In this paper we present new iterative algorithms in convex metric spaces. We show that these iterative schemes are convergent to the fixed point of a single-valued contraction operator. Then we make the comparison of their rate of convergence. Additionally, numerical examples for these iteration processes are given.
Cristian Daniel Alecsa
Babes-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania,
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania
convex metric space; fixed point; iterative algorithm; rate of convergence; convex combination.
C.-D. Alecsa, On new faster fixed point iterative schemes for contraction operators and comparison of their rate of convergence in convex metric spaces, International Journal Of Nonlinear Analysis And Applications, 8 (2019) no. 1, pp. 353-388.
International Journal of Nonlinear Analysis and Applications
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 M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Matematicki Vesnik 66 (2014) 223–234.
 R. Agarwal, D. O’Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Journal of Nonlinear and Convex Analysis 8 (2007) 61–79.
 S. Akbulut and M. Ozdemir, Picard iteration converges faster than Noor iteration for a class of quasi-contractive operators, Chiang Mai J.Sci. 39 (2012) 688–692.
 V. Berinde, Picard iteration converges faster than Man iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2014 (2004):1.
 M.R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, New York, 1999.
 H. Fukhar-ud-din and V. Berinde, Iterative methods for the class of quasi-contractive type operators and comparsion of their rate of convergence in convex metric spaces, Filomat 30 (2016) 223–230.
 K. Goebel and W. Kirk, Iteration processes for nonexpansive mappings, Topological Methods in Nonlinear Functional Analysis, Contemporary Mathematics 21, Amer. Math. Soc. Providence (1983), 115–123.
 F. Gursoy and V.Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, http://arxiv.org/abs/1403.2546
 H.V. Machado, A characterization of convex subsets of normed spaces, Kodai Math. Sem. Rep. 25 (1973) 307–320.
 R. Chugh, P.Malik and V.Kumar, On a new faster implicit fixed point iterative scheme in convex metric spaces, J. Function Spaces (2015), Article ID 905834.
 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) 217–226.
 S. Reich and I. Safrir, Nonexpansive iteration in hyperbolic spaces, Nonlinear. Anal. 15 (1990) 537–558.
 W. Sintunavarat and A. Pitea, On a new iteration scheme for numerical reckoning fixed points of Berinde mappings with convergence analysis, J. Nonlinear Science Appl. 9 (2016) 2553–2562.
 S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005) 506–517.
 W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep. 22 (1970) 142–149.
 L.A. Talman, Fixed points for condensing multifunctions in metric spaces with convex structure, Kodai Math. Sem. Rep. 29 (1977) 62–70.
 T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math. (Basel), 23 (1972) 292–298.