Linearized comparison theorems for a nonlinear delay 2D discrete dynamic systems

摘要

This paper deals with a class of a nonlinear delay 2D discrete dynamical systems of the form A/sub m+1,n/+A/sub m,n+1/-A/sub mn/+/spl Sigma//sub i=1//sup /spl kappa//Pi(m,n)f/sub i/(A/sub m+/spl sigma/i,n/+A/sub m,n+/spl sigma/i/)=0, (E). We show that if a positive solution exists, then a related minorant equation has a positive solution as well. Similar results for a dual equation are also obtained. In fact, we remark further that equation (E) may also be regarded as a discrete analog of partial differential equations of the form /spl part/u//spl part/x+/spl part/u//spl part/y-u(x,y)+/spl Sigma//sub i=1//sup /spl kappa//Pi(x,y)f/sub i/(u(x+/spl sigma//sub i/,y)+u(x,y+/spl sigma//sub i/))=0. Therefore, qualitative of (E) may yield useful information for this companion partial differential equation.