Mathematica Bohemica, Vol. 144, No. 1, pp. 69-83, 2019


Some convergence, stability and data dependency results for a Picard-S iteration method of quasi-strictly contractive operators

Müzeyyen Ertürk, Faik Gürsoy

Received October 12, 2017.   Published online June 19, 2018.

Abstract:  We study some qualitative features like convergence, stability and data dependency for Picard-S iteration method of a quasi-strictly contractive operator under weaker conditions imposed on parametric sequences in the mentioned method. We compare the rate of convergence among the Mann, Ishikawa, Noor, normal-S, and Picard-S iteration methods for the quasi-strictly contractive operators. Results reveal that the Picard-S iteration method converges fastest to the fixed point of quasi-strictly contractive operators. Some numerical examples are given to validate the results obtained herein. Our results substantially improve many other results available in the literature.
Keywords:  iteration method; quasi-strictly contractive operator; convergence; rate of convergence; stability; data dependency
Classification MSC:  47H09, 47H10, 54H25


References:
[1] H. Akewe, G. A. Okeke: Convergence and stability theorems for the Picard-Mann hybrid iterative scheme for a general class of contractive-like operators. Fixed Point Theory Appl. 2015 (2015), Paper No. 66, 8 pages. DOI 10.1186/s13663-015-0315-4 | MR 3343141 | Zbl 1312.47078
[2] V. Berinde: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory Appl. 2004 (2004), 97-105. DOI 10.1155/S1687182004311058 | MR 2086709 | Zbl 1090.47053
[3] V. Berinde: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics 1912. Springer, Berlin (2007). DOI 10.1007/978-3-540-72234-2 | MR 2323613 | Zbl 1165.47047
[4] V. Berinde: On a notion of rapidity of convergence used in the study of fixed point iterative methods. Creat. Math. Inform. 25 (2016), 29-40. MR 3558671 | Zbl 06762010
[5] V. Berinde, M. Păcurar: A fixed point proof of the convergence of a Newton-type method. Fixed Point Theory 7 (2006), 235-244. MR 2284596 | Zbl 1115.65053
[6] A. O. Bosede, B. E. Rhoades: Stability of Picard and Mann iteration for a general class of functions. J. Adv. Math. Stud. 3 (2010), 23-25. MR 2722440 | Zbl 1210.47093
[7] C. E. Chidume, J. O. Olaleru: Picard iteration process for a general class of contractive mappings. J. Niger. Math. Soc. 33 (2014), 19-23. MR 3235868 | Zbl 1341.47079
[8] H. Fukhar-ud-din, V. Berinde: Iterative methods for the class of quasi-contractive type operators and comparison of their rate of convergence in convex metric spaces. Filomat 30 (2016), 223-230. DOI 10.2298/FIL1601223F | MR 3498766 | Zbl 06749677
[9] F. Gürsoy: A Picard-S iterative method for approximating fixed point of weak-contraction mappings. Filomat 30 (2016), 2829-2845. DOI 10.2298/FIL1610829G | MR 3583408 | Zbl 06749928
[10] F. Gürsoy, V. Karakaya: A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. Avaible at https://arxiv.org/abs/1403.2546 (2014), 16 pages.
[11] F. Gürsoy, V. Karakaya, B. E. Rhoades: Data dependence results of new multi-step and S-iterative schemes for contractive-like operators. Fixed Point Theory Appl. 2013 (2013), Paper No. 76, 12 pages. DOI 10.1186/1687-1812-2013-76 | MR 3047130 | Zbl 06282865
[12] F. Gürsoy, A. R. Khan, H. Fukhar-ud-din: Convergence and data dependence results for quasi-contractive type operators in hyperbolic spaces. Hacet. J. Math. Stat. 46 (2017), 373-388. DOI 10.15672/HJMS.20174720334 | MR 36991880 | Zbl 06810307
