Singularly perturbed problems with double singularity (Q1277548)

The author deals with systems of linear differential equations \[ \varepsilon^{m_1}\bigl(\delta_0(x)+\varepsilon\bigr)^{m_2}{dy\over dx}=A(x,\varepsilon)y+f(x), \] with \(\delta_0(x)>0\) for \(x\in(0,1)\), \(\delta_0 (0)=\delta_0(1)=0\), \(m_1,m_2\) are positive integers (which determine the two singularities \(x=0\) and \(x=1\), respectively) and the matrix \(A(x,\varepsilon)\) possesses an absolutely and uniformly convergent expansion \(\sum^\infty_{k=0} A_k(x) \varepsilon^k\), in which \(A_0(x)\) has a simple nonzero stable spectrum. Asymptotic expansions for a bounded (as \(\varepsilon\to 0)\) solution \(y(x, \varepsilon)\) are proved in two theorems. In contrast to usual results, the structure of these asymptotic formulas does not depend only on the spectrum of \(A_0(x)\), even if \(m_1=m_2=1\).