scholarly journals Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property

2016 ◽  
Vol 135 (2) ◽  
pp. 397-430 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

RESTARTING PROJECTION METHODS FOR RATIONAL EIGENPROBLEMS ARISING IN FLUID‐SOLID VIBRATIONS

2008 ◽  
Vol 13 (2) ◽  
pp. 171-182 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Solid Structure ◽  
Minmax Characterization ◽  
Nonlinear Eigenvalue

For nonlinear eigenvalue problems T(λ)x = 0 satisfying a minmax characterization of its eigenvalues iterative projection methods combined with safeguarded iteration are suitable for computing all eigenvalues in a given interval. Such methods hit their limitations if a large number of eigenvalues is required. In this paper we discuss restart procedures which are able to cope with this problem, and we evaluate them for a rational eigenvalue problem governing vibrations of a fluid‐solid structure.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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