Investigation of One Linear Differential Equation by Using Generalized Functions with Values in a Banach Space

Abstract

We present a generalization of some facts of the theory of generalized functions of slow growth to the case of operator-valued test functions. We propose a construction of regular generalized functions with values in a Banach space. The results obtained are used for the description of slowly increasing solutions of linear homogeneous differential equations with shifted arguments and bounded operator coefficients in a Banach space.