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 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 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 Positive Solutions and Mann Iterative Algorithms for a Second-Order Nonlinear Difference Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-21
Author(s):  
Zeqing Liu ◽  
Xin Li ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Difference Equation ◽  
Iterative Method ◽  
Positive Solutions ◽  
Iterative Algorithms ◽  
Second Order ◽  
Nonlinear Difference Equation ◽  
Banach Fixed Point Theorem ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference

This paper deals with the second-order nonlinear neutral delay difference equationΔ(anh(xn-τ1n,xn-τ2n,…,xn-τmn)Δ(xn-qnxn-τ0))+f(n,xn-σ1n,xn-σ2n,…,xn-σkn)=bn,n≥n0. Using the Banach fixed point theorem, Mann iterative method with errors, and some new techniques, we prove the existence of uncountably many positive solutions and the convergence of the sequences generated by the Mann iterative method with errors relative to these solutions for the above equation. Six examples are included. Our results extend and improve essentially the known results in this field.

scholarly journals Positive Solutions of a Second-Order Nonlinear Neutral Delay Difference Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Zeqing Liu ◽  
Wei Sun ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference

The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equationΔ(a(n,ya1n,…,yarn)Δ(yn+bnyn-τ))+f(n,yf1n,…,yfkn)=cn,  ∀n≥n0. By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.

scholarly journals Oscillation Criteria of Solution for a Second Order Difference Equation with Forced Term

2010 ◽  
Vol 2010 ◽  
pp. 1-6
Author(s):  
Chen Huiqin ◽  
Jin Zhen
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Order Difference ◽  
Oscillation Criteria ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Forced Term

We will consider oscillation criteria for the second order difference equation with forced termΔ(anΔ(xn+λxn−τ))+qnxn−σ=rn(n≥0). We establish sufficient conditions which guarantee that every solution is oscillatory or eventually positive solutions converge to zero.

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.

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