Mengistu Goa (PhD)Gezahegn Anberber2025-08-172025-08-172025-06https://etd.aau.edu.et/handle/123456789/6831This study explores generalizations of nonexpansive mappings, focusing on two commuting mappings that satisfy the Bγ,μ condition. We used some algorithm for approximating a common fixed point on some class of generalized nonexpansive mappings and proved its strong convergence to a common _xed point. Our _ndings extend and enhance several recent results in the literature. We discussed the properties of generalized nonexpansive mappings, particularly emphasizing a sequence of commuting mappings that satisfy the B;_ condition. We proposed iterative algorithms for approximating a common _xed point for these sequences, demonstrating their convergence under mild assumptions on the parameters. Additionally, we introduced a pair of some class of generalized nonexpansive mappings and investigated the convergence and existence of common _xed points within this class. We applied the three-step iteration process of Abbas-Nazir for a pair mappings satisfying some class of generalized nonexpansive on a nonempty subset of a Banach space. This approach yielded results related to both strong and weak convergence, leading to the identi_cation of the common _xed point of the two mappings. Finally, we provided an example illustrating two mappings that satisfy the speci_ed conditions.en-USCommon Fixed Point ResultsSome Class of Generalized Nonexpansive MappingsBanach SpacesNonexpansive MappingsThesis