scholarly journals On a New Characterization of Some Class Nonlinear Eigenvalue Problem

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Lutfi Akin
Keyword(s):  
Asymptotic Behavior ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Normal Mode Analysis ◽  
Mode Analysis ◽  
Electrical System ◽  
Eigenvalues And Eigenfunctions ◽  
Complete Study ◽  
Nonlinear Eigenvalue

A normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. Faber made a fairly complete study of the existence and asymptotic behavior of eigenvalues and eigenfunctions, Green’s function, and expansion properties. We will investigate a new characterization of some class nonlinear eigenvalue problem.

scholarly journals On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Pavel Drábek
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Linear Case ◽  
Minimax Principle ◽  
Variational Characterization ◽  
Nonlinear Eigenvalue

We discuss nonlinear homogeneous eigenvalue problems and the variational characterization of their eigenvalues. We focus on the Ljusternik-Schnirelmann method, present one possible alternative to this method and compare it with the Courant-Fischer minimax principle in the linear case. At the end we present a special nonlinear eigenvalue problem possessing an eigenvalue which allows the variational characterization but is not of Ljusternik-Schnirelmann type.

scholarly journals Asymptotic Behavior of Solution to Nonlinear Eigenvalue Problem

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2064
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Numerical Approach ◽  
L1 Norm ◽  
Expansion Formula ◽  
Precise Asymptotic ◽  
Nonlinear Eigenvalue

We study the following nonlinear eigenvalue problem: −u″(t)=λf(u(t)),u(t)>0,t∈I:=(−1,1),u(±1)=0, where f(u)=log(1+u) and λ>0 is a parameter. Then λ is a continuous function of α>0, where α is the maximum norm α=∥uλ∥∞ of the solution uλ associated with λ. We establish the precise asymptotic formula for L1-norm of the solution ∥uα∥1 as α→∞ up to the second term and propose a numerical approach to obtain the asymptotic expansion formula for ∥uα∥1.

scholarly journals Rapid Characterization of Allosteric Networks with Ensemble Normal Mode Analysis

2016 ◽  
Vol 120 (33) ◽  
pp. 8276-8288 ◽  
Author(s):  
Xin-Qiu Yao ◽  
Lars Skjærven ◽  
Barry J. Grant
Keyword(s):  
Normal Mode ◽  
Normal Mode Analysis ◽  
Mode Analysis ◽  
Rapid Characterization

scholarly journals Lower bounds for Orlicz eigenvalues

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ariel Salort
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Lower Bounds ◽  
Dirichlet Boundary ◽  
Nonlinear Eigenvalue Problem ◽  
Dirichlet Boundary Conditions ◽  
Young Functions ◽  
New Strategies ◽  
Nonlinear Eigenvalue

<p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

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