## Abstract and Applied Analysis

### Unbounded Positive Solutions and Mann Iterative Schemes of a Second-Order Nonlinear Neutral Delay Difference Equation

#### Abstract

This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation ${\mathrm{\Delta }}^{2}\left({x}_{n}+{a}_{n}{x}_{n-\tau }\right)+\mathrm{\Delta }h\left(n,{x}_{{h}_{1n}},{x}_{{h}_{2n}},\dots ,{x}_{{h}_{kn}}\right)+f\left(n,{x}_{{f}_{1n}},{x}_{{f}_{2n}},\dots ,{x}_{{f}_{kn}}\right)={b}_{n},\forall n\ge {n}_{0}$. Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 245012, 12 pages.

Dates
First available in Project Euclid: 18 April 2013

https://projecteuclid.org/euclid.aaa/1366306758

Digital Object Identifier
doi:10.1155/2013/245012

Mathematical Reviews number (MathSciNet)
MR3034988

Zentralblatt MATH identifier
1266.65203

#### Citation

Liu, Zeqing; Hou, Xiaochuan; Ume, Jeong Sheok; Kang, Shin Min. Unbounded Positive Solutions and Mann Iterative Schemes of a Second-Order Nonlinear Neutral Delay Difference Equation. Abstr. Appl. Anal. 2013 (2013), Article ID 245012, 12 pages. doi:10.1155/2013/245012. https://projecteuclid.org/euclid.aaa/1366306758