Journal of Integral Equations and Applications

On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions

Fatima-Zahra Bensidhoum and Hacen Dib

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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.

Article information

Source
J. Integral Equations Applications, Volume 28, Number 4 (2016), 459-480.

Dates
First available in Project Euclid: 15 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1481792836

Digital Object Identifier
doi:10.1216/JIE-2016-28-4-459

Mathematical Reviews number (MathSciNet)
MR3582798

Zentralblatt MATH identifier
1358.34035

Subjects
Primary: 26A33: Fractional derivatives and integrals 34A08: Fractional differential equations
Secondary: 34B24: Sturm-Liouville theory [See also 34Lxx] 34B27: Green functions 34L05: General spectral theory 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 47G10: Integral operators [See also 45P05]

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

Citation

Bensidhoum, Fatima-Zahra; Dib, Hacen. On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions. J. Integral Equations Applications 28 (2016), no. 4, 459--480. doi:10.1216/JIE-2016-28-4-459. https://projecteuclid.org/euclid.jiea/1481792836


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