Tchaikovsky A. Classical solutions of linear first order differential equations with operator coefficients

Thesis deals with the existence and uniqueness of solutions of linear first order differential equations with operator coefficients relative to functions defined on a subset of the real axis and with values in complex Banach space. We construct and study a new class of operators which extends class sectorial operators - class G-sectorial operators. Sufficient conditions for existence and uniqueness of solution of the Cauchy problem for linear first order differential equations with G-sectorial operator coefficient and known H?lder function. New classes of known functions are found for which the Cauchy problem for linear differential equations with sectorial or G-sectorial operator coefficients has a solution of given smoothness. Sufficient conditions for existence of bounded on the axis solution of linear differential equations with unbounded operator coefficient and a delayed argument are obtained.