[13] R. H. Haghi, M. Postolache, S. Rezapour: On T-stability of the Picard iteration for generalized $\phi$-contraction mappings. Abstr. Appl. Anal. 2012 (2012), Article ID 658971, 7 pages. DOI 10.1155/2012/658971 | MR 2965457 | Zbl 1252.54035
[14] A. M. Harder, T. L. Hicks: Stability results for fixed point iteration procedures. Math. Jap. 33 (1988), 693-706. MR 0972379 | Zbl 0655.47045
[15] S. Ishikawa: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44 (1974), 147-150. DOI 10.2307/2039245 | MR 0336469 | Zbl 0286.47036
[16] V. Karakaya, K. Doğan, F. Gürsoy, M. Ertürk: Fixed point of a new three-step iteration algorithm under contractive-like operators over normed spaces. Abstr. Appl. Anal. 2013 (2013), Article ID 560258, 9 pages. DOI 10.1155/2013/560258 | MR 3147859 | Zbl 1364.47026
[17] V. Karakaya, F. Gürsoy, M. Ertürk: Some convergence and data dependence results for various fixed point iterative methods. Kuwait J. Sci. 43 (2016), 112-128. MR 3496310
[18] S. H. Khan: A Picard-Mann hybrid iterative process. Fixed Point Theory Appl. 2013 (2013), Paper No. 69, 10 pages. DOI 10.1186/1687-1812-2013-69 | MR 3053809 | Zbl 1317.47065
[19] A. R. Khan, F. Gürsoy, V. Karakaya: Jungck-Khan iterative scheme and higher convergence rate. Int. J. Comput. Math. 93 (2016), 2092-2105. DOI 10.1080/00207160.2015.1085030 | MR 3576658 | Zbl 06679732
[20] A. R. Khan, F. Gürsoy, V. Kumar: Stability and data dependence results for the Jungck-Khan iterative scheme. Turkish J. Math. 40 (2016), 631-640. DOI 10.3906/mat-1503-1 | MR 3486126
[21] A. R. Khan, V. Kumar, N. Hussain: Analytical and numerical treatment of Jungck-type iterative schemes. Appl. Math. Comput. 231 (2014), 521-535. DOI 10.1016/j.amc.2013.12.150 | MR 3174051
[22] W. R. Mann: Mean value methods in iteration. Proc. Am. Math. Soc. 4 (1953), 506-510. DOI 10.2307/2032162 | MR 0054846 | Zbl 0050.11603
[23] G. A. Okeke, J. K. Kim: Convergence and summable almost $T$-stability of the random Picard-Mann hybrid iterative process. J. Inequal. Appl. 2015 (2015), Paper No. 290, 14 pages. DOI 10.1186/s13660-015-0815-0 | MR 3399253 | Zbl 1351.47051
[24] M. O. Olatinwo, M. Postolache: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 218 (2012), 6727-6732. DOI 10.1016/j.amc.2011.12.038 | MR 2880328 | Zbl 1293.54033
[25] W. Phuengrattana, 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. MR 3065435 | Zbl 1294.47090
[26] E. Picard: Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. Journ. de Math. (4) 6 (1890), 145-210 (In French.). JFM 22.0357.02
[27] D. R. Sahu: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12 (2011), 187-204. MR 2797080 | Zbl 1281.47053
[28] O. Scherzer: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 194 (1995), 911-933. DOI 10.1006/jmaa.1995.1335 | MR 1350202 | Zbl 0842.65036
[29] Ş. M. Şoltuz, T. Grosan: Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory Appl. 2008 (2008), Article ID 242916, 7 pages. DOI 10.1155/2008/242916 | MR 2415408 | Zbl 1205.47059
[30] B. Xu, M. A. Noor: Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 267 (2002), 444-453. DOI 10.1006/jmaa.2001.7649 | MR 1888015 | Zbl 1011.47039
[31] I. Yildirim, M. Abbas, N. Karaca: On the convergence and data dependence results for multistep Picard-Mann iteration process in the class of contractive-like operators. J. Nonlinear Sci. Appl. 9 (2016), 3773-3786. MR 3517127 | Zbl 1350.47050

Affiliations:   Müzeyyen Ertürk, Faik Gürsoy, Department of Mathematics, Adiyaman University, Adiyaman 02040, Turkey, e-mail: merturk3263@gmail.com, faikgursoy02@hotmail.com


 
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