Iterative Algorithm for Solving Scalar Fractional Differential Equations with Riemann–Liouville Derivative and Supremum

Abstract

The initial value problem for a special type of scalar nonlinear fractional differential equation with a Riemann–Liouville fractional derivative is studied. The main characteristic of the equation is the presence of the supremum of the unknown function over a previous time interval. This type of equation is difficult to be solved explicitly and we need approximate methods for its solving. In this paper, initially, mild lower and mild upper solutions are defined. Then, based on these definitions and the application of the monotone-iterative technique, we present an algorithm for constructing two types of successive approximations. Both sequences are monotonically convergent from above and from below, respectively, to the mild solutions of the given problem. The suggested iterative scheme is applied to particular problems to illustrate its application.