Differential Equations in Banach Spaces with an Noninvertible Operator in the Principal Part and Nonclassical Initial Conditions

In this paper, we examine differential equations with nonclassical initial conditions and noninvertible operators in their principal parts. We find necessary and sufficient conditions for the existence of unbounded solutions with a pole of order p at points where the operator in the principal part of the differential equation is noninvertible. Based on the alternative Lyapunov–Schmidt method and Laurent expansions, we propose a two-stage method for constructing expansion coefficients of the solution in a neighborhood of a pole. We develop the techniques of skeleton chains of linear operators in Banach spaces and discuss its applications to the statement of initial conditions for differential equations. The results obtained develop the theory of degenerate differential equations. Illustrative examples are given.