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Necessary and Sufficient Conditions for Existence and Uniqueness of Solutions of Second‐Order Autonomous Differential Equations
Author(s) -
Pouso Rodrigo López
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610705006319
Subject(s) - uniqueness , mathematics , order (exchange) , interval (graph theory) , scalar (mathematics) , mathematical analysis , differential equation , pure mathematics , combinatorics , geometry , finance , economics
The main result ensures that the scalar problem x ″ = f ( x ), x (0) = x 0 , x ′(0) = x 1 , has a nonconstant locally W 2, 1 solution if and only if there exists a nontrivial interval J such that x 0 ∈ J , f ∈ L loc 1 ( J ) , x 1 2 + 2 ∫ x 0 y f ( s ) d s > 0for almost all y ∈ J andmax { 1 , | f |}x 1 2 + 2 ∫ x 0 ˙ f ( s ) d s∈ L loc 1 ( J )Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.