Positive Solutions and Iterative Approximations for a Nonlinear Two-Dimensional Difference System with Multiple Delays

Abstract

This paper studies the nonlinear two-dimensional difference system with multiple delays $\mathrm{\Delta }(x_n+p_{1n}{x}_{n-{\tau }_1})$ $+$ $f_1(n,x_{a_{1n}},\dots ,x_{a_{hn}},y_{b_{1n}},\dots ,y_{b_{kn}})=q_{1n},\mathrm{\Delta }(y_n+p_{2n}{y}_{n-{\tau }_2})+f_2(n,x_{c_{1n}},\dots ,x_{c_{hn}},y_{d_{1n}},\dots ,y_{d_{kn}})=q_{2n},n\ge n_0$. Using the Banach fixed point theorem and a few new analysis techniques, we show the existence of uncountably many bounded positive solutions for the system, suggest Mann iterative algorithms with errors, and discussthe error estimates between the positive solutions and iterative sequences generated by the Mann iterativealgorithms. Examples to illustrates the results are included.