In this paper, we apply the monotone iteration method to establish the existence of a positive solution for the fractional differential equation Dα 0+u(t) + q(t)f(t; u(t)) = 0; 0 < t < 1; together with the boundary conditions (BCs) u(0) = u’(0) = … = un-2(0) = 0; Dβ 0+u(1) = ∫1 0 h(s; u(s)) dA(s); where n > 2, n-1 < α ≤ n, β 2 [1; α - 1], Dα 0+ and Dβ 0+ are the standard Riemann-Liouville fractional derivatives of order α and β, respectively, and f; h: [0; 1] × [0,∞) → [0;∞) are continuous functions. The sufficient condition provided in this paper is new, interesting and easy to verify. Our conditions do not require the sublinearity or superlinearity on the nonlinear functions f and h at 0 or ∞. The paper is supplemented with examples illustrating the applicability of our result. © 2019, Razmadze Mathematical Institute. All rights reserved.