This paper proposes a novel hybrid iteration process, namely the Picard–CR iteration process. We apply the proposed iteration process for the numerical reckoning of fixed points of generalized \(\alpha \) -nonexpansive mappings. We establish weak and strong convergence results of generalized \(\alpha \) -nonexpansive mappings. This study demonstrates the superiority of the hybrid approach in terms of convergence speed. Moreover, we numerically compare the proposed iteration process with other well-known ones from the literature. In the comparison, we consider two problems: finding a fixed point of a generalized \(\alpha \) -nonexpansive mapping and finding roots of a complex polynomial. In the second problem, we use the so-called polynomiography in the analysis. The results showed that the proposed iteration scheme is better than other three-parameter iteration schemes from the literature. Using the proven fixed-point results, we also obtain solutions to fractional differential equations.
