Abstract
By a node of Sturm–Liouville problems, it means an interior zero of an eigenfunction. In this paper, when nodes are considered as implicitly defined nonlinear functionals of potentials from Lebesgue spaces, we will provide two basic results. One is that nodes are continuously Fréchet differentiable in potentials when the usual norms of potentials are considered. Moreover, the Fréchet derivatives will be given using the corresponding eigenfunctions. The other is that nodes are completely continuous in potentials when the weak topologies for potentials are considered. The latter means that nodes are continuously dependent on potentials in a very strong way, i.e., when a sequence of potentials is only weakly convergent to a potential, the nodes are still convergent to the corresponding node. These two results will be applied in future works to solve some optimization problems on nodes as well as the recovery of potentials from partial information on nodes.
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This work is supported by the National Natural Science Foundation of China (Grant no. 11790273).
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This author is supported by the National Natural Science Foundation of China (Grant no. 11790273)
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Guo, S., Zhang, M. On the Dependence of Nodes of Sturm–Liouville Problems on Potentials. Mediterr. J. Math. 19, 168 (2022). https://doi.org/10.1007/s00009-022-02100-8
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DOI: https://doi.org/10.1007/s00009-022-02100-8