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We consider a regular fractional Sturm-Liouville problem (FSLP):
\begin{align}
&& {\cal{L}}y(x)=D^{\alpha}_{b-}p(x)D^{\alpha}_{a+}y(x) = \lambda y(x) \label{s1}\\
&& y(a) =0 \quad y(b)=0,\label{s2}\\
&& y'(a) =0 \quad D^{\alpha}_{a+}y(x)|_{x=b}=0,\label{s3}
\end{align}
where order of Riemann-Liouville derivatives \(\alpha \in \left (1, 2\right)\) and \(p\) is an arbitrary positive function from the \(C[a,b]\)-space.
The Riemann-Liouville derivatives, respectively the left and the right, are the following
integro-differential operators
\begin{align}
&& D^{\alpha}_{a+}f(x):= \frac{1}{\Gamma(2-\alpha)} \frac{d^{2}}{dx^{2}} \int_{a}^{x} (x-v)^{1-\alpha}f(v)\;dv \nonumber\\
&& D^{\alpha}_{b-}f(x):= \frac{1}{\Gamma(2-\alpha)} \frac{d^{2}}{dx^{2}} \int_{x}^{b} (v-x)^{1-\alpha}f(v)\;dv. \nonumber
\end{align}
Using the inverse integral operator approach we prove the theorem below.
Theorem Fractional Sturm-Liouville problem \eqref{s1}-\eqref{s3}
has an infinite countable set of positive, simple eigenvalues: \(\Lambda_1 <\Lambda_2<....\) and the corresponding orthonormal set of
differentiable eigenfunctions
is a basis in the \(L^{2}(a,b)\) - space, provided \(\frac{3}{2}<\alpha<2\)
and \(p\in C[a,b]\) is an arbitrary positive function.
The obtained result extends methods of solving such problems, known from the classical Sturm-Liouville
theory, as well as recent results for problems with derivatives of fractional order \(\alpha\in (0,1)\) [1].
It also is an alternative approach to variational methods, effective in the case of FSLP
with Caputo fractional derivatives [2].
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References-
M. Klimek and M. Błasik, Regular Fractional Sturm-Liouville problem with
discrete spectrum: solutions and applications . To be published in: Proceedings of the 2014 International
Conference on Fractional Differentiation and Its Applications, Catania 23-25 June, Italy.
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M. Klimek, T. Odzijewicz and A. Malinowska, Variational methods for the fractional Sturm-Liouville problems , J. Math. Anal. Appl. 416, 2014, 402--426. dx.doi.org/10.1016/j.jmaa.2014.02.009.
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