Asymptotic integration of a linear homogeneous differential equation with degeneration and singular point in complex Banach space (Q2703305)

The author constructs partial asymptotic solutions to the linear differential equation in complex Banach space \({\mathcal E}\): NEWLINE\[NEWLINEx^{-h}B(x)(dy(x)/dx)=A(x)y(x); \quad h\in \mathbb{N};NEWLINE\]NEWLINE where \(A:{\mathcal E}\to{\mathcal E}\) and \(B:{\mathcal E}\to{\mathcal E}\) are linear operators; \(A_{0}=\lim_{x\to\infty}A(x)\), \(B_{0}=\lim_{x\to\infty}B(x)\) are Fredholm operators. The operator \(B_{0}\) has a zero eigenvalue, which corresponds to the \(A_{0}\)-Jordan set of vectors containing a finite number of Jordan chains with arbitrary finite order. The bundle of operators \((A_{0}-\omega B_{0})\) has eigenvalues which correspond to the \(B_{0}\)-Jordan set of vectors containing a finite number of Jordan chains with arbitrary finite order.