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We show the direct applicability of the Brouwer fixed point theorem for theexistence of equilibrium points and periodic solutions for differential systemson general domains satisfying geometric conditions at the boundary. We developa general approach for arbitrary bound sets and present applications to thecase of convex and star-shaped domains. We also provide an answer to a questionraised in a recent paper of Cid and Mawhin.

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains