Localization of Closed (or Periodic) Solutions of a Differential System with Concave Nonlinearities |
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Authors: | Sandqvist Allan; Andersen Kurt Munk |
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Institution: | 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 |
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Abstract: | Consider a scalar differential equation , where I is an open interval containing 0,T]. Assumethat f(t, x) is continuous with a continuous derivative , 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 therecan be either no, one or two closed solutions; that is, solutions (t) for which (0) = (T) If there are two closed solutions, thenthe greater has a negative characteristic exponent and the smallerhas a positive one. It is easily seen that this is equivalentto a statement on localization of closed solutions. It is shownhow this statement can be generalized to systems of differentialequations . The requirements are that the coordinate functions ) be continuous with continuous derivatives with respect to x1,x2, ...,xn, that the fj are weakly concave (or weakly convex)in , and that a certain condition pertaining to strict concavity (or strict convexity) is fulfilled.2000 Mathematics Subject Classification 34C25, 34C12. |
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