Left focal points for Caputo fractional differential equations

Let $\beta >0$ and assume $0<b\le \beta .$ Let $n\ge 2$ denote an integer and let $n-1<\alpha \le n.$ The theory of $u_0$-positive operators with respect to a cone in a Banach space was applied to study eigenvalue problems for left focal boundary value problems for Caputo fractional linear differential equations. Under suitable conditions, it was first established that there exists $0<\delta <\beta$ such that if $0<b<\delta ,$ the left focal boundary value problem had a unique solution, $u\equiv 0.$ Then, the left focal point of the left focal boundary value problem was defined and criteria was established to characterize the left focal point with respect to the spectral radius of an associated compact linear fractional integral operator. In order to establish the criteria, properties of related families of Green's functions were observed. The article closed with an application to a nonlinear boundary value problem.