A perturbation method for differential-algebraic systems. (Q1278051)

Consider the Cauchy problem \[ \begin{gathered} A(t)x'(t)+ B(t) x(t)= g(t)\quad\text{for }t\in (0,1),\\ x(0)= a,\end{gathered} \] where the entries of the \(n\times n\)-matrices \(A(t)\) and \(B(t)\) are real analytic functions of \(t\), \(g\) is a given sufficiently smooth function, \(\text{det\,}A(t)\equiv 0\). The author considers the perturbed problem \[ \begin{gathered} [A(t)+ \varepsilon(B(t)- A'(t))] x_\varepsilon'(t)+ [B(t)+ \varepsilon(B'(t)- A''(t))] x_\varepsilon(t)= g(t),\\ x_\varepsilon(0)= a\end{gathered} \] and derives conditions such that the estimate \[ \| x-x_\varepsilon\|= O(\varepsilon) \] holds.