Asymptotics for Eigenvalues of a Non-Linear Integral System

Edmunds, D.E. ; Lang, J.

Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 1, pp. 105-119.

Abstract (VO)

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 \to \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 \to \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.

Riassunto

Sia $I=[a, b]$ un sottinsieme di $\mathbb{R}$. Siano $1 q, p \infty$ siano $u$ e $v$ funzioni positive, con $u \in L_{p'}(I)$ e $v \in L_q(I)$. Sia $T \colon L_p(I) \to L_q(I)$ un operatore di Hardy definito nel modo seguente: \begin{equation*} (Tf)(x) = v(x) \int_a^x f(t)u(t) \, dt, \quad x\in I. \end{equation*} Dimostereremo che il comportamento asintotico degli autovalori $\lambda$ nel sistema integrale non lineare \begin{equation*} g(x) = (TF)(x) \qquad (f(x))_{(p)} = \lambda(T^{*}g_{(p)}))(x) \end{equation*} (dove, per esempio $t_{(p)} = |t|^{p-1}\operatorname{sgn}(t)$) è dato da \begin{align*} \lim_{n \to \infty}n\hat{\lambda}_n(T) = c_{p,q} \left( \int_I (uv)^r)^{1/r} \, dt \right)^{1/r}, \qquad \text{quando } 1 p q \infty \\ \lim_{n \to \infty} n\check{\lambda}_n(T) = c_{p,q} \left( \int_I (uv)^r \, dt \right)^{1/r} \qquad \text{quando } 1 q p \infty \end{align*} Qui $r = \frac{1}{p'} + \frac{1}{p}$, $c_{p,q}$ \`e una costante esplicita che dipende solo da $p$ e $q$, $\hat{\lambda}(T) = \max (sp_{n} (T, p, q))$, $\check{\lambda}_{n}(T) = \min(sp_{n}(T, p, q)$), dove $sp_{n}(T, p, q)$ rappresenta l'insieme di tutti gli autovalori $\lambda$ che corrispondono alle autofunzioni $g$ con $n$ zeri.

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