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Complexity
Volume 2017, Article ID 3720471, 7 pages
https://doi.org/10.1155/2017/3720471
Research Article

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

1Department of Mathematical Sciences, UAE University, P.O. Box 15551, Al Ain, Abu Dhabi, UAE
2Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia

Correspondence should be addressed to Mohammed Al-Refai; ea.ca.ueau@iaferla_m

Received 7 May 2017; Accepted 6 August 2017; Published 14 September 2017

Academic Editor: Abdelalim Elsadany

Copyright © 2017 Mohammed Al-Refai and Thabet Abdeljawad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.