In [Numer. Funct. Anal. Optim. 22, 767–773 (2001; Zbl 0999.47043
)], H.-K. Xu
and R. G. Ori
introduced an implicit iteration process for a finite family of nonexpansive mapping and proved the weak convergence of this process to a common fixed point of a finite family of nonexpansive maps in a Hilbert space. They posed the question: What assumptions would have to be made on the finite family of nonexpansive mappings and/or their parameters
to guarantee the strong convergence of the sequence
generated by their implicit iteration process? The authors prove the strong convergence of the sequence
in a much more general uniformly convex Banach space under the condition that one member of the finite family of nonexpansive mappings is semi-compact or any one of the contractive assumptions of Proposition 3.4 of their paper holds.