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In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form $$\left( {{P_{a,b}}} \right)\left\{ {_{u\left( 0 \right) = u\left( 1 \right) = 0,{D^{\alpha - 3}}u\left( 0 \right) = a,u'\left( 1 \right) = - b}^{{D^\alpha }u\left( x \right) + f\left( {x,u\left( x \right)} \right) = 0,x \in \left( {0,1} \right)}} \right.$$ , where 3 < α ≤ 4, D α is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) = −p(x)t σ , with σ ∈ (−1, 1) and p being a nonnegative continuous function that may be singular at x = 0 or x = 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Schauder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem (P 0,0). Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b > 0 and f(x, t) = t φ(x, t), with φ(x, t) being a nonnegative continuous function in (0, 1)×[0,∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem (P a,b ), which behaves like the unique solution of the homogeneous problem corresponding to (P a,b ). Some examples are given to illustrate the existence results.