Document Type : Research Paper

Authors

1 Department of Mathematics, University of Science and Technology Bannu, KPK Pakistan.

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

10.22130/scma.2018.62911.238

Abstract

In this paper, first we use an example to show the efficiency of $M$ iteration process introduced by Ullah and Arshad [4] for approximating fixed points of Suzuki generalized nonexpansive mappings. Then by using $M$ iteration process, we prove some strong and $\Delta -$convergence theorems for Suzuki generalized nonexpansive mappings in the setting of $CAT(0)$ Spaces. Our results are the extension, improvement and generalization of many known results in $CAT(0)$ spaces.

Keywords

Main Subjects

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

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

[3] M. Bridson and A. Heaflinger, Metric Space of Non-positive Curvature, Springer-Verlag, Berlin, 1999.

[4] F. Bruhat and J. Tits, Groupes reductifs sur un corps local. I, Donnees radicielles valuees Inst Hauts Etudea Sci. Publ. Math., 41 (1972), pp. 5-251.

[5] D. Burago, Y. Burago and S. Inavo, A course in Metric Geometry, Vol. 33, Americal Mathematical Socity, Providence, RI, 2001.

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

[7] S. Dhompongsa, W.A. Kirk, and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), pp. 762-772.

[8] S. Dhompongsa, W.A. Kirk, and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear and convex Anal., 8 (2007), pp. 35-45.

[9] S. Dhompongsa and B. Panyanak, On $Delta$-convergence theorem in $CAT(0)$ Spaces, Comput. Math. Appl., 56 (2008), pp. 2572-2579.

[10] A. Gharajelo and H. Dehghan, Convergence Theorems for Strict Pseudo-Contractions in $CAT(0)$ Metric Spaces, Filomat, 31 (2017), pp. 1967-1971.

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

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

[13] I. Karahan and M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Advances in Fixed Point Theory, 3 (2013), pp. 510-526.

[14] S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Th. Appl., 2013, Article ID 69 (2013).

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

[16] A. Moudafi, Krasnoselski-Mann iteration for hierarchical fixed point problems, Inverse Probl., 23 (2007), pp. 1635-1640.

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

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

[19] 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. Comp. Appl. Math., 235 (2011), pp. 3006-3014.

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

[21] D.R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, NonlinearAnalysis: Theory, Methods and Applications, 74 (2011), pp. 6012-6023.

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

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

[24] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki generalized nonexpansive mapping via new iteration process, Filomat, 32 (2018), pp. 187-196.

[25] R. Wangkeeree, H. Dehghan, Strong and $Delta$-convergence of Moudafi's iterative scheme in $CAT(0)$ spaces, J. Nonlinear Convex Anal., 16 (2015), 299-309.