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Anti‐periodic solutions for a gradient system with resonance via a variational approach
Author(s) -
Tian Yu,
Henderson Johnny
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200110
Subject(s) - mathematics , eigenvalues and eigenvectors , action (physics) , boundary value problem , mathematical analysis , resonance (particle physics) , dual (grammatical number) , periodic boundary conditions , order (exchange) , boundary values , principle of least action , classical mechanics , physics , art , literature , particle physics , quantum mechanics , finance , economics
In this paper, we investigate a second‐order resonance anti‐periodic boundary value problemq ̈ ( t ) + λ m q ( t ) + ∇ F ( t , q ( t ) ) = 0 , t ∈ [ 0 , T ] ,q ( 0 ) = − q ( T ) ,q ̇ ( 0 ) = − q ̇ ( T ) ,where λ m is the m ‐th eigenvalue of the corresponding eigenvalue problem. By using the dual least action principle, we obtain an existence result. In addition, we obtain the existence of 2T‐periodic solutions forq ̈ ( t ) + λ m q ( t ) + ∇ F ( t , q ( t ) ) = 0 , t ∈ R .