Premium
On the Eigenvalues of Half‐Linear Boundary Value Problems
Author(s) -
Eberhard Walter,
Elbert Árpád
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/(sici)1522-2616(200005)213:1<57::aid-mana57>3.0.co;2-j
Subject(s) - mathematics , eigenvalues and eigenvectors , sign (mathematics) , boundary value problem , mathematical analysis , value (mathematics) , function (biology) , weight function , boundary (topology) , boundary values , statistics , physics , quantum mechanics , evolutionary biology , biology
We consider the half‐linear boundary value problem$$x^{\prime \prime } \, \vert x^{\prime} \vert ^{n-1} + \lambda ^{{n+1}^{\ast}} q(t)x^{n^{\ast}} = 0 \, , \qquad x(0) = x(b) = 0$$where $x^{n^{\ast}} = \vert x \vert ^n {\rm sgn} x$ and the weight function q is assumed to change sign. We prove the existence of two sequences $\left ( \lambda ^+ _k \right )_{k \in {N}}$ , $\left ( \lambda ^- _k \right )_{k \in {N}}$ of eigenvalues and derive asymptotic estimates for $\lambda ^\pm _k$ as $k \to \infty$ .