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In this paper, we introduce new iterative methods for approximating common fixed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common fixed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common fixed points of two quasi-nonexpansive non-self mappings under appropriate conditions. Our results improve and generalize many of the results in the literature. Moreover, our findings will open the way forward for the study of iterative methods for finding common fixed points of two non-self mappings in Banach spaces more general than Hilbert spaces.

New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space