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

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.

Keywords

• Fractional boundary value problem
• smallest eigenvalues
• $u_0$-positive operator