Let be a Banach space and
a contraction mapping, where
is a closed set. Consider a sequence
and define the sequence
, by
, where
is a sequence of natural numbers. We highlight some general conditions so that the two sequences
and
are simultaneously convergent. Both cases: 1)
, for all
, and 2)
, for all
, are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to
sequences of mappings and some appropriate applications are also proposed.