On the oscillation of some partial difference equations

In this study, consist of five chapter. In the first chapter, information about oscillation of partial difference equations some studied before is given.In the second chapter, some main topics of oscillation of difference equations and partial difference equations are given and some theorems and lemmas concerning these concepts also are reminded.In the third chapter, necessary and sufficient conditions for the oscillation of the higher order linear partial difference equation with constant coefficient?_{m}^{r}?_{n}^{h}y_{m,n}+(-1)^{r+h+1}py_{m-?,n-?}=0are obtained, where m,n,?,??N, r,h?N?, p is a nonnegative real number. The forward partial differences ?_{m} and ?_{n} are defined as usual, i.e.?_{m}A_{m,n}=A_{m+1,n}-A_{m,n} and ?_{n}A_{m,n}=A_{m,n+1}-A_{m,n}.In the fourth chapter some oscillation criteria for the forced oscillation of a class of high order nonlinear partial difference equation?_{m}^{r}?_{n}^{h}y_{m,n}+p_{m,n}f(y_{m-?,n-?})=q_{m,n}are established, where m,n,?,??N, r,h?N?, p_{m,n} and q_{m,n} are double real sequences defined on N², xf(x)>0 for x?0, which includes the special case f(x)=|x|^{?}sgnx for ?>0. The forward partial differences ?_{m} and ?_{n} are defined as usual.In the last chapter relations between the oscillation discrete nonlinear delay wave equations of the form?_{n}²y_{m,n}+p_{n}?_{n}y_{m,n}+q_{n}f(y_{m,n-?})-r_{n}Ly_{m,n}=0, ?(m,n)??×N_{n?}and?_{n}²y_{m,n}+p_{n}?_{n}y_{m,n+1}+q_{n}f(y_{m,n-?})-r_{n}Ly_{m,n}=0, ?(m,n)??×N_{n?}and the oscillation of their linear limiting equations are investigated, where M?N_{n?}, ?={1,2,?,M}, ??N,{p_{n}},{q_{n}},{r_{n}} are sequences of real numbers, f is continuous and convex, uf(u)>0 for u?0 and Ly_{m,n} is the discrete Laplacian operator.