Fourth-Order Differential Equation with Deviating Argument

Abstract

We consider the fourth-order differential equation with middle-term and deviating argument $x^{(4)}(t)+q(t)x^{(2)}(t)+r(t)f(x(\phi (t)))=0$, 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.