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

Consider the higher order nonlinear partial difference equation of neutral type Δhn Δr m y(m,n) +cy(m− k,n− l) + F m,n, y(m− τ,n− σ) = 0, (∗) where h, r ∈ N(1), k, l, τ,σ ∈ N(0), c ∈ 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. (∗). © 2006 Elsevier Inc. All rights reserved