Chajkovs'kyj A. Bounded solutions of differential equations with argument's displacements in Banach space

Necessary and sufficient conditions of existence and uniqueness of the bounded solution of linear differential equation with argument's displacements in Banach space are given provided the equation's right side is bounded. Representation of the solution is found. A generalization of some facts of the theory of generalized functions of slow growth for the case of operator-valued basic functions is given. The construction of regular generalized functions with values in Banach space is proposed. Obtained results are used for description of slowly growing solutions of the linear differential equation with argument's displacement and bounded operator coefficient in Banach space provided the necessary and sufficient conditions of existence and uniqueness of the bounded solution are not valid. Sufficient conditions of existence of bounded solutions of non-linear differential equation with a bounded and unbounded operator coefficient in Banach space are given provided the right-hand side of equation is bounde d, and uniqueness of the solution in the given area is proved. Conditions are given for existence of several different bounded solutions.