Localization of closed (or periodic) solutions of a differential system with concave nonlinearities

Consider a scalar differential equation x˙=f(t,x),(t,x)∈I×R, where I is an open interval containing [0,T]. Assume that f(t, x) is continuous with a continuous derivative fx′(t,x), and weakly concave (or weakly convex) in x for all t ∈ I, though strictly concave (or strictly convex) for some t ∈ [0, T].

It is well known that in this case there can be either no, one or two closed solutions; that is, solutions ϕ(t) for which ϕ(0) = ϕ(T) If there are two closed solutions, then the greater has a negative characteristic exponent and the smaller has a positive one. It is easily seen that this is equivalent to a statement on localization of closed solutions.

It is shown how this statement can be generalized to systems of differential equations x˙_=f_(t,x_),(t,x_)∈I×Rn. The requirements are that the coordinate functions fj(t,x_)) be continuous with continuous derivatives with respect to x1, x2, …,xn, that the fj are weakly concave (or weakly convex) in x_, and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled. 2000 Mathematics Subject Classification 34C25, 34C12.