On regular growth and asymptotic stability (Q1093793)

The authors consider differential equations of the form (*) \(y''+q(x)y=0\) where q(x) is right-continuous, nondecreasing and such that \(\lim_{x\to X}q(x)=+\infty.\) They assume that for some solution y(x) of (*), \(\lim_{x\to X}y(x)\neq 0.\) Then, they prove that the growth of q(x) is ``irregular'', in a precise technical sense. From this fact, sufficient conditions for global asymptotic stability of the trivial solution of (*) can be easily obtained.