The rotation number approach to the periodic Fucik spectrum

[摘要] In this paper, we study the Fucik spectrum of the problem: (*) x + (lambda(+) + q(+)(t))x(+) +(lambda_ + q_(t))x_ = 0 with the 2pi-periodic boundary condition, where q+/-(t) are 2pi-periodic. After introducing a rotation number function rho(lambda(+),lambda_) for (*). we prove using the Hamiltonian structure and the positive homogeneity of (*) that for any positive integer n, the two boundary curves of the domain rho(-1)(n/2) in the (lambda(+),lambda(-))-plane are Fucik curves of (*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fucik spectrum with the Dirichlet condition. In particular, it remains open if the Fucik spectrum of (*) is composed of only these curves. (C) 2002 Fisevier Science (USA).

[发布日期] 2002-10-10