Oscillation And Existence Of Positive Solutions Of Delay Partial Difference Equations

Delay Partial Difference Equation is a new branch of mathematics only developed during the past ten years, so there are many problems to be solved. In this paper, the author is concerned with the problems that are to be solved in the field of Partial Difference Equations such as the following: the existence of oscillatory solutions of some partial difference equation, the asymptotic behavior of positive solutions of the same equation. Furthermore, the author gives the oscillation criteria of some partial difference equation with continuous variables.This paper is composed of five sections:The first section is preface. The author introduces the research and the development of partial difference equations, and gives the main questions which are studied.The second section is the research about the existence of oscillatory solution of the following equation:AA(xm,n - cxm-k,n-i) + f(m,n,xm-T,n-a) = 0.The third section is the research about the existence and asymptotic behavior of positive solutions of the above equation.The fourth section is the research about the nonexistence of monotone solution and the existence of positive solutions of the following neutral partial difference equationThe fifth section is the research about the oscillation criteria of the following nonlinear partial difference equation with continuous variablesA(x + a,y) + A(x,y + a) -A(x,y) + p(x,y)\A(x -Ti,y- ai)\asgnA(x - Ti,y - a\)+q(x, y)\A(x - T2, y - a2)f sgnA(x -r2,y- a2) = 0.