The asymptotic form of the solutions of the equation \(y^{(2n)}+(s_ 1y^{(n-1)})^{(n-1)}+...+(s_{n-1}y')'+s_ ny=0\) as \(x\to \infty\) is considered, where the \(s_ m\) are the product of a small factor \(\xi^ m\) and a periodic factor \(p_ m\), \(s_ m(x)=\xi^ m(x)p_ m(x)\quad (1\leq m\leq n),\) the mean value of \(p_ m\) being zero. The \(p_ m\) all have the same period which is taken to be \(2\pi\). In a previous paper [J. Lond. Math. Soc., II. Ser. 40, 507-518 (1989; Zbl 0707.34047)], the authors dealt with the situation where \(\xi\to 0\) slowly as, for example, when \(\xi (x)=x^{-\alpha}\) \((0<\alpha \leq 1)\). It was found that the solutions do not resemble those of the unperturbed equation \(y^{(2n)}=0.\)
Here the interest is in the case where \(\xi (x)=x^{-1}\eta (x)\) and \(\eta\) tends to zero slowly as, for example, when \(\eta (x)=(\log x)^{- \gamma}(\log \log x)^{-\delta}\quad (\gamma \geq 0,\quad \delta \geq 0).\) Subject to suitable conditions on \(\eta\), it is found that there are solutions \[ (*)\quad y_ k(x)\sim \exp (\int^{x}_{a}t^{-1}\nu_ k(t)dt), \] where the \(\nu_ k\) are the eigenvalues of a certain perturbation of the diagonal matrix \(\Lambda =diag(0,1,...,2n-1)\). A specific example of (*) is where \(s_ m=0\) (1\(\leq m\leq n-1)\) and \(x^{- 1}\eta^{4n}\) is L(a,\(\infty)\). Then \[ y_ k(x)\sim x^{k-1} \exp ((- 1)^{n+k+1}c\{(2n-k)!(k-1)!\}^{-1}\int^{x}_{a}t^{- 1}\eta^{2n}(t)dt), \] where \[ c=(2\pi)^{-1}\int^{\pi}_{- \pi}\{p_ n^{(-n)}(t)\}^ 2dt \] and (-n) refers to an n-fold integral of \(p_ n\). These results generalise long-standing ones due to P. Hartman and A. Wintner for the second-order case \(n=1\) [Am. J. Math. 75, 717-730 (1953)].