Global Bifurcation in 2m-Order Generic Systems of Nonlinear Boundary Value Problems. (18th December 2012) - Han, Xiaoling; Xu, Jia; Dai, Guowei
Abstract : We consider the systems of ( - 1 ) m u ( 2 m ) = λ u + λ v + u f ( t, u, v ),    t ∈ ( 0, 1 ),    u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0, and 0 ≤ i ≤ m - 1,    ( - 1 ) m v ( 2 m ) = μ u + μ v + v g ( t, u, v ),    t ∈ ( 0, 1 ),    v ( 2 i ) ( 0 ) = v ( 2 i ) ( 1 ) = 0,    0 ≤ i ≤ m - 1, where λ, μ ∈ R are real parameters. f, g : [ 0, 1 ] × R 2 → R are C k, k ≥ 3 functions and f ( t, 0, 0 ) = g ( t, 0, 0 ) = 0, t ∈ [ 0, 1 ] . It will be shown that if the functions, f and g are "generic" then the solution set of the systems consists of a countable collection of 2-dimensional, C k manifolds.