The Milne problem for a periodic medium (Q2387988)

The author considers the linear system of differential equations \[ \begin{aligned} {dx\over d\tau} &= -Ax+ L^+(\tau) x+ L^-(\tau)y,\\ {dy\over d\tau} &= -Ay+ L^-(\tau)x+ L^+(\tau)y,\quad 0\leq \tau<\infty,\end{aligned}\tag{1} \] where \(x(\tau)\) and \(y(\tau)\) are the unknown \(m\)-vector functions, \(A\) is a constant \(m\times m\)-diagonal-matrix with positive diagonal entries \(a_i\) and the \(L^\pm= (l^\pm_{ij}(\tau))\) are nonnegative indecomposable (for each \(\tau\)) continuous \(m\times m\)-matrix functions related to \(A\) by the conditions \[ \sum^m_{j=1} (l^+_{ij}(\tau)+ l^-_{ij}(\tau))\leq \lambda a_i,\quad 0< \lambda\leq 1,\;i= 1,\dots, m,\tag{2} \] \[ L^\pm(\tau)= MG^\pm(\tau);\quad G^\pm(\tau= 1)= G^\pm(\tau)= (G^+(\tau))^T,\tag{3} \] where \(M\) is a diagonal matrix with positive diagonal entries and the \(G^\pm(\tau)\) are nonnegative symmetric periodic matrix functions. The author studies the following two problems: (A) Under the conditions (2), (3) and the supplementary condition: ``\(x(0)= \alpha\), \(y(\tau)\) is bounded as \(\tau\to+\infty\)'', the author proves that system (1) has a solution and presents a numerical algorithm for finding the solution. (B) In the conservative case \(\sum^n_{j=1} (l^+_{ij}(\tau)+ l^-_{ij}(\tau))= a_i\), \(i= 1,\dots, m\), the author constructs a positive solution \((x(\tau), y(\tau))\) of system (1) defined on the half-line \(0<\tau<+\infty\) and satisfying the condition \(x(0)= 0\) and the asymptotics \(x(\tau)= O(\tau)\), \(y(\tau)= O(\tau)\), \(\tau\to+\infty\).