Abstract.
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, x 0 and x 1, thus allowing singularities in the equation. Several other basic topics such as uniqueness, continuation, extremality and periodicity are studied in our general framework.
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The research of J. Á. Cid and R. L. Pouso is partially supported by Ministerio de Educación y Ciencia, Spain, project MTM2004-06652-C03-01, and by Xunta de Galicia, Spain/FEDER, project PGIDIT05PXIC20702PN.
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Cid, J.Á., Pouso, R.L. & Enguiça, R.R. Sharp Conditions for the Existence of Solutions of Second-Order Autonomous Differential Equations. MedJM 4, 191–214 (2007). https://doi.org/10.1007/s00009-007-0112-3
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DOI: https://doi.org/10.1007/s00009-007-0112-3