In this paper, we study the existence, uniqueness, and other qualitative properties of the solution of the differential equation with initial conditions of fractional order involving the Caputo fractional derivative. The tool employed in the analysis is based on the application of a new three steps iteration process introduced by V. Karakaya, Y. Atalan, K. Dogan, and NH. Bouzara [23] (i. e. V. Karakaya, Y. Atalan, K. Dogan and N. El Houda Bouzara, Some fixed point results for a new three steps iteration process in Banach spaces, Fixed Point Theory, 18, 2(2017), 625–640). The new three steps iteration method, due to simplicity and fastness, has attracted our attention and hence, it is used in this paper. Furthermore, the study of various properties such as dependence on initial given conditions, the closeness of solutions, and dependence on parameters and functions involved therein. The results obtained are illustrated through an example. The conclusion section is also included.