Abstract
The present paper obtains the infimum of the densities for vibrating string equations in terms of the gap and ratio of the first two eigenvalues. The main result in this paper can be viewed as a new version of the Lyapunov inequality involving the first two eigenvalues. Some new estimates of the gap and ratio are obtained. Furthermore, the classical Lyapunov inequality can be deduced by the main conclusion.
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Acknowledgements
The authors gratefully acknowledge the referee for his/her comments and suggestions. This research was partially supported by the National Natural Science Foundation of China (Grants 11771253 and 11601277) and the Natural Science Foundation of Shandong Province (Grants ZR2016AM20, ZR2019MA038 and ZR2019MA050).
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Qi, J., Li, J. & Xie, B. Extremal Problems of the Density for Vibrating String Equations with Applications to Gap and Ratio of Eigenvalues. Qual. Theory Dyn. Syst. 19, 12 (2020). https://doi.org/10.1007/s12346-020-00351-y
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DOI: https://doi.org/10.1007/s12346-020-00351-y