On conditions of non-quasiregularity of second-order singular differential operators in a space of vector functions (Q5950850)

The author deals with a system of linear differential second-order equations \[ \{P(x)y'(x)\}' = Q(x)y(x),\tag{1} \] with \(x\in [0,\infty)\), \(y\in \mathbb{R}^n\), \(P(x)\) and \(Q(x)\) are \(n\times n\)-matrices (nonsymmetric, in general) and \(\{P(x)\}^{-1}\) and \(Q(x)\) are assumed to be Lebesgue integrable on each segment \( [0,b]\), \( 0<b<\infty\). Together with system (1) the conjugated system \[ \{P^*(x)y'(x)\}' = Q^*(x)y(x)\tag{\(1^*\)} \] is considered. The author establishes conditions under which at least one of the systems (1) or \((1)^*\) possesses a solution \(y(x)\) such that the integral \( \int\limits_0^\infty\bigg\{\sum\limits_{k=1}^n |y_k(x)|^p\bigg\} \) diverges for some \(p\), \(1\leq p<\infty\).