I propose a rather difficult question concerning the boundedness of solutions to a "weakly-coupled" pair of linear differential equations.
Let $f$ and $g$ be two continous real-valued functions defined on the non-negative reals such that
$$\int_0^{\infty}|f-g|$$
converges. We consider the differential equations
(E1): $y''+f(x)y=0$
(E2): $y''+g(x)y=0$
Let $S_{E1}$ and $S_{E2}$ denote the set of solutions defined over the non-negative reals. of $E1$ and $E2$ repsectively.
Suppose that every solution $\phi$ of (E1) is bounded and has a bounded derivative.
I want to demonstrate two things:
1. Every solution $\psi$ of (E2) is bounded and has a bounded derivative.
2. There exists an isomorphism $\Phi$ from $S_{E1}$ to $S_{E2}$ such that for every $\phi \in S_{E1}$, $\psi=\Phi(\phi)$ satisfies
$$\lim _{x \to \infty} \psi(x)-\phi(x)=\lim _{x \to \infty} \psi'(x)-\phi'(x)=0$$