The Set of First-Order Differential Equations with Periodic or Bounded Solutions

objective of this note is the announcement of two results of Ambrosetti-Prodi type concerning the existence of periodic (respectively bounded) solu-tions of the first order differential equation x ′ = f(t, x) 1. Periodic solutions Let us fix a real number T> 0 and define C as the set of all continuous functions f: R × R → R such that: A1) f(t, x) is T-periodic in t. A2) f(t, x) is locally Lipschitz continuous in x. A3) f(t, x) is concave in x and there exists t0 = t0(f) ∈ R such that f(t0, x) is strictly concave in x. A4) lim|x|→ ∞ f(t, x) = − ∞ uniformly on t ∈ R. In C we shall consider the topology of uniform convergence on compact sets. We also define C0 as the subset of C consisting of all points f such that the equation x ′ = f(t, x) (1) has a unique T-periodic solution. ∗Partially supported by D.G.E.S. PB96-1462 ∗∗Partially supported by CDCHT, Universidad de los Andes 293 294 j.l. bravo, m. fernández, a. tine