Controllability and Accumulation of Errors Arising in a General Iteration Method

Yıl 2021, Cilt: 4 Sayı: 2, 77 - 82, 01.06.2021

Faik Gürsoy , Abdul Rahim Khan , Kadri Doğan

https://doi.org/10.33401/fujma.812274

Cited By: 1

Öz

In this paper, we propose and analyze a three-step general iteration method which is a special case of an iteration method proposed in (S. Thianwan and S. Suantai, Convergence criteria of a new three-step iteration with errors for nonexpansive nonself-mappings, Comput. Math. Appl. 52 (2006), 1107-1118). Here we intend to study directly the accumulation, estimation and control of random errors in the newly proposed general iteration method. We give conditions under which the accumulated-error in our iteration method is bounded and controllable in a permissible range.

Anahtar Kelimeler

Error Analysis, Random Errors, TS iteration method

Kaynakça

• [1] C. Garodia, I. Uddin, A new iterative method for solving split feasibility problem, J. Appl. Anal. Comput., 10(3) (2020), 986-1004.
• [2] C. Garodia, I. Uddin, A new fixed-point algorithm for finding the solution of a delay differential equation, AIMS Math., 5 (4) (2020), 3182-3200.
• [3] E. Hacıo˘glu, F. Gürsoy, S. Maldar, Y. Atalan, G. V. Milovanovic, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning, Appl. Numer. Math., 167 (2021), 143–172.
• [4] S. Maldar, F. Gürsoy, Y. Atalan, M. Abbas, On a three-step iteration process for multivalued Reich-Suzuki type a nonexpansive and contraction mappings, J. Appl. Math. Comput., (2021). https://doi.org/10.1007/s12190-021-01552-7.
• [5] S. Maldar, Y. Atalan, K. Doğan, Comparison rate of convergence and data dependence for a new iteration method, Tbilisi Math. J., 13(4) (2020), 65–79.
• [6] S. Maldar, An examination of data dependence for Jungck-type iteration method, Erciyes Univ. J. Inst. Sci. Tech., 36 (3) (2020), 374–384.
• [7] E. Hacıoğlu, V. Karakaya, Existence and convergence for a new multivalued hybrid mapping in CAT(k) spaces, Carpathian J. Math., 33(3) (2017), 319–326.
• [8] E. Hacıoğlu, V. Karakaya, Some fixed point results for a multivalued generalization of generalized hybrid mappings in CAT(k)-spaces, Konuralp J. Math., 6(1) (2018), 26–34.
• [9] E. Hacıoğlu, V. Karakaya, A new contraction-like multivalued mapping on geodesic spaces, Sci. Stud. Res. Ser. Math. Inform., 29(1) (2019), 89–102.
• [10] F. Gürsoy, K. Doğan, A. R. Khan, Direct estimate of accumulated errors for a general iteration method, Math. Adv. Pure Appl. Sci. (MAPAS), 2(2019), 19–24.
• [11] Y. Xu, Z. Liu, On estimation and control of errors of the Mann iteration process, J. Math. Anal. Appl., 286 (2003), 804-806.
• [12] Y. Xu, Z. Liu, S. M. Kang, Accumulation and control of random errors in the Ishikawa iterative process in arbitrary Banach space, Comput. Math. Appl., 61 (2011), 2217-2220.
• [13] S. Thianwan, S. Suantai, Convergence criteria of a new three-step iteration with errors for nonexpansive nonself-mappings, Comput. Math. Appl., 52 (2006), 1107-1118.
• [14] K. Nammanee, S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces, Appl. Math. Comput., 187 (2007), 669-679.
• [15] K. Nammanee, M. A. Noor, S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 314 (2006), 320-334.
• [16] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000) 217–229.
• [17] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
• [18] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
• [19] S. M. Şoltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory A., 2008 (2008),1-7.