Geodesic
On the number of real eigenvalues of a~certain boundary-value problem for a~second-order equation with fractional derivative
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 137-155

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The asymptotics as $\alpha\to0+$ of the number of real eigenvalues $\lambda_n(\alpha)$ of the problem $y''(x)+\lambda D_{0}^{\alpha}y(x)=0$, $0$, $y(0)=y(1)=0$, is found. The minimization of real eigenvalues was carried out. It is proved that $\lim\limits_{\alpha\to0+}\lambda_n(\alpha)=(\pi n)^2$. 
@article{FPM_2006_12_6_a8,
     author = {A. Yu. Popov},
     title = {On the number of real eigenvalues of a~certain boundary-value problem for a~second-order equation with fractional derivative},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {137--155},
     publisher = {mathdoc},
     volume = {12},
     number = {6},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a8/}
}
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A. Yu. Popov. On the number of real eigenvalues of a~certain boundary-value problem for a~second-order equation with fractional derivative. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 137-155. http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a8/