Abstract
By comparing rate of convergence for some iteration methods, we show that the coefficients have important role in rate of convergence for quasi-contractions in Picard, Mann, Ishikawa, Noor, Agarwal and Thakur iteration methods. Using the Matlab, we provide some numerical examples for our results.
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References
Abbas M, Nazir T (2014) A new faster iteration process applied to constrained minimization and feasibility problems. Mat Vesnik 66(2):223–234
Agarwal RP, O’Regan D, Sahu DR (2007) Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J Nonlinear Conv Anal 8(1):61–79
Berinde V (2004) Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl Article ID 716359:9
Berinde V (2007) Iterative approximation of fixed points. Springer, New York (2007)
Chugh R, Kumar S (2013) On the rate of convergence of some new modified iterative schemes. Am J Comput Math 3:270–290
De La Sen M (2010) Stable iteration procedures in metric spaces which generalize a Picard-type iteration. Fixed Point Theory Appl Article ID 953091:15
De La Sen M, Karapinar E (2014) On a cyclic Jungck modified \(TS\)-iterative procedure with application examples. Appl Math Comput 233:383–397
Fathollahi Sh, Ghiura A, Postolache M, Rezapour Sh (2015) A comparative study on the convergence rate of some iteration methods involving contractive mappings. Fixed Point Theory Appl 2015:234
Ishikawa S (1974) Fixed points by a new iteration method. Proc Am Math Soc 44:147–150
Liu QH (1990) A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings. J Math Anal Appl 146:301–305
Lj B (1974) Ciric, a generalization of Banach’s contraction principle. Proc Am Math Soc 45(2):267–273
Mann WR (1953) Mean value methods in iteration. Proc Am Math Soc 4:506–510
Noor MA (2000) New approximation schemes for general variational inequalities. J Math Anal Appl 251:217–229
Olaleru JO (2007) A comprison of Picard and Mann iteration for quasi contraction maps. Fixed Point Theory 8(1):87–95
Ozturk Celikler F (2014) Convergence analysis for a modified SP iteration method. Sci World J Article ID 840504:5
Rhoades BE, Xue Z (2010) Comparison of the rate of convergence among Picard, Mann, Ishikawa, and Noor iterations applied to quasicontractive maps. Fixed Point Theory Appl. Article ID 169062:12
Rhoades BE (1977) A comparison of various definitions of contractive mappings. Trans Am Math Soc 226:257–290
Thakur D, Singh BS, Postolache M (2014) New iteration scheme for numerical reckoning fixed points of nonexpansive mappings. J Ineq Appl 2014:328
Thakur BS, Thakur D, Postolache M (2016) A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings. Appl Math Comput 275:147–155
Xu H (1992) A Note on the lshikawa iteration scheme. J Math Anal Appl 167:582–587
Acknowledgements
The authors express their gratitude to dear unknown referees for their helpful suggestions which improved final version of this paper. The authors supported by Azarbaijan Shahid Madani University.
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Fathollahi, S., Rezapour, S. Efficacy of Coefficients on Rate of Convergence of Some Iteration Methods for Quasi-Contractions. Iran J Sci Technol Trans Sci 42, 1517–1523 (2018). https://doi.org/10.1007/s40995-017-0390-y
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DOI: https://doi.org/10.1007/s40995-017-0390-y