Abstract
In this work, we prove the weak and strong convergence of a sequence generated by a modified S-iteration process for finding a common fixed point of two G-nonexpansive mappings in a uniformly convex Banach space with a directed graph. We also give some numerical examples for supporting our main theorem and compare convergence rate between the studied iteration and the Ishikawa iteration.
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Banach, S.: Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales. Fund. Math 3, 133–181 (1922)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex. Anal. 8, 61–79 (2007)
Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc 136(4), 1359–1373 (2008)
Aleomraninejad, S.M.A., Rezapour, S., Shahzad, N.: Some fixed point result on a metric space with a graph. Topol. Appl 159, 659–663 (2012)
Alfuraidan, M.R.: Fixed points of monotone nonexpansive mappings with a graph. Fixed Point Theory Appl. doi:10.1186/s13663-015-0299-0 (2015)
Alfuraidan, M.R., Khamsi, M.A.: Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory Appl. doi:10.1186/s13663-015-0294-5 (2015)
Tiammee, J., Kaewkhao, A., Suantai, S.: On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs. Fixed Point Theory Appl. doi:10.1186/s13663-015-0436-9 (2015)
Tripak, O.: Common fixed points of G-nonexpansive mappings on Banach spaces with a graph. Fixed Point Theory Appl. doi:10.1186/s13663-016-0578-4 (2016)
Shahzad, S., Al-Dubiban, R.: Approximating common fixed points of nonexpansive mappings in Banach spaces. Georgian Math. J. 13(3), 529–537 (2006)
Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16(12), 1127–1138 (1991)
Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43(1), 153–159 (1991)
Suantai, S.: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings. J. Math. Anal. Appl 331, 506–517 (2005)
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York (2009)
Rhoades, B.E.: Comments on two fixed point iteration method. J. Math. Anal. Appl. 56(2), 741–750 (1976)
Phuengrattana, W., Suantai, S.: On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math. 235, 3006–3014 (2011)
Burden, R.L., Faires, J.D.: Numerical Analysis, 9th edn. Brooks/Cole Cengage Learning, Boston (2010)
Berinde, V.: Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare (2002)
Phuengrattana, W., Suantai, S.: Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach spaces. Thai J. Math. 11, 217–226 (2013)
Acknowledgements
The authors would like to thank the referees for valuable comments and suggestions for improving this work. R. Suparatulatorn and S. Suantai would like to thank the Thailand Research Fund under the project RTA5780007, the Royal Golden Jubilee (RGJ) Ph.D. Programme and Chiang Mai University. W. Cholamjiak would like to thank University of Phayao.
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Suparatulatorn, R., Cholamjiak, W. & Suantai, S. A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs. Numer Algor 77, 479–490 (2018). https://doi.org/10.1007/s11075-017-0324-y
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DOI: https://doi.org/10.1007/s11075-017-0324-y