Equivalence of Picard‐type Hybrid Iterative Algorithms for Contractive Mappings
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Asian Network for Scientific Information
Abstract
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Background and Objective: Fixed point iterative algorithms are designed to be applied in solving
equations arising in physical formulation but there is no systematic study of numerical aspects of these
iterative algorithms. The Picard, Mann, Ishikawa, Noor and multi step iterative algorithms are the
commonly used iterative algorithms in proving fixed point convergence and stability results of different
classes of mappings. The objectives of this study therefore were: (1) To develop a Picard‐type hybrid
iterative algorithm called Picard‐Mann, Picard‐Ishikawa, Picard‐Noor and Picard‐multistep iterative
algorithms, (2) Prove equivalence of convergence theorems using these algorithms for a general class of
mappings in a normed linear space and (3) Provide numerical examples to justify the applicability of the
algorithms. Materials and Methods: Analytical method was used to prove the main theorem, while
numerical method was to demonstrate the application of the equivalence results. Results: Strong
convergence, equivalence and numerical results constitute the main results of this study. Conclusion:
The results obtained from this study showed that the Picard‐type hybrid iterative algorithms have good
potentials for further applications, especially in terms of rate of convergence.
Keywords
QA Mathematics