Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

Furmose Mendy; John T Mendy; Jatta Bah; Gabriel Mendy

doi:doi:10.11648/j.ijtam.20230902.12

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Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

Furmose Mendy, John T Mendy, Jatta Bah, Gabriel Mendy

Published in International Journal of Theoretical and Applied Mathematics (Volume 9, Issue 2)

Received: 3 May 2023 Accepted: 18 May 2023 Published: 1 November 2023

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Abstract

Viscosity’s implicit algorithm for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors in different settings in Hilbert and Banach space. In most cases, they consider the following study of viscosity implicit double midpoint, generalized viscosity in the class of nonexpansive and asymptotically nonexpansive mappings. The implicit midpoint rule can effectively solve ordinary differential equations. Meanwhile, many authors have used viscosity iterative algorithms for finding common fixed points for nonlinear operators and solutions of variational inequality problems. Recently, the convergence rate and comparison viscosity implicit iterative algorithm has been studied widely. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences {ω_n} and {ξ_n} converge to the same fixed point of an asymptotically nonexpansive mapping in Hilbert spaces without comparison. It is also proven that {ω_n} and {ξ_n} converge strongly to the same solution, which also solves the variational inequality problem. The results presented in this paper improve and extend some recent corresponding results in the literature.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 9, Issue 2)
DOI	10.11648/j.ijtam.20230902.12
Page(s)	14-22
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Viscosity, Hilbert Space, Asymptotically Nonexpansive Mapping, Fixed Point

References

[1]	Aibinu, M.O. and Kim, J.K., 2020. On the rate of convergence of viscosity implicit iterative algorithms. Nonlinear Funct. Anal. Appl, 25 (1), pp. 135-152.
[2]	F.E. Browder, Existence of periodic solutions for nonlinear equations of evolution, Proc. Natl. Acad. Sci. USA 53 (5) (1965), 1100- 1103.
[3]	H.K. Xu, M.A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl. 41 (2015).
[4]	H.K. Xu, Iterative algorithms for nonlinear op erators, J. Lond. Math. So c. 66 (2) (2002) 240-256.
[5]	K.Go eb el, W.A. Kirk, Topics in Me tric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990).
[6]	L.-C. Zhao, S.-S. Chang, C.-F. Wen, Viscosity approximation methods for the implicit midp oint rule of asymptotically nonexpansive mappings in Hilbert spaces, J. Nonlinear Sci. Appl. 9 (6) (2016) 4478-4488.
[7]	Mendy, J.T. and Shukla, R., 2022. Viscosity like implicit methods for zeros of monotone operators in Banach spaces. Khayyam Journal of Mathematics, 8 (1), pp. 53-72.
[8]	Mendy, J.T., 2020. The viscosity iterative algorithms for the implicit double midpoint rule of nonexpansive mappings in Hilbert spaces. American Journal of Mathematical Analysis, 8 (1), pp. 1-8.
[9]	Mendy, S.B., Mendy,J. T. and Jobe, A., 2021. The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. European Journal of Mathematical Analysis, 1, pp. 19-33.
[10]	S.F.A. Naqvi, M.S. Khan, On the viscosity rule for common fixed points of two nonexpansive mappings in Hilbert spaces, Open J. Math. Sci. 1 (1) (2017) 111-125.
[11]	S. He, Y. Mao, Z. Zhou, J.Q. Zhang, The generalized viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces, Applied Mathematical Science 11 (12) (2017) 549-560.
[12]	Sijun He, Yingdong Mao, Zheng Zhou, and Jian- Qiang Zhang. The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilb ert Spaces, Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718
[13]	Y. Ke, C. Ma, The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Appl. 190 (2015).

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APA Style

Mendy, F., T Mendy, J., Bah, J., Mendy, G. (2023). Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. International Journal of Theoretical and Applied Mathematics, 9(2), 14-22. https://doi.org/10.11648/j.ijtam.20230902.12

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ACS Style

Mendy, F.; T Mendy, J.; Bah, J.; Mendy, G. Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. Int. J. Theor. Appl. Math. 2023, 9(2), 14-22. doi: 10.11648/j.ijtam.20230902.12

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AMA Style

Mendy F, T Mendy J, Bah J, Mendy G. Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. Int J Theor Appl Math. 2023;9(2):14-22. doi: 10.11648/j.ijtam.20230902.12

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@article{10.11648/j.ijtam.20230902.12,
  author = {Furmose Mendy and John T Mendy and Jatta Bah and Gabriel Mendy},
  title = {Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {9},
  number = {2},
  pages = {14-22},
  doi = {10.11648/j.ijtam.20230902.12},
  url = {https://doi.org/10.11648/j.ijtam.20230902.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20230902.12},
  abstract = {Viscosity’s implicit algorithm for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors in different settings in Hilbert and Banach space. In most cases, they consider the following study of viscosity implicit double midpoint, generalized viscosity in the class of nonexpansive and asymptotically nonexpansive mappings. The implicit midpoint rule can effectively solve ordinary differential equations. Meanwhile, many authors have used viscosity iterative algorithms for finding common fixed points for nonlinear operators and solutions of variational inequality problems. Recently, the convergence rate and comparison viscosity implicit iterative algorithm has been studied widely. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences {ωn} and {ξn} converge to the same fixed point of an asymptotically nonexpansive mapping in Hilbert spaces without comparison. It is also proven that {ωn} and {ξn} converge strongly to the same solution, which also solves the variational inequality problem. The results presented in this paper improve and extend some recent corresponding results in the literature.
},
 year = {2023}
}

