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Erschienen in:
2013 | OriginalPaper | Buchkapitel
2. Background Material and Notation
verfasst von : Todd Kapitula, Keith Promislow
Erschienen in: Spectral and Dynamical Stability of Nonlinear Waves
Verlag: Springer New York
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Abstract
This chapter provides an overview of the background material assumed in the remainder of the book. We state major results and provide sketches of the less technical proofs, particularly where the ideas presented are instrumental in subsequent constructions. The first topic is the theory of linear systems of ordinary differential equations (ODEs). Much of this material is standard for first-year graduate courses in ODEs, such as presented in Hartman [114], Perko [234]. This is followed by a review of the basic theory of functional analysis as applied to linear partial differential equations; further details can be found in the standard references Evans [81], Kato [162]. We finish the general overview by discussing the point spectrum in the context of the Sturm–Liouville theory for second-order operators. These operators have a one-to-one relationship between the ordering of the eigenvalues and the number of zeros for the associated eigenfunctions, which is extremely useful in applications.