Abstract. For α ∈ (1, 2], the singular fractional boundary value problemsatisfying the boundary conditionsRiemann-Liouville derivatives of order α, β and µ respectively, is considered. Here f satisfies a local Carathéodory condition, and f (t, x, y) may be singular at the value 0 in its space variable x. Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given.

We establish the existence of smallest eigenvalues for the fractional linear boundary value problems

Let b > 0. Let 1 < α ≤ 2. The theory of u 0 -positive operators with respect to a cone in a Banach space is applied to study the conjugate boundary value problem for Riemann-Liouville fractional linear differential equations D α 0+ u + λp(t)u = 0, 0 < t < b, satisfying the conjugate boundary conditions u(0) = u(b) = 0. The first extremal point, or conjugate point, of the conjugate boundary value problem is defined and criteria are established to characterize the conjugate point. As an application, a fixed point theorem is applied to give sufficient conditions for existence of a solution of a related boundary value problem for a nonlinear fractional differential equation.

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