Abstract
We consider finite dimensional nonlinear eigenvalue problems of the typeAu=λFu whereA is a matrix and(Fu) i =f(u i ),i=1, ...,m. These may be thought of as discretizations of a corresponding boundary value problem. We show that positive, spurious solution branches of the discrete equations (which have been observed in some cases in [1, 7]) typically arise iff increases sufficiently strong and ifA −1 has at least two positive columns of a certain type. We treat in more detail the casesf(u)=e u andf(u)=u α where also discrete bifurcation diagrams are given.
Zusammenfassung
Es werden endlich dimensionale, nichtlineare Eigenwertprobleme der FormAu=λFu mit einer MatrixA und einem Feld(Fu) i =f(u i ),i=1, ...,m betrachtet. Diese können als Diskretisierung eines entsprechenden Randwertproblems angesehen werden. Wir zeigen, daß diese diskreten Gleichungen dann zusätzliche, positive Lösungszweige (welche in [1,7] beobachtet wurden) aufweisen, wennf hinreichend stark wächst undA −1 mindestens zwei positive Spalten von einem bestimmten Typ besitzt. Ausführlicher, werden die Fällef(u)=e u undf(u)=u α behandelt, für die auch diskrete Verzweigungsdiagramme angegeben werden.
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Beyn, W.J., Lorenz, J. Spurious solutions for discrete superlinear boundary value problems. Computing 28, 43–51 (1982). https://doi.org/10.1007/BF02237994
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DOI: https://doi.org/10.1007/BF02237994