Fourth-order nonlinear oscillations of difference equations

The authors consider the fourth-order difference equation Delta (2) (r(n)Delta (2)y(n)) + f (n, y(n)) = 0, n is an element of N (n(0) ), (*) where f(n, u) may be classified as superlinear, sublinear: strongly superli near, and strongly sublinear and Sigma (infinity)(n=n0) (n/r(n)) < infinity . In superlinear and sublinear cases, necessary and sufficient conditions a re obtained for the existence of nonoscillatory solutions of (*) with a spe cified asymptotic behavior. Further, in strongly superlinear and strongly s ublinear cases, necessary and sufficient conditions are given for all solut ions of (*) to be oscillatory. (C) 2001 Elsevier Science Ltd. All rights re served.