Some new criteria for the oscillation of all solutions of third order nonlinear difference equations of the formwith ∞ a −1/α (n) < ∞ are established.

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By using variational methods and critical point theory, the authors establish the existence of infinitely many weak solutions for impulsive differential inclusions involving two parameters and the p-Laplacian and having Dirichlet boundary conditions.

The authors consider the third order neutral delay difference equation with positive and negative coefficients ∆(a n ∆(b n ∆(x n + px n−m))) + p n f (x n−k) − q n (x n−l) = 0, n ≥ n 0 , and give some new sufficient conditions for the existence of nonoscillatory solutions. Banach's fixed point theorem plays a major role in the proofs. Examples are provided to illustrate their main results.