A Picard-Mann hybrid iterative process
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In this article, a hybrid of Picard and Mann iterative processes is proposed, which converges faster than all of Picard, Mann, and Ishikawa iterative process in the sense of Berinde (Iterative Approximation of fixed points, 2002) for contractions.Abstract:
We introduce a new iterative process which can be seen as a hybrid of Picard and Mann iterative processes. We show that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of Berinde (Iterative Approximation of Fixed Points, 2002) for contractions. We support our analytical proof by a numerical example. We prove a strong convergence theorem with the help of our process for the class of nonexpansive mappings in general Banach spaces and apply it to get a result in uniformly convex Banach spaces. Our weak convergence results are proved when the underlying space satisfies Opial’s condition or has Frechet differentiable norm or its dual satisfies the Kadec-Klee property.read more
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References
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Journal ArticleDOI
Mean value methods in iteration
TL;DR: In this article, it is shown that the Schauder fixpoint theorem can play a somewhat analogous role in the theory of divergent iteration processes, and that the same methods can be used to prove that a given problem has a solution.
Journal ArticleDOI
Weak convergence of the sequence of successive approximations for nonexpansive mappings
Journal ArticleDOI
Fixed points by a new iteration method
TL;DR: In this article, it was shown that a certain sequence of points which is iteratively defined converges always to a fixed point of a lipschitzian pseudo-contractive map.
Journal ArticleDOI
Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process
Kok-Keong Tan,Hong-Kun Xu +1 more
Book
Iterative Approximation of Fixed Points
TL;DR: The Picard Iteration, the Krasnoselskij Iteration and the Mann Iteration are pre-requisites of fixed point iterations as mentioned in this paper, as well as the Ishikawa Iteration.