Oscillation for Partial Difference Equations I

Here, we shall provide sufficient conditions for the oscillation of all solutions of the partial difference equation 23.1 $$u(k + 1,\ell ) + \beta (k,\ell )u(k,\ell + 1) - \delta (k,\ell )u(k,\ell ) + P(k,\ell ,u(k - \tau ,\ell - \upsilon )) = Q(k,\ell ,u(k - \tau ,\ell - \upsilon )),k \in N({k_0}),\ell \in N({\ell _0})$$ where τ, ν are non-negative integers, and β(k, ℓ), δ(k, ℓ) are functions such that for all large k and ℓ $$\beta \left( {k,l} \right) \geqslant \beta > 0\,and\,\delta \left( {k,l} \right) \leqslant \delta \left( { > 0} \right).$$ We note that δ(k, ℓ) is allowed to be negative. Functions P and Q are defined on N(k₀) × N(ℓ₀) × ℝ.