Periodic Solutions of the Riccati Equation in Banach Spaces

Abstract

Abstract. In this paper we study the problem of the existence and the construction of periodic solutions of the Riccati equation with continuous periodic coefficients defined on the real line with values in Banach space. IntroductionThe theory of invariant manifolds, especially the theory of center manifolds, yields an important contribution to the study of some systems of differential equations [2]- [11]. The main theorems of the invariant manifold theory have been already proved, generally speaking, for quasi-linear systems with a block-diagonal structure of their linear parts.For this reason an important problem arises how to construct a transformation, which transforms an arbitrary system of differential equations to such a form. However, this transformation significantly complicates the nonlinear part of the system. On the other hand, some results of the invariant manifold theory may be obtained for so-called systems of special form, which are differential systems with block-triangular structure of the linear part [4;5], [11]. In this connection a crucial question arises: how to construct a transformation, which transforms this system to a system of special form. To this end, we must construct a solution of the corresponding differential Riccati equation. We also note that systems of special form can be easily transformed to systems with block-diagonal structure of their linear parts. If the given and obtained systems are periodic, then the solution of the Riccati equation is also periodic. This motivates the study of periodic solutions of the Riccati equation.1991 Mathematics Subject Classification: 34A05, 34C25, 34G20.