Exact multiplicity for periodic solutions of a first-order differential equation

Consider the differential equation x + f (t,x) = h(t), (1) where h(t) is a 1-periodic continuous function and f (t,x) ∈ C3 is concave-convex in x and 1-periodic in t . We obtain the complete structure of 1-periodic solutions by means of singularity theory. More precisely, we show that the image of singularities F(Σ) consists of a codimension-1 manifold that divides the C(R/Z) into two open sets A1,A3. (i) Equation (1) has a unique 1-periodic solution for h(t) ∈ A1, (ii) the equation has exactly three 1-periodic solutions for h(t) ∈ A3. Furthermore, if the image of cusp singularities F(C) is a codimension-1 manifold of F(Σ) the differential equation has exactly two solutions for h(t) ∈ F(Σ) \ F(C), and has a unique 1-periodic solution for h(t) ∈ F(C).  2004 Elsevier Inc. All rights reserved.