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The ratio of eigenvalues of the Dirichlet eigenvalue problem for equations with one-dimensional \(p\)-Laplacian. (English) Zbl 1215.34034

The authors investigate the ratio of the eigenvalues \(\{\lambda_n\}_{n\geq 1}\) of the boundary value problem
\[ \big(\Phi(x')\big)'+c(t)\Phi(x)= \lambda\Phi(x) \]
with a one-dimensional \(p\)-Laplacian \((\Phi(x'))'=(|x'|^{p-2}(x'))'\), \(p>1\), a nonnegative differentiable function \(c\), and Dirichlet boundary conditions. They show that
\[ \frac{\lambda_n}{\lambda_m}\leq \frac{n^p}{m^p},\quad n>m. \]
This extends the result of an article by M. Horváth and M. Kiss [Proc. Am. Math. Soc. 134, No. 5, 1425–1434 (2006; Zbl 1098.34067)], where the linear case \(p=2\) was considered.

MSC:

34B24 Sturm-Liouville theory
34B09 Boundary eigenvalue problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47E05 General theory of ordinary differential operators

Citations:

Zbl 1098.34067
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Full Text: DOI

References:

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