scholarly journals On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Zeqing Liu ◽  
Heng Wu ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Third Order ◽  
Order Difference ◽  
Iterative Schemes ◽  
Order Difference Equation ◽  
Delay Difference

The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

scholarly journals Positive Solutions for a Third Order Nonlinear Neutral Delay Difference Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-33
Author(s):  
Zeqing Liu ◽  
Xiaochen Wang ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Difference Equation ◽  
Positive Solutions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Third Order ◽  
Delay Difference

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper.

scholarly journals On Positive Solutions of a Fourth Order Nonlinear Neutral Delay Difference Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-29 ◽  
Author(s):  
Zeqing Liu ◽  
Xiaoping Zhang ◽  
Shin Min Kang ◽  
Young Chel Kwun
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Point Theorem ◽  
Fourth Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Delay Difference

The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.

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 Global behavior of a third order difference equation

2012 ◽  
Vol 43 (3) ◽  
pp. 375-384
Author(s):  
Raafat Abo-zeid
Keyword(s):  
Difference Equation ◽  
Global Stability ◽  
Positive Solutions ◽  
Global Behavior ◽  
Real Numbers ◽  
Third Order ◽  
Order Difference ◽  
Order Difference Equation ◽  
The Difference

The aim of this paper is to investigate the global stability and periodic nature of the positive solutions of the difference equation\[x_{n+1}=\frac{A+Bx_{n-1}}{C+Dx_{n}x_{n-2}},\qquad n=0,1,2,\ldots\] where $A,B$ are nonnegative real numbers and$C, D>0$.

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