Fourt-order nonlinear oscillations of difference equations

The authors consider the fourth-order difference equation where f(n, u) may be classified as superlinear, sublinear, strongly superlinear, and strongly sublinear and Σ∞n=n0 (n/rn) < ∞. In superlinear and sublinear cases, necessary and sufficient conditions are obtained for the existence of nonoscillatory solutions of (*) with a specified asymptotic behavior. Further, in strongly superlinear and strongly sublinear cases, necessary and sufficient conditions are given for all solutions of (*) to be oscillatory.