Open Access
2016 On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions
Fatima-Zahra Bensidhoum, Hacen Dib
J. Integral Equations Applications 28(4): 459-480 (2016). DOI: 10.1216/JIE-2016-28-4-459

Abstract

The present work deals with some spectral properties of the problem

\medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$,

\vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip

\noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda $ is a scalar parameter.

First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the corresponding eigenfunctions form a complete orthonormal system in the Hilbert space $L^{2}_q[a,b]$. Then, we investigate some asymptotic properties of the spectrum as $\alpha \underset {\lt }{\rightarrow } 1$. We give, in particular, the asymptotic expansion of the first eigenvalue.

Citation

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Fatima-Zahra Bensidhoum. Hacen Dib. "On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions." J. Integral Equations Applications 28 (4) 459 - 480, 2016. https://doi.org/10.1216/JIE-2016-28-4-459

Information

Published: 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1358.34035
MathSciNet: MR3582798
Digital Object Identifier: 10.1216/JIE-2016-28-4-459

Subjects:
Primary: 26A33 , 34A08
Secondary: 34B24 , 34B27 , 34L05 , 34L10 , 34L15 , 47G10

Keywords: fractional Green's function , fractional Sturm-Liouville problem , Hilbert-Schmidt operators , min-max principle , Right- and left-sided Riemann-Liouville fractional derivatives

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.28 • No. 4 • 2016
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