### Abstract

The number of solutions of the differential equation ẍ(t)+g(x(t)) = 0 subject to the boundary conditions x(t_{1}) = x_{1}, x(t_{2}) = x_{2} is studied. The set of the points (t_{1}, t_{2}, x_{1}, x_{2}) for which the boundary value problem has i solutions is denoted by Σ_{i}(g). These sets for i = 0, 1, 2,... form a partition of ℝ^{4} which is called a focal decomposition. Two differential equations are called focally equivalent if there is a homeomorphism of ℝ^{4} 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 language | English |
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Pages (from-to) | 519-533 |

Number of pages | 15 |

Journal | IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications) |

Volume | 71 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 1 2006 |

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### Keywords

- Boundary value problem
- Convex nonlinearity
- Focal decomposition

### ASJC Scopus subject areas

- Applied Mathematics