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The zeros of the solutions of the fractional oscillation equation

We consider the zeros of the solution ℧α(t) = Eα(−tα), 1 < α < 2, of the fractional oscillation equation in terms of the Mittag-Leffler function, and give a wholly and clarified description for these zeros. We find that the number of zeros can be any finite number: 1, 2, 3, 4, ..., not necessarily an odd number. When the number of zeros of ℧α(t) is an even number, ℧α(t) has a critical zero. All of the values of α for which ℧α(t) has an even number of zeros constitute a countable set S. For each α ∈ (1, 2) \ S, ℧α(t) has an odd number of zeros. These results are a supplement and a perfecting for the existed related documents. We also show that the eigenvalue problems are related with the zeros of the Mittag-Leffler functions.