Viscosity s-iteration algorithm for finding common fixed point of nonexpansive mappings and application to nonexpansive semigroups

Adeolu Taiwo; Timilehin Opeyemi Alakoya; Lateef Olakunle O. Jolaoso; Oluwatosin Temitope Mewomo

doi:10.18514/MMN.2023.4024

Viscosity s-iteration algorithm for finding common fixed point of nonexpansive mappings and application to nonexpansive semigroups

Adeolu Taiwo, Timilehin Opeyemi Alakoya, Lateef Olakunle O. Jolaoso, Oluwatosin Temitope Mewomo^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We study the problem of finding the common element of the set of fixed points of two nonexpansive mappings in Hilbert spaces. Some previous attempts in this direction make some restrictive assumptions which may be difficult to check in practice on the generated sequence. In this paper, we introduce a viscosity S-iteration scheme for finding the common fixed point of two nonexpansive mappings. Under some very mild conditions, we obtain a strong convergence theorem for the sequence generated by our algorithm. We apply our main result to approximating common fixed points of nonexpansive semigroups. We also provide numerical examples to support our main results and illustrate the efficiency and effectiveness of our algorithm by comparing with some existing algorithms in literature. This work generalizes and improves some existing works in the literature in this direction.

Original language	English
Pages (from-to)	1581-1603
Number of pages	23
Journal	Miskolc Mathematical Notes
Volume	24
Issue number	3
DOIs	https://doi.org/10.18514/MMN.2023.4024
Publication status	Published - 01 Jul 2023
Externally published	Yes

Bibliographical note

Funding Information:
The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The third author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS, IMU and NRF.

Funding Information:
The first author was supported by International Mathematical Union (IMU) Breakout Graduate Fellowship. The third author was supported by Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS). The fourth author was supported in part by the National Research Foundation (NRF), South Africa, Grant No. 119903.

Publisher Copyright:
© 2023 Miskolc University Press

Keywords

Hilbert space
nonexpansive semgroup
S-iteration
viscosity approximation algorithm

ASJC Scopus subject areas

Analysis
Algebra and Number Theory
Numerical Analysis
Discrete Mathematics and Combinatorics
Control and Optimization

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10.18514/MMN.2023.4024Licence: Unspecified

Cite this

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title = "Viscosity s-iteration algorithm for finding common fixed point of nonexpansive mappings and application to nonexpansive semigroups",

abstract = "We study the problem of finding the common element of the set of fixed points of two nonexpansive mappings in Hilbert spaces. Some previous attempts in this direction make some restrictive assumptions which may be difficult to check in practice on the generated sequence. In this paper, we introduce a viscosity S-iteration scheme for finding the common fixed point of two nonexpansive mappings. Under some very mild conditions, we obtain a strong convergence theorem for the sequence generated by our algorithm. We apply our main result to approximating common fixed points of nonexpansive semigroups. We also provide numerical examples to support our main results and illustrate the efficiency and effectiveness of our algorithm by comparing with some existing algorithms in literature. This work generalizes and improves some existing works in the literature in this direction.",

keywords = "Hilbert space, nonexpansive semgroup, S-iteration, viscosity approximation algorithm",

author = "Adeolu Taiwo and Alakoya, {Timilehin Opeyemi} and Jolaoso, {Lateef Olakunle O.} and Mewomo, {Oluwatosin Temitope}",

note = "Funding Information: The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The third author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS, IMU and NRF. Funding Information: The first author was supported by International Mathematical Union (IMU) Breakout Graduate Fellowship. The third author was supported by Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS). The fourth author was supported in part by the National Research Foundation (NRF), South Africa, Grant No. 119903. Publisher Copyright: {\textcopyright} 2023 Miskolc University Press",

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AU - Alakoya, Timilehin Opeyemi

AU - Jolaoso, Lateef Olakunle O.

AU - Mewomo, Oluwatosin Temitope

N1 - Funding Information: The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The third author acknowledges with thanks the bursary and financial support from Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS) Doctoral Bursary. The fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS, IMU and NRF. Funding Information: The first author was supported by International Mathematical Union (IMU) Breakout Graduate Fellowship. The third author was supported by Department of Science and Innovation and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DSI-NRF COE-MaSS). The fourth author was supported in part by the National Research Foundation (NRF), South Africa, Grant No. 119903. Publisher Copyright: © 2023 Miskolc University Press

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N2 - We study the problem of finding the common element of the set of fixed points of two nonexpansive mappings in Hilbert spaces. Some previous attempts in this direction make some restrictive assumptions which may be difficult to check in practice on the generated sequence. In this paper, we introduce a viscosity S-iteration scheme for finding the common fixed point of two nonexpansive mappings. Under some very mild conditions, we obtain a strong convergence theorem for the sequence generated by our algorithm. We apply our main result to approximating common fixed points of nonexpansive semigroups. We also provide numerical examples to support our main results and illustrate the efficiency and effectiveness of our algorithm by comparing with some existing algorithms in literature. This work generalizes and improves some existing works in the literature in this direction.

AB - We study the problem of finding the common element of the set of fixed points of two nonexpansive mappings in Hilbert spaces. Some previous attempts in this direction make some restrictive assumptions which may be difficult to check in practice on the generated sequence. In this paper, we introduce a viscosity S-iteration scheme for finding the common fixed point of two nonexpansive mappings. Under some very mild conditions, we obtain a strong convergence theorem for the sequence generated by our algorithm. We apply our main result to approximating common fixed points of nonexpansive semigroups. We also provide numerical examples to support our main results and illustrate the efficiency and effectiveness of our algorithm by comparing with some existing algorithms in literature. This work generalizes and improves some existing works in the literature in this direction.

KW - Hilbert space

KW - nonexpansive semgroup

KW - S-iteration

KW - viscosity approximation algorithm

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DO - 10.18514/MMN.2023.4024

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JO - Miskolc Mathematical Notes

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IS - 3

ER -

Viscosity s-iteration algorithm for finding common fixed point of nonexpansive mappings and application to nonexpansive semigroups

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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