Functional Perturbations of Second Order Differential Equations

Conditions are given which imply that the functional differential equation \[(r(t)x'(t))' + q(t)x(t) = f(t,x(g(t)))\] has a solution $\bar x$ which behaves for large t in a precisely defined way like a given solution $\bar y$ of the ordinary differential equation \[(r(t)y')' + q(t)y = 0\] It is not assumed that $g(t) - t$ is sign-constant, and $f(t,u)$ need only be defined and continuous on a subset of the $(t,u)$ plane which is near the curve $u = \bar y(g(t))$ in an appropriate sense for large t. The integral smallness conditions on $f(t,u)$ permit some of the improper integrals in question to converge conditionally. Separate treatments are given for the cases where the unperturbed equation is oscillatory or nonoscillatory. The results are new even in the case where $g(t) = t$.