Fourth-order differential equation with deviating argument

Abstract

We consider the fourth-order differential equation with middle-term and deviating argument x (4) (t)+q(t) x (2) (t)+r(t)f(x(ℓ(t) ))=0, in case when the corresponding second-order equation h +q(t)h=0 is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too. Copyright © 2012 M. Bartuek et al.