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

Let n is an element of N, n >= 2, beta > 0 fixed, and 0 < b <= beta. For n - 1 < alpha <= 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, ..., 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.