Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Abstract

Let $n\in \mathbb {N}$, $n\ge 2$, $\beta >0$ fixed, and $0<b\le \beta$. For $n-1<\alpha \le n$, we look to classify extremal points for the fractional differential equation $D_{0^+}^{\alpha }u+p(t) u=0$, satisfying the boundary conditions $u^{(i)}(0)=0$, $i=0,\ldots ,n-2$, $D_{0^+}^\gamma u(b)=0$, where p(t) is a continuous nonnegative function on $[0,\beta ]$ which does not vanish identically on any nondegenerate compact subinterval of $[0,\beta ]$. Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.