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TY  - JOUR
T1  - Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
AU  - Furmose Mendy
AU  - John T Mendy
AU  - Jatta Bah
AU  - Gabriel Mendy
Y1  - 2023/11/01
PY  - 2023
N1  - https://doi.org/10.11648/j.ijtam.20230902.12
DO  - 10.11648/j.ijtam.20230902.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 14
EP  - 22
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20230902.12
AB  - Viscosity’s implicit algorithm for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors in different settings in Hilbert and Banach space. In most cases, they consider the following study of viscosity implicit double midpoint, generalized viscosity in the class of nonexpansive and asymptotically nonexpansive mappings. The implicit midpoint rule can effectively solve ordinary differential equations. Meanwhile, many authors have used viscosity iterative algorithms for finding common fixed points for nonlinear operators and solutions of variational inequality problems. Recently, the convergence rate and comparison viscosity implicit iterative algorithm has been studied widely. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences {ωn} and {ξn} converge to the same fixed point of an asymptotically nonexpansive mapping in Hilbert spaces without comparison. It is also proven that {ωn} and {ξn} converge strongly to the same solution, which also solves the variational inequality problem. The results presented in this paper improve and extend some recent corresponding results in the literature.

VL  - 9
IS  - 2
ER  -

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Author Information

Furmose Mendy

Department of Mathematics, University of the Gambia, Brikama Campus, The Gambia

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John T Mendy

Department of Mathematics, University of the Gambia, Brikama Campus, The Gambia; Department of Mathematics, University of L’Aquila , L’Aquila, Italy

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Jatta Bah

Department of Mathematics, University of the Gambia, Brikama Campus, The Gambia

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Gabriel Mendy

Department of Mathematics, University of the Gambia, Brikama Campus, The Gambia

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Plain Text BibTeX RIS

APA Style

Mendy, F., T Mendy, J., Bah, J., Mendy, G. (2023). Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. International Journal of Theoretical and Applied Mathematics, 9(2), 14-22. https://doi.org/10.11648/j.ijtam.20230902.12

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ACS Style

Mendy, F.; T Mendy, J.; Bah, J.; Mendy, G. Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. Int. J. Theor. Appl. Math. 2023, 9(2), 14-22. doi: 10.11648/j.ijtam.20230902.12

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AMA Style

Mendy F, T Mendy J, Bah J, Mendy G. Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces. Int J Theor Appl Math. 2023;9(2):14-22. doi: 10.11648/j.ijtam.20230902.12

Copy | Download

@article{10.11648/j.ijtam.20230902.12,
  author = {Furmose Mendy and John T Mendy and Jatta Bah and Gabriel Mendy},
  title = {Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {9},
  number = {2},
  pages = {14-22},
  doi = {10.11648/j.ijtam.20230902.12},
  url = {https://doi.org/10.11648/j.ijtam.20230902.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20230902.12},
  abstract = {Viscosity’s implicit algorithm for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors in different settings in Hilbert and Banach space. In most cases, they consider the following study of viscosity implicit double midpoint, generalized viscosity in the class of nonexpansive and asymptotically nonexpansive mappings. The implicit midpoint rule can effectively solve ordinary differential equations. Meanwhile, many authors have used viscosity iterative algorithms for finding common fixed points for nonlinear operators and solutions of variational inequality problems. Recently, the convergence rate and comparison viscosity implicit iterative algorithm has been studied widely. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences {ωn} and {ξn} converge to the same fixed point of an asymptotically nonexpansive mapping in Hilbert spaces without comparison. It is also proven that {ωn} and {ξn} converge strongly to the same solution, which also solves the variational inequality problem. The results presented in this paper improve and extend some recent corresponding results in the literature.
},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
AU  - Furmose Mendy
AU  - John T Mendy
AU  - Jatta Bah
AU  - Gabriel Mendy
Y1  - 2023/11/01
PY  - 2023
N1  - https://doi.org/10.11648/j.ijtam.20230902.12
DO  - 10.11648/j.ijtam.20230902.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 14
EP  - 22
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20230902.12
AB  - Viscosity’s implicit algorithm for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors in different settings in Hilbert and Banach space. In most cases, they consider the following study of viscosity implicit double midpoint, generalized viscosity in the class of nonexpansive and asymptotically nonexpansive mappings. The implicit midpoint rule can effectively solve ordinary differential equations. Meanwhile, many authors have used viscosity iterative algorithms for finding common fixed points for nonlinear operators and solutions of variational inequality problems. Recently, the convergence rate and comparison viscosity implicit iterative algorithm has been studied widely. Under suitable conditions imposed on the control parameters, it is shown in this paper that certain two implicit iterative sequences {ωn} and {ξn} converge to the same fixed point of an asymptotically nonexpansive mapping in Hilbert spaces without comparison. It is also proven that {ωn} and {ξn} converge strongly to the same solution, which also solves the variational inequality problem. The results presented in this paper improve and extend some recent corresponding results in the literature.

VL  - 9
IS  - 2
ER  -

Copy | Download