Ronto A. Initial and periodic problems for functional differential equations

The existence and uniqueness of a solution of the Cauchy problem and periodic problem for functional differential equations in finite-dimensional and abstract Banach spaces are investigated. We establish general theorems on functional differential inequalities that prove to be convenient for the investigation of the unique solvability of the initial value problem for many-dimensional systems of linear functional differential equations of a general form. We establish new efficient conditions sufficient for the existence of Green's operator of the Cauchy problem for many-dimensional systems of linear functional differential equations with argument subjected to deviations of an arbitrary character (not necessarily purely retarded or advanced). The conditions obtained are formulated directly in terms of the given equation and are unimprovable in a certain sense. We also give additional conditions guaranteeing the positivity of Green's operator. We obtain efficient ! conditions necessary for the existence of non-constant periodic solutions of non-linear autonomous difference equations with globally Lipschitzian non-linearity in a Banach space. The Lipschitz condition is understood in terms of a certain subadditive cone-valued mapping in the given space. It is shown that, for such difference equations, the smaller is the Lipschitz constant, the larger are the periods of all the non-constant periodic solutions. The notion of an abstract problem of the periodic type is introduced for a functional differential equation in a Banach space. We obtain some general conditions under which problems of this kind are non-critical. For this purpose, we suggest a way to reduce the study of such problems to that of certain families of initial value problems.