A decomposition theorem for bounded solutions and the existence of periodic solutions to periodic differential equations
A decomposition theorem which describes the bounded uniformly continuous mild solutions to the equation \[ \frac{dx}{dt}=A(t)x + f(t),\quad x\in X,\quad t\in \mathbb{R},\tag{1} \] is given. Here, \(X\) is a Banach space, \(A(\cdot)\) is a \(\tau\)-periodic operator on \(X\) and \(f\) is an \(X\)-valued \(\tau\)-periodic continuous function. The authors formulate the result: Let \(u(\cdot)\) be a bounded uniformly continuous mild solution to (1). Then it is of the form \[ u(t) = u_0(t)+\sum_{k=1}^Ne^{i\lambda_k t}u_k(t), \] where \(u_0\) is a bounded uniformly continuous mild \(\tau\)-periodic solution to the inhomogeneous equation (1), \(u_k\), \(k=1,\dots{},N\), are \(\tau\)-periodic solutions to (1) with \(f=-i\lambda_k u_k\), respectively, \(v(t) = \sum_{k=1}^Ne^{i\lambda_k t}u_k(t)\) is a quasi-periodic mild solution to the homogeneous equation \[ \frac{dx}{dt}=A(t)x,\quad t\in \mathbb{R}, \tag{2} \] and \(\lambda_1,\dots{},\lambda_N\) are the basic oscillatory exponents of (2). As a corollary, the existence of such a solution implies the existence of a \(\tau\)-periodic bounded uniformly continuous mild solution to (1) and a formula for its Fourier coefficients. Similar results as the first one and of its consequence, are proved for abstract functional-differential equations with finite delay \(x_t\), \[ \frac{dx}{dt}=Ax+F(t)x_t + f(t), \quad t\in \mathbb{R}, \] where \(A\) generates a compact \(C_0\)-semigroup, \(F(t)\) and \(f\) are \(\tau\)-periodic and continuous. Finally, an example of an inhomogeneous parabolic equation is considered.
abstract functional-differential equation
scientific article; zbMATH DE number 1417564