New three-step iteration process and fixed point approximation in Banach spaces

Subject Areas : Fixed point theory

K. Ullah ¹ , M. Arshad ²

1 - Department of Mathematics, International Islamic University H-10, 44000- Islamabad, Pakistan
2 - Department of Mathematics, International Islamic University H-10, 44000- Islamabad, Pakistan

Received: 2017-10-25 Accepted : 2018-04-18 Published : 2018-06-01

Keywords: Banach space, Suzuki generalized nonexpansive mapping, Iteration process, weak convergence, strong convergence, contraction mapping,

Abstract :

‎In this paper we propose a new iteration process‎, ‎called the $K^{\ast }$ iteration process‎, ‎for approximation of fixed‎‎points‎. ‎We show that our iteration process is faster than the existing well-known iteration processes using numerical examples‎. ‎Stability of the $K^{\ast‎}‎$ iteration process is also discussed‎. ‎Finally we prove some weak and strong convergence theorems for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces‎. ‎Our results are the extension‎, ‎improvement and generalization of many well-known results in the literature of iterations in‎‎fixed point theory‎.

References:

[1] M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesn. 66 (2014), 223-234.

[2] R. P. Agarwal, D. O'Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61-79.

[3] V. Berinde, Iterative approximation of fixed points, Springer, Berlin, 2007.

[4] R. Chugh, V. Kumar, S. Kumar, Strong Convergence of a new three step iterative scheme in Banach spaces, Amer. J. Comp. Math. 2 (2012), 345-357.

[5] K. Goebel, W. A. Kirk, Topic in metric fixed point theory, Cambridge University Press, 1990.

[6] F. Gursoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2 (2014).

[7] A. M. Harder, Fixed point theory and stability results for fixed point iteration procedures, Ph.D. Thesis, University of Missouri-Rolla, Missouri, 1987.

[8] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.

[9] I. Karahan, M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Adv. Fixed. Point. Theor. 3 (3) (2013), 510-526.

[10] V. Karakaya, F. Gursoy, M. Erturk, Comparison of the speed of convergence among various iterative schemes, arXiv:1402.6080 (2014).

[11] S. H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Theory Appl. (2013), 2013:69.

[12] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.

[13] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229.

[14] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597.

[15] W. Phuengrattana, Approximating fixed points of Suzuki-generalized nonexpansive mappings, Nonlinear
Anal. 5 (2011), 583-590.

[16] W. Phuengrattana, S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comp. Appl. Math. 235 (2011), 3006-3014.

[17] B. E. Rhoades, Some fixed point iteration procedures, Int. J. Math. Math. Sci. 14 (1991), 1-16.

[18] B. E. Rhoades, Fixed point iterations using innite matrices (Fixed Points, Algorithms and Applications), Academic Press Inc, 1977.

[19] D. R. Sahu, A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, Nonlinear Anal. 74 (2011), 6012-6023.

[20] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc. 43 (1991), 153-159.

[21] H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380.

[22] S. M. Soltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. (2008), 2008:242916.

[23] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088-1095.

[24] B. S. Thakur, D. Thakur, M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, Appl. Math. Comp. 275 (2016), 147-155.

[25] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.