Journal of Integral Equations and Applications

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

Fatima-Zahra Bensidhoum and Hacen Dib

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 15 December 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

Export citation


  • O.P. Agrawal, Generalized variational problems and euler-lagrange equations, Comp. Math. Appl. 59 (2010), 1852–1864.
  • E. Bas, Fundamental spectral theory of fractional singular Sturm-Liouville operator, J. Funct. Spaces Appl. 2013 (2013), 1–7.
  • ––––, The inverse nodal problem for the fractional diffusion equation, Acta Sci. Tech. 37 (2015), 251–257.
  • E. Bas, R. Yilmazer and E.S. Panakhov, Fractional solutions of Bessel equation with $N$-method, The Scientific World J. Math. Anal. 2013, (2013), 1–8.
  • M.S. Birman and M.Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publishing Company, Amsterdam, 1987.
  • N. Dunford and J.T. Schwartz, Linear operators, Vol. II, Spectral theory, self-adjoint operators in Hilbert space, Interscience Publishers, New York, 1963.
  • M. Klimek and O.P. Agrawal, Fractional Sturm-Liouville problem, Comp. Math. Appl. 66 (2013), 795–812.
  • M. Klimek and O.P. Agrawal, On a regular fractional Sturm-Liouville problem with derivatives of order in $(0,1)$, Slovakia IEEE Explore Digital Library, (2012), 3284–289.
  • M. Klimek and M. Blasik, Regular Sturm-Liouville problem with Riemann-Liouville derivatives of order in $(1,2)$: Discrete spectrum, solutions and applications, Lect. Notes Electr. Eng. 320 (2015), 25–36.
  • ––––, Regular fractional Sturm-Liouville problem with discrete spectrum: Solutions and applications, IEEE (2014), 1–6.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
  • N.N. Lebedev, Special functions and their applications, Prentice-Hall, Inc., New York, 1965.
  • M. Mckiernan, On the $n$th derivative of composite functions, Math. Notes 63 (1956), 331–333.
  • K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley and Sons, New York, 1993.
  • I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  • M. Rivero, J.J. Trujillo and M.P. Velasco, A fractional approach to the Sturm-Liouville problem, Centr. Eur. J. Phys. 11 (2013), 1246–1254.
  • S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives, Theory and applications, Gordon and Breach Science Publishers, Amsterdam, 1993.
  • V.K. Tuan, Inverse problem for fractional diffusion equation, Fract. Calc. Appl. Anal. 14 (2011), 31–55.
  • J. Weidmann, Linear operators in Hilbert spaces, Springer-Verlag, New York, 1980.