Iterative techniques for the initial value problem for Caputo fractional differential equations with non-instantaneous impulses

Two types of algorithms for constructing monotone successive approximations for solutions to initial value problems for a scalar nonlinear Caputo fractional differential equation with non-instantaneous impulses are given. The impulses start abruptly at some points and their action continue on given finite intervals. Both algorithms are based on the application of lower and upper solutions to the problem. The first one is a generalization of the monotone iterative technique and it requires an application of the Mittag-Leffler function with one and two parameters. The second one is easier from a practical point of view and is applicable when the right hand sides of the equation are monotone. We prove that the functional sequences are convergent and their limits are minimal and maximal solutions of the problem. An example is given to illustrate the results. (C) 2018 Elsevier Inc. All rights reserved.