On Solutions to Fractional Iterative Differential Equations with Caputo Derivative

In this paper, we are concerned with two points. First, the existence and uniqueness of the iterative fractional differential equation ${}^c{D}^{\alpha }cx\left(t\right)=f\left(t,x\left(t\right),x\left(g\left(x\left(t\right)\right)\right)\right)$ are presented using the fixed-point theorem by imposing some conditions on $f$ and $g$ . Second, we proposed the iterative scheme that converges to the fixed point. The convergence of the iterative scheme is proved, and different iterative schemes are compared with the proposed iterative scheme. We prepared algorithms to implement the proposed iterative scheme. We have successfully applied the proposed iterative scheme to the given iterative differential equations by taking examples for different values of $\alpha$ .