Il'chenko Y. The existence of solutions of differential equations with a shift argument and unbounded operator coefficients in a Banach space.

Thesis deals with the question of existence and uniqueness of bounded solutions of abstract differential equations of the first order. Necessary conditions for the existence and uniqueness of bounded solution on the axis of abstract linear differential equation of the first order with one shift of argument and unbounded operator coefficient for an arbitrary bounded right side were found, and also sufficient conditions for a class of bounded functions were received. For presentation of the solution of abstract differential equation of the first order with a shift of argument and two unbounded coefficients some analogue of the Green function was built, written as a series. For a homogeneous linear differential equation with a shift of argument in the space for the case of degenerate unbounded operator coefficients with block structure, whose spectrum can be placed in very general way, we found the conditions under which solutions exist. For the case of two-dimensional blocks these solutions were clearly written out. The linear inhomogeneous differential equation in a Banach space with a strongly -positive operator coefficient was solved by analytical methods. The behavior of -power operators was studied in detail and Cauchy problem for the linear inhomogeneous differential equation in a Banach space with a -power operator coefficient was solved.