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

A Picard-S hybrid type iteration method for solving a differential equation with retarded argument