The equivalence of some iteration schemes with their errors for uniformly continuous strongly successively pseudo-contractive operators
Keywords:
Modified Mann-Ishikawa Iterations (with errors), Modified Noor-Multistep Iteration(with errors), Uniformly Continuous Maps, Strongly Successively Pseudocontractive Maps
Abstract
Some iteration schemes may converge faster for certain types of functions or structures in
an arbitrary space. In this paper, we show that the convergence of modified Mann iteration,
modified Mann iteration with errors, modified Ishikawa iteration, modified Ishikawa iteration
with errors, modified Noor iteration, modified Noor iteration with errors, modified multistep
iteration and modified multistep iteration with errors are equivalent for uniformly continuous
strongly successively pseudo-contractive maps in an arbitratry real Banach space. The results
generalize and extend the results of several authors, including Huang and Bu [1], Rhoades and
Soltuz [2â4] and improve the results of Huang et al. [5].
References
[1] Huang, Z. Y., & Bu, F. (2007). The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successive pseudocontractive mappings without Lipschitzian assumption, Journal of Mathematical Analysis and Applications, 325(1): 586-594.
[2] Rhoades, B. E., & Soltuz, S. M. (2003). The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operation, Journal of Mathematical Sciences, 42: 2645-2651.
[3] Rhoades, B. E., & Soltuz, S. M. (2004). The equivalence between the convergence of Ishikawa and Mann iterations for an assymtotically non expansive in the intermediate sense and strongly successively pseudo-contractive maps, Journal of Mathematical Analysis and Applications, 289: 266-278.
[4] Rhoades, B. E., & Soltuz, S. M. (2004). The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Analysis: Theory, Methods and Applications, 58: 219-228.
[5] Huang, Z. Y., Bu, F. W., & Noor, M. A. (2006). On the equivalence of the convergence criteria between modified Mann-Ishikawa and multistep iterations with errors for successively strongly pseudo contractive operators, Appl. Math. Comput., 181: 641-647.
[6] Tijani, K. R., & Olayemi, S. A. (2021). Approximate fixed point results for rational-type contraction mapping, International Journal of Mathematical Sciences and Optimization: Theory and Application, 7(1): 76-86.
[7] Okeke, G. A., Olaleru, J. O., & Olatinwo, M. O. (2020). Existence and approximation of fixed point of a nonlinear mapping satisfying rational type contractive inequality condition in complex-valued Banach spaces, International Journal of Mathematical Sciences and Optimization: Theory and Application, 7(1): 707-717.
[8] Eke, K. S., Olaoluwa, H. O., & Olaleru, J. O. (2024). Common Fixed Point Results for Asymptotic Quasi-Contraction Mappings in Quasi-Metric Spaces, International Journal of Mathematical Sciences and Optimization: Theory and Application, 10(1): 115-127.
[9] Ajibade, F. D., Nkwuda, F. M., Joshua, H., Fajusigbe, T. P., & Oshinubi, K. (2023). Investigation of the Fâ Algorithm on Strong Pseudocontractive Mappings and Its Application. Axioms, 12, 1041.
[10] Sahu, C. K., Srivastava, S. C., & Biswas, S. (2020). History, Development And Application of Pseudocontractive Mapping With Fixed Point Theory. International Journal of Mathematics Trends and Technology, 66(4):13-16.
[11] Zegeye, H., & Wega, G. B. (2021). f-pseudo-pseudocontractive mappings and their f-fixed points in Banach spaces. Journal of Nonlinear and Convex Analysis, 22(11): 2479-2498.
[12] Yang, L. P. (2007). The equivalence of Ishikawa-Mann and multi-step iterations in Banach space, A Journal of Chinese Universities, 16(1): 83-91.
[13] Olaleru, J. O., & Odumosu, M. O. (2009). On the equivalence of some iteration schemes with their errors, Scientia Magna, 5(2): 77-91.
[14] Goebel, K. (1969). An elementary proof of the fixed point theorem of Browder and Kirk. The Michigan Mathematical Journal 16(4).
[15] Xu, B. L., & Noor, M. A. (2002). Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces. J.Math. Anal.Appl., 267, 444-453.
[16] Huang, Z. Y. (2007). Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively pseudo contractive mappings without Lipschitzian assumptions, J. Math. Anal. Appl., 329, 935-947.
[17] Mann, W. R. (1953). Mean value methods in iteration method. Proc. Amer. Math. Soc., 4(3):506-510.
