Asymptotics for Eigenvalues of a Non-Linear Integral System

Edmunds, D.E., and Lang, J.. "Asymptotics for Eigenvalues of a Non-Linear Integral System." Bollettino dell'Unione Matematica Italiana 1.1 (2008): 105-119. <http://eudml.org/doc/290492>.

@article{Edmunds2008,
abstract = {Let $I=[a, b] \subset \mathbb\{R\}$, let $1 < q, p < \infty$, let $u$ and $v$ be positive functions with $u \in L_\{p'\}(I)$ e $v \in L_q(I)$ and let $T \colon L_p(I) \to L_q(I)$be the Hardy-type operator given by \begin\{equation*\} (Tf)(x) = v(x) \int \_a^x f(t)u(t) \, dt, \quad x\in I. \end\{equation*\} We show that the asymptotic behavior of the eigenvalues$\lambda$of the non-linear integral system \begin\{equation*\} g(x) = (TF)(x) \qquad (f(x))\_\{(p)\} = \lambda (T^\{*\}g\_\{(p)\}))(x) \end\{equation*\} (where, for example,$t_\{(p)\} = |t|^\{p-1\}\operatorname\{sgn\}(t)$is given by \begin\{align*\} &\lim \_\{n \rightarrow \infty \}n\hat\{\lambda \}\_n(T) = c\_\{p,q\} \left( \int \_I (uv)^r)^\{1/r\} \, dt \right)^\{1/r\}, \qquad \text\{for \} 1 < p < q < \infty \\ &\lim \_\{n \rightarrow \infty \} n\check\{\lambda \}\_n(T) = c\_\{p,q\} \left( \int \_I (uv)^r \, dt \right)^\{1/r\} \qquad \text\{for \} 1 < q < p < \infty \end\{align*\} Here$r = \frac\{1\}\{p'\} + \frac\{1\}\{p\}$, $c_\{p,q\}$ is an explicit constant depending only on $p$ and $q$, $\hat\{\lambda\}(T) = \max (sp_\{n\} (T, p, q))$, $\check\{\lambda\}_\{n\}(T) = \min(sp_\{n\}(T, p, q))$ where $sp_\{n\}(T, p, q)$ stands for the set of all eigenvalues $\lambda$ corresponding to eigenfunctions $g$ with $n$ zeros.},
author = {Edmunds, D.E., Lang, J.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {105-119},
publisher = {Unione Matematica Italiana},
title = {Asymptotics for Eigenvalues of a Non-Linear Integral System},
url = {http://eudml.org/doc/290492},
volume = {1},
year = {2008},
}

TY - JOUR
AU - Edmunds, D.E.
AU - Lang, J.
TI - Asymptotics for Eigenvalues of a Non-Linear Integral System
JO - Bollettino dell'Unione Matematica Italiana
DA - 2008/2//
PB - Unione Matematica Italiana
VL - 1
IS - 1
SP - 105
EP - 119
AB - Let $I=[a, b] \subset \mathbb{R}$, let $1 < q, p < \infty$, let $u$ and $v$ be positive functions with $u \in L_{p'}(I)$ e $v \in L_q(I)$ and let $T \colon L_p(I) \to L_q(I)$be the Hardy-type operator given by \begin{equation*} (Tf)(x) = v(x) \int _a^x f(t)u(t) \, dt, \quad x\in I. \end{equation*} We show that the asymptotic behavior of the eigenvalues$\lambda$of the non-linear integral system \begin{equation*} g(x) = (TF)(x) \qquad (f(x))_{(p)} = \lambda (T^{*}g_{(p)}))(x) \end{equation*} (where, for example,$t_{(p)} = |t|^{p-1}\operatorname{sgn}(t)$is given by \begin{align*} &\lim _{n \rightarrow \infty }n\hat{\lambda }_n(T) = c_{p,q} \left( \int _I (uv)^r)^{1/r} \, dt \right)^{1/r}, \qquad \text{for } 1 < p < q < \infty \\ &\lim _{n \rightarrow \infty } n\check{\lambda }_n(T) = c_{p,q} \left( \int _I (uv)^r \, dt \right)^{1/r} \qquad \text{for } 1 < q < p < \infty \end{align*} Here$r = \frac{1}{p'} + \frac{1}{p}$, $c_{p,q}$ is an explicit constant depending only on $p$ and $q$, $\hat{\lambda}(T) = \max (sp_{n} (T, p, q))$, $\check{\lambda}_{n}(T) = \min(sp_{n}(T, p, q))$ where $sp_{n}(T, p, q)$ stands for the set of all eigenvalues $\lambda$ corresponding to eigenfunctions $g$ with $n$ zeros.
LA - eng
UR - http://eudml.org/doc/290492
ER -