Some convergence and data dependence results for various fixed point iterative methods
Keywords:
Convergence, data dependence of fixed points, equivalence of convergence, iterative methods, rate of convergenceAbstract
We have compared rate of convergence among various iterative methods. Also, wehave established an equivalency result between convergence of two recently introducediterative methods and we prove a data dependence result for one of them.
References
Agarwal, R., O’Regan, D. & Sahu, D. (2007) Iterative construction of fixed points of nearly
asymptotically nonexpansive mappings. Journal of Nonlinear and Convex Analysis, 8(1):61-79.
Berinde, V. (2007) Iterative approximation of fixed points. Berlin: Springer.
Berinde, V. (2004) Picard iteration converges faster than Mann iteration for a class of quasi-contractive
operators. Fixed Point Theory and Applications, (2):97-105.
Chugh, R., Kumar V. & Kumar, S. (2012) Strong convergence of a new three step iterative scheme in
Banach spaces. American Journal of Computational Mathematics, 2(4):345-357.
Hussain, N., Rafiq, A., Damjanović, B. & Lazović, R. (2011) On rate of convergence of various iterative
schemes. Fixed Point Theory and Applications. (1):45.
Ishikawa, S. (1974) Fixed points by a new iteration method. Proceedings of the American Mathematical
Society, 44(1):147-150.
Karahan, I. & Ozdemir, M. (2013) A general iterative method for approximation of fixed points and
their applications. Advances in Fixed Point Theory, 3(3):510-526.
Karakaya, V., Doğan, K., Gürsoy, F. & Ertürk, M. (2013) Fixed point of a new three-step iteration
algorithm under contractive-like operators over normed spaces. Abstract and Applied Analysis,
-9.
Mann, W. (1953) Mean value methods in iteration. Proceedings of the American Mathematical Society,
(3):506-510.
Noor, M. (2000) New approximation schemes for general variational inequalities. Journal of Mathematical
Analysis and Applications, 251(1):217-229.
Phuengrattana, W. & Suantai, S. (2011) On the rate of convergence of Mann, Ishikawa, Noor and SPiterations
for continuous functions on an arbitrary interval. Journal of Computational and Applied
Mathematics, 235(9):3006-3014.
Picard, E. (1890) Mémoiresur la théorie des équations aux dérivé espartielle set la méthode des
approximations successives. Journal de Mathématiques pures et appliquées, 6:145-210.
Rhoades, B. & Şoltuz, Ş. (2004) The equivalence between the convergences of Ishikawa and Mann
iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively
pseudo contractive maps. Journal of Mathematical Analysis and Applications, 289(1):266-278.
Sahu, D. (2011) Applications of the S-iteration process to constrained minimization problems and split
feasibility problems. Fixed Point Theory, 12(1):187-204.
Şoltuz, Ş. & Grosan, T. (2008) Data dependence for Ishikawa iteration when dealing with contractive-like
operators.Fixed Point Theory and Applications, (1):242916.
Weng, X. (1991) Fixed point iteration for local strictly pseudo contractive mapping. Proceedings of the
American Mathematical Society, 113: =727-731.
Xue, Z. (2007) Remarks of equivalence among Picard, Mann, and Ishikawa iterations in normed spaces.
Fixed Point Theory and Applications, (1):1-5.
Xue, Z. (2008) The comparison of the convergence speed between Picard, Mann, Krasnoselskij and
Ishikawa iterations in Banach spaces. Fixed Point Theory and Applications, 2 :1-6.
