On periodic solutions of abstract differential equations (Q1599212)

Here, the existence of periodic solutions to the linear abstract differential equation of arbitrary integer order \(N>0\) in a Banach space \(E\) is studied, \[ u_{t}^{N}(t)=Au(t), \qquad u(t+T)=u(t), \quad t\in \mathbb{R}, \] where \(A\) is a closed linear operator with domain \(D(A)\) (not necessarily dense in \(E\)) and \(T>0\) is a given number. The equivalent boundary value problem on the finite interval \([0,T]\) is \[ u_{t}^{N}(t)=Au(t),\quad 0\leq t \leq T, \qquad u^{j}(0)=u^{j}(T), \quad j=0,\dots,N-1. \] The authors study the zero and nonzero periodic classical solutions to the considered problem under conditions on eigenvalues of the operator \(A\) and in terms of the Fourier series with respect to the eigenfunctions of this operator.