Multiple periodic solutions of small vector fields on differentiable manifolds (Q1336343)

Let \(M\) be an \(m\)-dimensional smooth manifold in \(\mathbb{R}^ k\). Consider the system of differential equations \[ x' = \varepsilon u(x,t), \tag{1.1} \] where \(\varepsilon \neq 0\) is a small parameter, and \(u : M \times \mathbb{R} \to \mathbb{R}^ k\) is continuous and 1-periodic in \(t \in \mathbb{R}\). The main purpose of this paper is to prove the existence of multiple 1- periodic solutions of (1.1). The author presents several existence theorems in this topic under various conditions. Then, in the last part of the paper, he considers (1.1) on more general spaces, namely on closed ANRs in \(\mathbb{R}^ k\).