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Journal of the London Mathematical Society 2005 71(2):397-414; doi:10.1112/S0024610705006319
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© The London Mathematical Society

Necessary and Sufficient Conditions for Existence and Uniqueness of Solutions of Second-Order Autonomous Differential Equations

Rodrigo López Pouso

Departamento de Análise Matemática, Universidade de Santiago de Compostela 15782 Santiago de Compostela, Spain rodrigolp{at}usc.es

Received 26 February 2004. Revision received 26 July 2004.

The main result ensures that the scalar problem x'' = f(x), x(0) = x0, x'(0) = x1, has a nonconstant locally W2, 1 solution if and only if there exists a nontrivial interval J such that x0 isin J, Formula for almost all y isin J and

Formula

Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.


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