ON THE COMPARISON OF ISHIKAWA-TYPE ITERATIVE PROCESSES FOR CONTRACTION MAPPINGS IN BANACH SPACES WITH GRAPHS
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Abstract
It is well-known that there are many results on convergence to a common fixed point of contraction mappings for iterative processes with graphs. However, there are few results related to comparing the convergence rates of these iterative processes. In this paper, we consider the comparisons with respect to convergence rates of several Ishikawa-type iterative processes to a common fixed point of contraction mappings in Banach spaces with graphs. More specifically, we propose some sufficient conditions to ensure that an iteration process is faster than the other one. Our work improves the results of a recent one related to the comparison of convergence rates of two-step and three-step iteration sequences. Roughly speaking, the assumptions we make are better for parameter ranges than the previous results. An interesting point is that our paper opens up the new idea of building an optimal hypothesis to obtain a comparison of the convergence rates of general iterative processes.
Keywords
Banach spaces with graphs, contraction mappings, convergence rate, Ishikawa-type iterative processes
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References
Aleomraninejad, S. M. A., Rezapour, S., & Shahzad, N. (2012). Some fixed point result on a metric space with a graph. Topol. Appl., 159(3), 59-663.
Dotson, W. G. (1970). On the Mann iterative process. Trans. Amer. Math. Soc., 149, 655-673.
Robert Mann,W. (1953). Mean value methods in iteratio. Proc. Amer. Math. Soc., 4, 506-510.
Ishikawa, S.(1974). Fixed points by a new iteration method. Proc. Amer. Math. Soc., 44, 147-150.
Jachymski, J. (2008). The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc., 136(4) (2008), 1359-1373.
Kalinde, A. K., & Rhoades, B. E. (1992). Fixed point Ishikawa iterations. J. Math. Anal. Appl., 170 (2), 600-606.
Nguyen, V., D., & Nguyen, T., H. (2020). Convergence of a new three-step iteration process to common fixed points of three G-nonexpansive mappings in Banach spaces with directed graphs. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 114, 140. http://doi.org/10.1007/s13398-020-00872-w
Suparatulatorn, R., Cholamjiak, W., & Suantai, S. (2018). A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer. Algorithms, 77(2), 479-490. https://doi.org/10.1007/s11075-017-0324-y
Thianwan, T., & Yambangwai, D. (2019). Convergence analysis for a new two-step iteration process for G-nonexpansive mappings with directed graphs. J. Fixed Point Theory Appl., 21(44), 1-16.
Tripak, O. (2016). Common fixed points of G-nonexpansive mappings on Banach spaces with a graph. Fixed Point Theory Appl. Art., 87, 1-8.