Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments

•We introduce monotone sequences of lower and upper solutions.•The existence result of solution are established.•We introduce a numerical iterative scheme.

By means of the monotone iterative technique and the method of lower and upper solutions, we consider the nonlinear boundary value problems with Riemann–Liouville fractional derivative and deviating arguments, introduce two well-defined monotone sequences of lower and upper solutions which converge uniformly to the actual solution of the problem, and then the existence result of solution for the problems are established. A numerical iterative scheme is introduced to obtain an accurate approximate solution for the problem. As an application, an example is presented to demonstrate the accuracy and efficiency of the new approach.