Existence of Bounded Positive Solutions for Partial Difference Equations with Delays

Abstract

This paper deals with solvability of the third-order nonlinear partial difference equation with delays ${\Delta }_n\left(a_{m,n}{\Delta }_m^2\right(x_{m,n}+b_{m,n}{x}_{m-{\tau }_0,n-{\sigma }_0}\left)\right)+f(m,n,x_{m-{\tau }_{1,m},n-{\sigma }_{1,n}},\dots ,x_{m-{\tau }_{k,m},n-{\sigma }_{k,n}})=c_{m,n},m\ge m_0,n\ge n_0$. With the help of the Banach fixed-point theorem, the existence results of uncountably many bounded positive solutions for the partial difference equation are given; some Mann iterative schemes with errors are suggested, and the error estimates between the iterative schemes and the bounded positive solutions are discussed. Three nontrivial examples illustrating the results presented in this paper are also provided.