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

TL;DR

This paper introduces the Picard-S iteration method, demonstrating its faster convergence compared to existing methods for fixed points of contraction mappings and its application to solving differential equations with retarded arguments.

Contribution

The paper presents a new iteration method called Picard-S, proving its convergence speed, equivalence to other methods, and applicability to differential equations with retarded arguments.

Findings

Picard-S iteration converges faster than existing methods for contraction mappings.

The method is applicable to solving differential equations with retarded arguments.

A data dependence result for fixed points is established.

Abstract

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.