FIRST EXTREMAL POINT COMPARISON FOR A FRACTIONAL BOUNDARY VALUE PROBLEM WITH A FRACTIONAL BOUNDARY CONDITION

Let n >= 2 be a natural number, and let n - 1 < alpha <= n and 0 < gamma <= alpha - 1 be real numbers. Let beta > 0 and b is an element of (0, beta]. We compare first extremal points of the differential equations D-0+(alpha) u+p(t)u = 0, D(0+)(alpha)u+q(t)u = 0, t is an element of (0, beta), each of which satisfies the boundary conditions u((i))(0) = 0, i = 0, 1,..., n - 2, D(0+)(gamma)u(b) = 0. While it is assumed that q is nonnegative, no sign restrictions are put on p. The fact that the associated Green's function G(b; t, s) is nonnegative and increasing with respect to b plays an important role in the analysis.