Electron. J. Diff. Equ., Vol. 2014 (2014), No. 43, pp. 1-10.

Existence and comparison of smallest eigenvalues for a fractional boundary-value problem

Paul W. Eloe, Jeffrey T. Neugebauer

Abstract:
The theory of $u_0$-positive operators with respect to a cone in a Banach space is applied to the fractional linear differential equations
$$ D_{0+}^{\alpha} u+\lambda_1p(t)u=0\quad\text{and}\quad D_{0+}^{\alpha} u+\lambda_2q(t)u=0, $$
$0< t< 1$, with each satisfying the boundary conditions $u(0)=u(1)=0$. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained.

Submitted August 22, 2013. Published February 10, 2014.
Math Subject Classifications: 26A33
Key Words: Fractional boundary value problem; smallest eigenvalues; $u_0$-positive operator.

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Paul W. Eloe
Department of Mathematics, University of Dayton
Dayton, OH 45469, USA
email: peloe1@udayton.edu
Jeffrey T. Neugebauer
Department of Mathematics and Statistics
Eastern Kentucky University
Richmond, KY 40475, USA
email: jeffrey.neugebauer@eku.edu

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