HORNG-JAAN LI ◽  
SUI-SUN CHENG
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Nonlinear Difference Equation ◽  
Monotone Solutions

Necessary conditions as well as sufficient conditions for the eventually positive solutions of a class of nonlinear difference equation to be monotone are derived.

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 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.

scholarly journals Existence of positive solutions for certain partial difference equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12
Author(s):  
Binggen Zhang ◽  
Qiuju Xing
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Partial Difference Equation ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Existence Of Positive Solutions

We give some sufficient conditions for the existence of positive solutions of partial difference equationaAm+1,n+1+bAm,n+1+cAm+1,n−dAm,n+Pm,nAm−k,n−1=0by two different methods.

scholarly journals Boundedness and Global Attractivity of a Higher-Order Nonlinear Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiu-Mei Jia ◽  
Wan-Tong Li
Keyword(s):  
Difference Equation ◽  
Global Attractor ◽  
Positive Solutions ◽  
Local Stability ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Higher Order ◽  
Nonlinear Difference Equation ◽  
Positive Equilibrium ◽  
Prime Period

We investigate the local stability, prime period-two solutions, boundedness, invariant intervals, and global attractivity of all positive solutions of the following difference equation: , , where the parameters and the initial conditions . We show that the unique positive equilibrium of this equation is a global attractor under certain conditions.

scholarly journals Boundedness and asymptotic behavior of solutions of a forced difference equation

1994 ◽  
Vol 17 (2) ◽  
pp. 397-400 ◽  
Author(s):  
John R. Graef ◽  
Paul W. Spikes
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Nonlinear Difference Equation

The authors consider the nonlinear difference equation?[yn+pnyn-h]+qnf(yn-k)=rnwhere?yn=yn+1-yn,{pn},{qn}, and{rn}are real sequences, anduf(u)>0foru?0. Sufficient conditions for boundedness and convergence to zero of certain types of solutions axe given. Examples illustrating the results are also included.

scholarly journals Asymptotic Behavior of Solutions to Half-Linearq-Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Pavel Řehák
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Special Limit ◽  
Bounded Functions ◽  
Necessary And Sufficient

We derive necessary and sufficient conditions for (some or all) positive solutions of the half-linearq-difference equationDq(Φ(Dqy(t)))+p(t)Φ(y(qt))=0,t∈{qk:k∈N0}withq>1,Φ(u)=|u|α−1sgn⁡uwithα>1, to behave likeq-regularly varying orq-rapidly varying orq-regularly bounded functions (that is, the functionsy, for which a special limit behavior ofy(qt)/y(t)ast→∞is prescribed). A thorough discussion on such an asymptotic behavior of solutions is provided. Related Kneser type criteria are presented.

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