Classification of positive convex functions according to focal equivalence

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The number of solutions of the differential equation ẍ(t)+g(x(t)) = 0 subject to the boundary conditions x(t1) = x1, x(t2) = x2 is studied. The set of the points (t1, t2, x1, x2) for which the boundary value problem has i solutions is denoted by Σi(g). These sets for i = 0, 1, 2,... form a partition of ℝ4 which is called a focal decomposition. Two differential equations are called focally equivalent if there is a homeomorphism of ℝ4 mapping their focal decompositions into each other. Here the equations with convex positive g are classified according to focal equivalence. It is shown that there are at least four classes, and the focal decomposition is determined by the asymptotic properties of g at infinity.

Original languageEnglish
Pages (from-to)519-533
Number of pages15
JournalIMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)
Issue number4
Publication statusPublished - Aug 1 2006



  • Boundary value problem
  • Convex nonlinearity
  • Focal decomposition

ASJC Scopus subject areas

  • Applied Mathematics

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