We consider the class of generalized α-nonexpansive mappings in a setting of Banach spaces. We prove existence of fixed point and convergence results for these mappings under the K∗-iterative process. The weak convergence is obtained with the help of Opial’s property while strong convergence results are obtained under various assumptions. Finally, we construct two numerical examples and connect our K∗-iterative process with them. An application to solve a fractional differential equation (FDE) is also provided. It has been eventually shown that the K∗- iterative process of this example gives more accurate numerical results corresponding to some other iterative processes of the literature. The main outcome is new and improves some known results of the literature.

We obtain strong convergence theorems of two modifications of Mann iteration processes with errors in the doubly sequence setting. Furthermore, we establish some weakly convergence theorems for doubly sequence Mann's iteration scheme with errors in a uniformly convex Banach space by a Frechét differentiable norm.