Solution of the homogeneous Cauchy problem for an equation with a Fredholm operator multiplying the derivative

Abstract

This paper studies properties of irregular operator pencils with Fredholm operator on Banach spaces. A relationship between the uniqueness of a solution of a certain algebraic equation and that of the Cauchy problem is established. Complete conditions on the initial values for the existence of a solution are given. Solutions are constructed in the cases of unique and nonunique solvability. Consider the Cauchy problem for the equation