Quasiregular asymptotic behavior of the solution of a singularly perturbed Cauchy problem for linear systems of differential matrix equations (Q881208)

This paper considers linear systems of differential matrix systems \[ \varepsilon\dot Z= A(t)Z+ ZB(t),\quad Z(0,\varepsilon)= Z^0\tag{\(*\)} \] and its nonhomogeneous equations \[ \varepsilon\dot Z= A(T)Z+ ZB(t)+ F(t),\quad Z(0,\varepsilon)= Z^0,\tag{\(**\)} \] where \(A,B,F\in C^\infty[0,1]\), \(A(t)\), \(B(t)\), \(F(t)\) and \(Z\) are \(n\times n\)-matrices, and the spectra \(\{\lambda_{Aj}(t)\}^n_1\) and \(\{\lambda_{Bj}(t)\}^n_1\) fo the matrices \(A(t)\) and \(B(t)\) fulfill the conditions \[ \begin{gathered} \sigma_{Ajk}(t)= \lambda_{Aj}(t)- \lambda_{Ak}(t)\neq 0,\quad\text{Re}\,\lambda_{Aj}(t)\leq 0,\\ \sigma_{Bjk}(t)= \lambda_{Bj}(t)- \lambda_{Bk}(t)\neq 0,\quad\text{Re\,} \lambda_{Bj}(t)\leq 0\end{gathered} \] (\(j\neq k\), \(j,k= 1,2,\dots, n\), \(t\in [0,1]\)). The author proposes a new method to study the quasiregular asymptotic behavior of the solution to \((*)\) and \((**)\), respectively.