In this paper, we prove that a three-step iteration process is stable for contractive-like mappings. It is also proved analytically and numerically that the considered process converges faster than some remarkable iterative processes for contractive-like mappings. Furthermore, some convergence results are proved for the mappings satisfying Suzuki’s condition (C) in uniformly convex Banach spaces. A couple of nontrivial numerical examples are presented to support the main results and the visualization is showed by Matlab. Finally, by utilizing our main result the solution of a nonlinear fractional differential equation is approximated.
In this paper, we introduce a new scheme and prove convergence results for nonexpansive mappings as well as for weak contractions in the frame of Banach spaces. Moreover, we prove analytically and numerically that the proposed scheme converges to a fixed point of a weak contraction faster than some known and leading schemes. Further, we prove that the new scheme is almost stable with respect to weak contraction. For supporting the main results, we give a couple of nontrivial numerical examples, and the visualization is shown by using the Matlab program. Finally, the solution of a nonlinear fractional differential equation is approximated by operating the main result of the paper.
In this article, inspired by Berinde (Approximating fixed points of enriched nonexpansive mappings by Krasnoselskii iteration in Hilbert spaces, Carpathian J. Math. 35(3), 2019, 293-304), we define and study a new enriched class of mappings which includes many other contractive type mappings. We prove an existence result for the fixed point of newly introduced mapping. We also estimate fixed points of the proposed mapping via newly modified Mann iteration. In the process, some convergence results are also obtained for the proposed class of mappings in Uniformly convex Banach space.
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