Existence Of Uncountable Many Solutions And Mann Iterative Algorithms Of A Nonlinear Two-dimensional Third Order Difference Equations

The purpose of this paper is to study the below nonlinear two-dimensional third order difference system with multiple delaysIn this article the Banach fixed point theorem and a few new techniques from nonlinear analysis are used to study the difference equations above, especially the existence of uncountable many bounded positive solutions of this difference equations. And then discuss the Mann iterative algorithms with error estimates and obtain the error estimates between the positive solutions and iterative sequences. The article is composed by five parts. In the first part, an introduction is proposed, which points out the development process of nonlinear delay difference equations over the past decade. And put forward to the sufficient conditions of the equations in this paper if the equations have solutions. The second part of this paper are preparations and lemmas, they will mainly be used to define some symbols in the proof of the theorems and at the same time be used in describing the theorems. The seven theorems in part three are the core part of this paper. The goal is to study the existence of uncountable many positive solutions for the equations and establish their iterative approximations and discuss the error estimates between the positive solutions and iterative approximations. Different conditions and mappings are built as the coefficients of the difference equations are different. In the fourth part of the paper, five examples are given as the the application of the theorems in part three. The fifth part of this article is the references.