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On the Solution Structure of Nonlinear Hill's Equation I, Global Results
Author(s) -
Mrziglod Thomas
Publication year - 1997
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.19971880114
Subject(s) - mathematics , nonlinear system , mathematical analysis , spectrum (functional analysis) , bifurcation , infinity , boundary value problem , a priori and a posteriori , interval (graph theory) , a priori estimate , bifurcation theory , initial value problem , combinatorics , philosophy , physics , epistemology , quantum mechanics
The solution structure in W 2,p (ℝ) × ℝ of nonlinear Hill's equation is discussed in full detail. In a recent article, the existence of unbounded solution components was shown for values of the branching parameter in the gaps of the continuous spectrum of the linearized problem. This result is the starting point of further investigations concerning the existence region of the corresponding solution components. In particular, new phenomena such as asymptotic bifurcation from infinity at a specific parameter value inside of each gap of the spectrum can be shown, if the nonlinearity satisfies a growth condition. The main assumption is the concentration of the nonlinearity to a compact interval which allows the reduction to an equivalent nonlinear Sturm ‐ Liouville problem with parameter dependent boundary conditions, if the parameter does not belong to the continuous spectrum. Extending this problem to all real parameter values makes it possible to get information about the existence region of the solution components with the help of a priori bounds for solutions of the Sturm‐Liouville problem.