On perturbation of unstable second order linear differential equations

Abstract:

In connection with a conjecture of J. M. Bownds, conditions will be given on the fundamental system of the solutions of the unstable differential equation $y'' + a(t)y = 0$ which assure that the differential equation $x'' + a(t)x = g(t,x,x’)$ has a solution with the property \[ \lim \sup (|x(t)| + |x’(t)|) = \infty \quad {\text {as }}t \to \infty ,\] provided that $g(t,x,x’)$ is “sufficiently small".