Oscillation for Partial Difference Equations III

Here, we shall offer sufficient conditions for the oscillation of all solutions of the partial difference equations 25.1 $$u\left( {k - 1,\ell } \right) + \beta \left( {k,\ell } \right)u\left( {k,\ell - 1} \right) - \delta \left( {k,\ell } \right)u\left( {k,\ell } \right) + P\left( {k,\ell ,u\left( {k + \tau ,\ell + v} \right)} \right) = Q\left( {k,\ell ,u\left( {k + \tau ,\ell + v} \right)} \right),k \in N\left( {{k_0}} \right),\ell \in N\left( {{\ell _0}} \right)$$ and 25.2 $$u\left( {k - 1,\ell } \right) + \beta \left( {k,\ell } \right)u\left( {k,\ell - 1} \right) - \delta \left( {k,\ell } \right)u\left( {k,\ell } \right) + \sum\limits_{i = 1}^\sigma {{P_i}} \left( {k,\ell ,u\left( {k + {\tau _i},\ell + {v_i}} \right)} \right) = \sum\limits_{i = 1}^\sigma {{Q_i}} \left( {k,\ell ,u\left( {k + {\tau _i},\ell + {v_i}} \right)} \right),k \in N\left( {{k_0}} \right),\ell \in N\left( {{\ell _0}} \right)$$ where τ, v, τ _i , v _i , 1 ≤ i ≤ σ are non-negative integers, functions β(k, ℓ), δ(k, ℓ) satisfy the same conditions as in Section 23, and the functions P, Q, P _i , Q _i , 1 ≤ i ≤ σ are defined on N(k₀) × N(ℓ₀) × ℝ.