Existence of bounded and unbounded nonoscillatory solutions of nonlinear partial difference equations

Consider the higher order nonlinear partial difference equation of neutral typeequation(∗)ΔnhΔmr(y(m,n)+cy(m−k,n−l))+F(m,n,y(m−τ,n−σ))=0, where h,r∈N(1)h,r∈N(1), k,l,τ,σ∈N(0)k,l,τ,σ∈N(0), c∈Rc∈R and F:N×N×R→R. In this paper, we first establish the discrete Arzela–Ascoli's theorem. Next, we obtain some sufficient conditions for the existence of bounded and unbounded nonoscillatory solution of Eq. (∗).