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Bulletin of the London Mathematical Society 2005 37(2):213-223; doi:10.1112/S002460930400390X
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© London Mathematical Society

Localization of Closed (or Periodic) Solutions of a Differential System with Concave Nonlinearities

Allan Sandqvist and Kurt Munk Andersen

Department of Mathematics, Technical University of Denmark DK 2800 Kongens Lyngby, Denmark A.Sandqvist{at}mat.dtu.dk, K.M.Andersen{at}mat.dtu.dk

Received 5 September 2003. Revision received 24 February 2004.

Consider a scalar differential equation Formula, where I is an open interval containing [0,T]. Assume that f(t, x) is continuous with a continuous derivative Formula, and weakly concave (or weakly convex) in x for all t isin I, though strictly concave (or strictly convex) for some t isin [0, T]. It is well known that in this case there can be either no, one or two closed solutions; that is, solutions {phi}(t) for which {phi}(0) = {phi}(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 Formula. The requirements are that the coordinate functions Formula) be continuous with continuous derivatives with respect to x1, x2, ...,xn, that the fj are weakly concave (or weakly convex) in Formula, and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled. 2000 Mathematics Subject Classification 34C25, 34C12.


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