Mediterranean Journal of Mathematics

, Volume 4, Issue 2, pp 191–214 | Cite as

Sharp Conditions for the Existence of Solutions of Second-Order Autonomous Differential Equations

  • José Ángel Cid
  • Rodrigo López Pouso
  • Ricardo Roque Enguiça


We prove general existence results for \(x^{\prime\prime} = f(x)g(x^{\prime}), x(0) = x_{0}, x^{\prime}(0) = x_{1},\) where f and g need not be continuous or monotone. Moreover neither f nor g need be bounded around, respectively, x0 and x1, thus allowing singularities in the equation. Several other basic topics such as uniqueness, continuation, extremality and periodicity are studied in our general framework.

Mathematics Subject Classification (2000).

34A12 34A36 


Second-order differential equations discontinuous differential equations existence uniqueness 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • José Ángel Cid
    • 1
  • Rodrigo López Pouso
    • 2
  • Ricardo Roque Enguiça
    • 3
  1. 1.Departamento de MatemáticasUniversidad de JaénJaénSpain
  2. 2.Departamento de Análise Matemática, Facultade de MatemáticasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  3. 3.Área Científica de MatemáticaInstituto Superior de Engenharia de LisboaLisboaPortugal

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