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

Authors:	José Ángel Cid Rodrigo López Pouso Ricardo Roque Enguiça

Institution:	(1) Departamento de Matemáticas, Universidad de Jaén, Campus Las Lagunillas, Jaén, Spain;(2) Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain;(3) área Científica de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1-1950-062 Lisboa, Portugal

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. 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.

Keywords:	34A12 34A36
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