SPECTRAL CRITERIA FOR PERIODIC AND ALMOST PERIODIC S OLUTIONS

The problem of our primary concern in this chapter is to find spectral conditions for the existence of almost periodic solutions of periodic equations. Although the theory for periodic equations can be carried out parallelly to that for autonomous equations, there is always a difference between them. This is because that in gen­ eral there is no Floquet representation for the monodromy operators in the infinite dimensional case . Section 1 will deal with evolution semigroups acting on invariant function spaces of AP(X) . Since, originally, this technique is intended for nonau­ tonomous equations we will treat equations with as much nonautonomousness as possible, namely, periodic equations . The spectral conditions are found in terms of spectral properties of the monodromy operators . Meanwhile, for the case of au­ tonomous equations these conditions will be stated in terms of spectral properties of the operator coefficients. This can be done in the framework of evolution semi­ groups and sums of commuting operators in Section 2. Section 3 will be devoted to the critical case in which a fundamental technique of decomposition is presented. In Section 4 we will present another, but traditional, approach to periodic solutions of abstract functional differential equations . The remainder of the chapter will be devoted to several extensions of these methods to discrtete systems and nonlinear equations. As will be shown in Section 5, many problems of evolution equations can be studied through discrete systems with less sophisticated notions .