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 ${\mathrm{\Delta }}^2(x_n+a_n{x}_{n-\tau })+\mathrm{\Delta }h(n,x_{h_{1n}},x_{h_{2n}},\dots ,x_{h_{kn}})+f(n,x_{f_{1n}},x_{f_{2n}},\dots ,x_{f_{kn}})=b_n,\forall n\ge n_0$. 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.