Document Type : Research Paper
1 Department of Mathematics, Faculty of Science and Arts, Aksaray University, Aksaray Turkey.
2 Department of Mathematical Engineering,Y\i ld\i z Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey.
In the present paper, we show that $S^*$ iteration method can be used to approximate fixed point of almost contraction mappings. Furthermore, we prove that this iteration method is equivalent to CR iteration method and it produces a slow convergence rate compared to the CR iteration method for the class of almost contraction mappings. We also present table and graphic to support this result. Finally, we obtain a data dependence result for almost contraction mappings by using $S^*$ iteration method and in order to show validity of this result we give an example.
 V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl., 2 (2004), pp. 97-105.
 V. Berinde, On the Approximation of Fixed Points of Weak Contractive Mappings, Carpathian J. Math., 19 (2003), pp. 7-22.
 S.S. Chang, Y.J. Cho, and J.K. Kim, The equivalence between the convergence of modified Picard, modified Mann, and modified Ishikawa iterations, Math. Comput. Model., 37 (2003), pp. 985-991.
 R. Chugh, V. Kumar, and S. Kumar, Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces, Amer. J. Comput. Math., 2 (2012), pp. 345-357.
 N. Hussain, A. Rafiq, B. Damjanovic, and R. Lazovic, On rate of convergence of various iterative schemes, Fixed Point Theory Appl., 2011 (2011), pp. 1-6.
 S. Ishikawa, Fixed Point By a New Iteration Method, Proc. Amer. Math. Soc., 44 (1974), pp. 147-150.
 I. Karahan and M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Adv. Fixed Point Theory, 3 (2013), pp. 510-526.
 F. Gursoy, V. Karakaya, and B.E. Rhoades, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl., 2013 (2013), pp. 1-12.
 V. Karakaya, K. Dogan, F. Gursoy, and M. Erturk, Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces, Abstr. Appl. Anal., 2013 (2013), pp. 1-9.
 V. Karakaya, Y. Atalan, K. Dogan, and NEH. Bouzara, Some Fixed Point Results for a New Three Steps Iteration Process in Banach Spaces, Fixed Point Theory, 18 (2017), pp. 625-640.
 V. Karakaya, Y. Atalan, K. Dogan, and NEH. Bouzara, Convergence Analysis for a New Faster Iteration Method, Istanbul Commerce University Journal of Science, 15 (2016), pp. 35-53.
 W.R. Mann, Mean Value Methods in Iteration, Proc. Amer. Math. Soc., 4 (1953), pp. 506-510.
 M.A. Noor, New Approximation Schemes for General Variational Inequalities, J. Math. Anal. Appl., 251 (2000), pp. 217-229.
 W. Phuengrattana and S. Suantai, Comparison of the Rate of Convergence of Various Iterative Methods for the Class of Weak Contractions in Banach Spaces, Thai J. Math., 11 (2013), pp. 217-226.
 W. Phuengrattana and S. Suantai, On the Rate of Convergence of Mann, Ishikawa, Noor and SP Iterations for Continuous Functions on an Arbitrary Interval, J. Comput. Appl. Math., 235 (2011), pp. 3006-3014.
 E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., 6 (1890), pp. 145-210.
 B.E. Rhoades and S.M. Soltuz, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Anal., 58 (2004), pp. 219-228.
 S.M. Soltuz and T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl., 2008 (2008), pp. 1-7.
 X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113 (1991), pp. 727-731.