The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

Abstract

Abstract In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has n − 1 $n-1$ zero in ( 0 , π ) $( 0,\pi ) $ for n ∈ N $n\in \mathbb{N}$ . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

Keywords

• Conformable fractional derivatives
• Sturm–Liouville problem
• Eigenvalue gap
• Eigenvalue ratio