Localization Theorems for Nonlinear Eigenvalue Problems
نویسندگان
چکیده
Abstract. Let T : Ω → C be a matrix-valued function that is analytic on some simplyconnected domain Ω ⊂ C. A point λ ∈ Ω is an eigenvalue if the matrix T (λ) is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.
منابع مشابه
Locating Real Eigenvalues of a Spectral Problem in Fluid-solid Type Structures
Exploiting minmax characterizations for nonlinear and nonoverdamped eigenvalue problems, we prove the existence of a countable set of eigenvalues converging to ∞ and inclusion theorems for a rational spectral problem governing mechanical vibrations of a tube bundle immersed in an incompressible viscous fluid. The paper demonstrates that the variational characterization of eigenvalues is a power...
متن کاملSubcritical Perturbations of Resonant Linear Problems with Sign-changing Potential
We establish existence and multiplicity theorems for a Dirichlet boundary-value problem at resonance. This problem is a nonlinear subcritical perturbation of a linear eigenvalue problem studied by Cuesta, and includes a sign-changing potential. We obtain solutions using the Mountain Pass lemma and the Saddle Point theorem. Our paper extends some recent results of Gonçalves, Miyagaki, and Ma.
متن کاملLp Embedding and Nonlinear Eigenvalue Problems for Unbounded Domains
Let R denote real iV-dimensional Euclidean space. Then it is a well-known fact that the imbedding of the Sobolev space Wi,2(R) in LP(R ) is bounded for 2g>pS2N/(N-2), but is definitely not compact. Consequently the theory of critical points for general isoperimetric variational problems defined over arbitrary unbounded domains in R has been little investigated despite its importance. Indeed the...
متن کاملQuadratic Eigenproblems of Restricted Rank — Remarks on a Paper of Conca, Duran and Planchard
In [2] Conca et al. stated two inclusion theorems for quadratic eigenvalue problems the proof of which are not complete. In this note we demonstrate by simple examples that the assertions as they stand are false. Taking advantage of an appropriate enumeration for eigenvalues of nonlinear eigenproblems we adjust the results.
متن کاملMountain pass and linking type sign-changing solutions for nonlinear problems involving the fractional Laplacian
where ⊂Rn (n≥ 2) is a bounded smooth domain, s ∈ (0, 1), (– )s denotes the fractional Laplacian, λ is a real parameter, the nonlinear term f satisfies superlinear and subcritical growth conditions at zero and at infinity. When λ≤ 0, we prove the existence of a positive solution, a negative solution and a sign-changing solution by combing minimax method with invariant sets of descending flow. Wh...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- SIAM Review
دوره 57 شماره
صفحات -
تاریخ انتشار 2013