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Unbounded solution components for nonlinear Hill's equations

Published online by Cambridge University Press:  14 November 2011

Thomas Mrziglod
Thomas Mrziglod
Affiliation:
Mathematisches Institut, Universität zu Köln, D-50923 Köln, Germany
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Abstract

A class of nonlinear Hill's equations on ℝ is considered, where the nonlinearity is concentrated on a compact interval [−N, N]. For values of the parameter λ not in the spectrum of the linearised equation (which is purely continuous) an equivalent nonlinear Sturm–Liouville problem on [−N, N] with parameter-dependent boundary conditions at x = ± N is given. Extending this problem to all real values of the parameter in a suitable way makes it possible to prove the existence of unbounded solution components for both the extended Sturm–Liouville problem and the original problem. The complicated structure of the extended problem results in new phenomena. For example, the number of zeros of different functions in the same solution component may be different.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 6 , 1995 , pp. 1131 - 1167
Copyright
Copyright © Royal Society of Edinburgh 1995

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