Convergence to common fixed points of averaged mappings without commutativity assumption in Hilbert spaces

We introduce a new iteration scheme for averaged mappings in Hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings. The main theorem extends a recent result of Shimizu and Takahashi [T. Shimizu, W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997) 71–83].