Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 62, pp. 1-14.

Title: Smallest eigenvalues for boundary value problems of two term
       fractional differential operators depending on fractional boundary conditions
       
Authors: Paul W. Eloe (Univ. of Dayton, Dayton, OH, USA)
         Jeffrey T. Neugebauer (Eastern Kentucky Univ., Richmond, KY, USA)

Abstract: 
 Let n≥2 be an integer, and let n-1<α≤n. We consider eigenvalue problems
 for two point n-1 ,1 boundary value problems
 $$\displaylines{
 D_{0+}^{\alpha} u+a(t)u+\lambda p(t)u=0, \quad 0<t<1, \cr
 u^{(i)}(0)=0, \quad i=0,1,\dots,n-2, \quad D_{0^+}^{\beta} u(1)=0,
 }$$
 where 0&lte; &beta; &lte;n-1 and $D_{0+}^{\alpha}$ and $D_{0+}^{\beta}$ denote standard
 Riemann-Liouville differential operators. We prove the existence of smallest positive
 eigenvalues and then obtain comparisons of these smallest eigenvalues as functions of
 both p and &beta;.

Submitted October 14, 2020. Published July 07, 2021.
Math Subject Classifications: 26A33, 34A08, 34A40, 26D20
Key Words: Riemann-Liouville fractional differential equation;
           boundary value problem; principal eigenvalue; fractional boundary conditions