Abstract
We consider a Sturm–Liouville problem with Dirichlet boundary conditions and a weighted integral condition on the potential which may have singularities of different orders at the end-points of the interval (0, 1). In this article we give one extra integral condition which is required for existence of the first eigenvalue of this problem. We find the values of parameters of the weighted integral condition, for which the first eigenvalue exists. We use the variational method for finding the first eigenvalue. Showing that the first eigenvalue is not greater than \(\pi ^2\), we prove that for \(0<\gamma <1\), \(\alpha , \beta >2\gamma -1\), the upper estimate for the first eigenvalue is strictly less than \(\pi ^2\).
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Ezhak, S., Telnova, M. (2020). On Conditions on the Potential in a Sturm– Liouville Problem and an Upper Estimate of its First Eigenvalue. In: Pinelas, S., Graef, J.R., Hilger, S., Kloeden, P., Schinas, C. (eds) Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. https://doi.org/10.1007/978-3-030-56323-3_37
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DOI: https://doi.org/10.1007/978-3-030-56323-3_37
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