  We introduce a new iteration method called Picard-S iteration. We show that
the Picard-S iteration method can be used to approximate fixed point of
contraction mappings. Also, we show that our new iteration method is equivalent
and converges faster than CR iteration method for the aforementioned class of
mappings. Furthermore, by providing an example, it is shown that the Picard-S
iteration method converges faster than all Picard, Mann, Ishikawa, Noor, SP,
CR, S and some other iteration methods in the existing literature when applied
to contraction mappings. A data dependence result is proven for fixed point of
contraction mappings with help of the new iteration method. Finally, we show
that the Picard-S iteration method can be used to solve differential equations
with retarded argument.
