Research Article
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Year 2022, , 35 - 39, 30.04.2022
https://doi.org/10.33187/jmsm.993823

Abstract

References

  • [1] W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506-510.
  • [2] I. Ishikawa, Fixed point by a new iteration method, Proc. Am. Math. Soc. 44 (1974), 147-150.
  • [3] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.
  • [4] R. P. Agarwal, D. O’Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8(1) (2007), 61-79.
  • [5] M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik, 66(2) (2014), 223-234.
  • [6] F. G¨ursoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, (2014), Preprint arXiv 1403.2546, 1-16.
  • [7] B. S. Thakur, D. Thakur, M. Postolache, A new iteration scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
  • [8] K. Ullah, M. Arshad, Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat, 32(1) (2018), 187-196.
  • [9] K. Aoyama, F. Kohsaka, Fixed point theorem for a-nonexpansive mappings in Banach spaces, Nonlinear Analy., 74(13) (2011), 4378-4391.
  • [10] M. Bas¸arır, A. S¸ ahin, On the strong and D-convergence of S-iteration process for generalized nonexpansive mappings on CAT(0) space, Thai J. Math., 12(3) (2014), 549-559.
  • [11] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414.
  • [12] N. Hussain, K. Ullah, M. Arshad, Fixed point approximation of Suzuki generalized non-expansive mappings via new faster iteration process, J. Nonlinear Convex Anal., 19 (2018), 1383-1393.
  • [13] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Ame. Math.Soc., 73 (1967), 591-597.
  • [14] R. Pant, R. Shukla, Approximating fixed points of generalized a-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 38(2) (2017), 248-266.
  • [15] A. S¸ ahin, Some new results of M-iteration process in hyperbolic spaces, Carpathian J. Math., 35(2) (2019), 221-232.
  • [16] A. S¸ ahin, M. Bas¸arır, Some convergence results of the K*-iteration process in CAT(0) spaces , In: J. L. Cho, Y.L. Jleli, M. Mursaleen, B.Samet, C.Vetro, (Eds.), Advances in Metric Fixed Point Theory and Applications, Springer, Singapore, (2021).
  • [17] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340(2) (2008), 1088-1095.
  • [18] J. Schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.
  • [19] H. F. Senter, W. G. Dotson Jr., Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Soc. 44 (1974), 375-380.
  • [20] K. Ullah, F. Ayaz, J. Ahmad, Some convergence results of M iterative process in Banach spaces, Asian-European Journal of Mathematics (2021), 2150017, 12 pages, doi:10.1142/S1793557121500170.

Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process

Year 2022, , 35 - 39, 30.04.2022
https://doi.org/10.33187/jmsm.993823

Abstract

In this paper we introduce a new iteration process for approximation of fixed points. We numerically compare convergence behavior of our iteration process with other iteration process like M-iteration process. We also prove weak and strong convergence theorems for generalized $\alpha-$nonexpansive mappings by using new iteration process. Furthermore we give an example for generalized $\alpha-$nonexpansive mapping but does not satisfy $(C)$ condition.

References

  • [1] W. R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953), 506-510.
  • [2] I. Ishikawa, Fixed point by a new iteration method, Proc. Am. Math. Soc. 44 (1974), 147-150.
  • [3] M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.
  • [4] R. P. Agarwal, D. O’Regan, D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8(1) (2007), 61-79.
  • [5] M. Abbas, T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Mat. Vesnik, 66(2) (2014), 223-234.
  • [6] F. G¨ursoy, V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, (2014), Preprint arXiv 1403.2546, 1-16.
  • [7] B. S. Thakur, D. Thakur, M. Postolache, A new iteration scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings, Appl. Math. Comput., 275 (2016), 147-155.
  • [8] K. Ullah, M. Arshad, Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat, 32(1) (2018), 187-196.
  • [9] K. Aoyama, F. Kohsaka, Fixed point theorem for a-nonexpansive mappings in Banach spaces, Nonlinear Analy., 74(13) (2011), 4378-4391.
  • [10] M. Bas¸arır, A. S¸ ahin, On the strong and D-convergence of S-iteration process for generalized nonexpansive mappings on CAT(0) space, Thai J. Math., 12(3) (2014), 549-559.
  • [11] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414.
  • [12] N. Hussain, K. Ullah, M. Arshad, Fixed point approximation of Suzuki generalized non-expansive mappings via new faster iteration process, J. Nonlinear Convex Anal., 19 (2018), 1383-1393.
  • [13] Z. Opial, Weak convergence of successive approximations for nonexpansive mappings, Bull. Ame. Math.Soc., 73 (1967), 591-597.
  • [14] R. Pant, R. Shukla, Approximating fixed points of generalized a-nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 38(2) (2017), 248-266.
  • [15] A. S¸ ahin, Some new results of M-iteration process in hyperbolic spaces, Carpathian J. Math., 35(2) (2019), 221-232.
  • [16] A. S¸ ahin, M. Bas¸arır, Some convergence results of the K*-iteration process in CAT(0) spaces , In: J. L. Cho, Y.L. Jleli, M. Mursaleen, B.Samet, C.Vetro, (Eds.), Advances in Metric Fixed Point Theory and Applications, Springer, Singapore, (2021).
  • [17] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340(2) (2008), 1088-1095.
  • [18] J. Schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.
  • [19] H. F. Senter, W. G. Dotson Jr., Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Soc. 44 (1974), 375-380.
  • [20] K. Ullah, F. Ayaz, J. Ahmad, Some convergence results of M iterative process in Banach spaces, Asian-European Journal of Mathematics (2021), 2150017, 12 pages, doi:10.1142/S1793557121500170.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Seyit Temir 0000-0001-9056-2354

Öznur Korkut This is me

Publication Date April 30, 2022
Submission Date September 10, 2021
Acceptance Date December 27, 2021
Published in Issue Year 2022

Cite

APA Temir, S., & Korkut, Ö. (2022). Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process. Journal of Mathematical Sciences and Modelling, 5(1), 35-39. https://doi.org/10.33187/jmsm.993823
AMA Temir S, Korkut Ö. Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process. Journal of Mathematical Sciences and Modelling. April 2022;5(1):35-39. doi:10.33187/jmsm.993823
Chicago Temir, Seyit, and Öznur Korkut. “Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process”. Journal of Mathematical Sciences and Modelling 5, no. 1 (April 2022): 35-39. https://doi.org/10.33187/jmsm.993823.
EndNote Temir S, Korkut Ö (April 1, 2022) Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process. Journal of Mathematical Sciences and Modelling 5 1 35–39.
IEEE S. Temir and Ö. Korkut, “Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process”, Journal of Mathematical Sciences and Modelling, vol. 5, no. 1, pp. 35–39, 2022, doi: 10.33187/jmsm.993823.
ISNAD Temir, Seyit - Korkut, Öznur. “Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process”. Journal of Mathematical Sciences and Modelling 5/1 (April 2022), 35-39. https://doi.org/10.33187/jmsm.993823.
JAMA Temir S, Korkut Ö. Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process. Journal of Mathematical Sciences and Modelling. 2022;5:35–39.
MLA Temir, Seyit and Öznur Korkut. “Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process”. Journal of Mathematical Sciences and Modelling, vol. 5, no. 1, 2022, pp. 35-39, doi:10.33187/jmsm.993823.
Vancouver Temir S, Korkut Ö. Approximating Fixed Points of Generalized $\alpha-$Nonexpansive Mappings by the New Iteration Process. Journal of Mathematical Sciences and Modelling. 2022;5(1):35-9.

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