[18] Ishikawa, S. (1974). Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1): 147-150.
[19] Noor, M. A. (2007). Some developments in general variational inequalities, Appl. Math. Comput., 15(2): 199-277.
[20] Noor, M. A., Rassias, T. M., & Huang, Z. H. (2002). Three-step iterations for nonlinear accretive operator equations, J. Math.Anal.Appl., 274, 59-68.
[2] Rhoades, B. E., & Soltuz, S. M. (2003). The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operation, Journal of Mathematical Sciences, 42: 2645-2651.
[3] Rhoades, B. E., & Soltuz, S. M. (2004). The equivalence between the convergence of Ishikawa and Mann iterations for an assymtotically non expansive in the intermediate sense and strongly successively pseudo-contractive maps, Journal of Mathematical Analysis and Applications, 289: 266-278.
[4] Rhoades, B. E., & Soltuz, S. M. (2004). The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Analysis: Theory, Methods and Applications, 58: 219-228.
[5] Huang, Z. Y., Bu, F. W., & Noor, M. A. (2006). On the equivalence of the convergence criteria between modified Mann-Ishikawa and multistep iterations with errors for successively strongly pseudo contractive operators, Appl. Math. Comput., 181: 641-647.
[6] Tijani, K. R., & Olayemi, S. A. (2021). Approximate fixed point results for rational-type contraction mapping, International Journal of Mathematical Sciences and Optimization: Theory and Application, 7(1): 76-86.
[7] Okeke, G. A., Olaleru, J. O., & Olatinwo, M. O. (2020). Existence and approximation of fixed point of a nonlinear mapping satisfying rational type contractive inequality condition in complex-valued Banach spaces, International Journal of Mathematical Sciences and Optimization: Theory and Application, 7(1): 707-717.
[8] Eke, K. S., Olaoluwa, H. O., & Olaleru, J. O. (2024). Common Fixed Point Results for Asymptotic Quasi-Contraction Mappings in Quasi-Metric Spaces, International Journal of Mathematical Sciences and Optimization: Theory and Application, 10(1): 115-127.
[9] Ajibade, F. D., Nkwuda, F. M., Joshua, H., Fajusigbe, T. P., & Oshinubi, K. (2023). Investigation of the Fâ Algorithm on Strong Pseudocontractive Mappings and Its Application. Axioms, 12, 1041.
[10] Sahu, C. K., Srivastava, S. C., & Biswas, S. (2020). History, Development And Application of Pseudocontractive Mapping With Fixed Point Theory. International Journal of Mathematics Trends and Technology, 66(4):13-16.
[11] Zegeye, H., & Wega, G. B. (2021). f-pseudo-pseudocontractive mappings and their f-fixed points in Banach spaces. Journal of Nonlinear and Convex Analysis, 22(11): 2479-2498.
[12] Yang, L. P. (2007). The equivalence of Ishikawa-Mann and multi-step iterations in Banach space, A Journal of Chinese Universities, 16(1): 83-91.
[13] Olaleru, J. O., & Odumosu, M. O. (2009). On the equivalence of some iteration schemes with their errors, Scientia Magna, 5(2): 77-91.
[14] Goebel, K. (1969). An elementary proof of the fixed point theorem of Browder and Kirk. The Michigan Mathematical Journal 16(4).
[15] Xu, B. L., & Noor, M. A. (2002). Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces. J.Math. Anal.Appl., 267, 444-453.
[16] Huang, Z. Y. (2007). Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively pseudo contractive mappings without Lipschitzian assumptions, J. Math. Anal. Appl., 329, 935-947.
[17] Mann, W. R. (1953). Mean value methods in iteration method. Proc. Amer. Math. Soc., 4(3):506-510.
[18] Ishikawa, S. (1974). Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1): 147-150.
[19] Noor, M. A. (2007). Some developments in general variational inequalities, Appl. Math. Comput., 15(2): 199-277.
[20] Noor, M. A., Rassias, T. M., & Huang, Z. H. (2002). Three-step iterations for nonlinear accretive operator equations, J. Math.Anal.Appl., 274, 59-68.
Published
2025-02-28
How to Cite
Odumosu, M. O., Olaleru, J. O., & Ayodele, I. O. (2025). The equivalence of some iteration schemes with their errors for uniformly continuous strongly successively pseudo-contractive operators. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 11(1), 32-44. Retrieved from http://ijmso.unilag.edu.ng/article/view/2456
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