scholarly journals Global solvability for a second order nonlinear neutral delay difference equation

2009 ◽  
Vol 57 (4) ◽  
pp. 587-595 ◽  
Author(s):  
Zeqing Liu ◽  
Yuguang Xu ◽  
Shin Min Kang
Keyword(s):  
Difference Equation ◽  
Global Solvability ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference

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 Solvability of a Second Order Nonlinear Neutral Delay Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-24 ◽  
Author(s):  
Zeqing Liu ◽  
Liangshi Zhao ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Difference Equation ◽  
Point Theorem ◽  
Existence Results ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
New Techniques ◽  
Delay Difference

This paper studies the second-order nonlinear neutral delay difference equationΔ[anΔ(xn+bnxn−τ)+f(n,xf1n,…,xfkn)]+g(n,xg1n,…,xgkn)=cn,n≥n0. By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. Ten examples are given to illustrate the results presented in this paper.

scholarly journals Oscillation and nonoscillation of nonlinear neutral delay difference equations

2007 ◽  
Vol 38 (4) ◽  
pp. 323-333 ◽  
Author(s):  
E. Thandapani ◽  
P. Mohan Kumar
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Oscillation And Nonoscillation

In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation$$ \Delta^2 (x_n-p_nx_{n-k}) + q_nf(x_{n-\ell}) = 0, ~~n \ge n_0 $$where $ \{p_n\} $ and $ \{q_n\} $ are non-negative sequences with $ 0$

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