Abstract
This article proves some theorems to approximate fixed point of Zamfirescu operators on normed spaces for some two-step iterative schemes, namely, Picard-Mann iteration, Ishikawa iteration, S-iteration, and Thianwan iteration, with their errors. We compare the aforementioned iterations using numerical approach; the results show that S-iteration converges faster than other iterations followed by Picard-Mann iteration, while Ishikawa iteration is the least in terms of convergence rate. These results also suggest the best among two-step iterative fixed point schemes in the literature.
Highlights
Fixed point theory takes a large amount of literature, since it provides useful tools to solve many problems that have applications in different fields like engineering, economics, chemistry, game theory, and so forth
We compare the aforementioned iterations using numerical approach; the results show that S-iteration converges faster than other iterations followed by Picard-Mann iteration, while Ishikawa iteration is the least in terms of convergence rate
In the last four decades, numerous papers were published on the iterative approximation of fixed points of self- and non-self-contractive type operators in metric spaces, Hilbert spaces, or several classes of Banach spaces, while, for strict contractive type operators, the Picard iteration can be used to approximate the unique fixed point, for operators satisfying slightly weaker contractive type conditions, instead of Picard iteration, which does not generally converge; it was necessary to consider other fixed point iteration procedures
Summary
Fixed point theory takes a large amount of literature, since it provides useful tools to solve many problems that have applications in different fields like engineering, economics, chemistry, game theory, and so forth. To find fixed points is not an easy task; that is why we use iterative methods for computing them. The Krasnoselskij iteration, the Mann iteration, and the Ishikawa iteration are certainly the most studied of these fixed point iteration procedures. Other iterations which have been studied are Implicit Mann, Implicit Ishikawa, Thianwan, S-iteration, and hybrid Picard-Mann iterations. Wahab and Rauf [1] obtained some results on a faster implicit hybrid Kirk-multistep schemes for contractive type operators, to mention but a few. Our aim in this paper is to establish the convergence and convergence rate of some two-step iterative schemes with errors using Zamfirescu operator in Banach spaces
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