The existence of positive solutions was studied for a class of mixed fractional p-Laplacian operator equations with both Riemann-Liouville derivatives and Caputo derivatives under Riemann-Stieltjes integral boundary conditions. According to the Riemann-Stieltjes integral properties, several conclusions were obtained about the existence of multiple positive solutions of boundary value problems by using the fixed point theorem and monotone iterative method. Iterative sequences were established for finding the approximate solutions of such boundary value problems. Finally, some examples were presented to demonstrate the applicability of the conclusions.