Asymptotic integration of delay differential systems (Q5905539)

For the system with delays \((*)\;x'(t)=[\Lambda+V(t)]x(t)+L(t,x_ t)\), \(x\in E^ n\), in which among others \(\Lambda=\text{diag}(\lambda_ 1,\dots,\lambda_ n)\), \(V(t)=\text{diag}(v_ 1(t),\dots,v_ n(t))\), \(L(t,\cdot)\) is linear and continuous, asymptotic formulas for the solutions \(x(t)\) are obtained. Each solution \(x(t_ 0,\varphi)\) of \((*)\) has the asymptotic form \(x(t_ 0,\varphi)(t)=X(t,t_ 0)[c(t_ 0,\varphi)+f(t_ 0,\varphi)(t)]\), where \(X(t,t_ 0)\) is an \(n\times n\) matrix, \(c(t_ 0,\varphi)\) is a constant vector in \(E^ n\), and \(f(t_ 0,\varphi)(t)\in E^ n\). The functions \(X\), \(f\) and \(c_ 0\) are described in the paper.