Modified forward-backward splitting method for split equilibrium, variational inclusion, and fixed point problems

Victor A. Uzor; Timilehin O. Alakoya; Oluwatosin T. Mewomo

doi:10.23952/asvao.5.2023.1.07

Modified forward-backward splitting method for split equilibrium, variational inclusion, and fixed point problems

Victor A. Uzor
, Timilehin O. Alakoya
, Oluwatosin T. Mewomo^*

^*Corresponding author for this work

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

9 Citations (Scopus)

Abstract

In the recent time, the problem of finding common solutions of fixed point problems (FPPs) of nonlinear mappings and optimization problems (OPs) has received great research attention due to its potential applications to mathematical models whose constraints can be expressed as the FPPs and OPs. In this paper, we study the problem of finding a common solution of a split equilibrium problem (SEP), a variational inclusion problem (VIP) and the FPP with a finite family of multivalued demicontractive mappings. We propose a new inertial iterative method, which employs the forward-backward splitting technique together with the viscosity method for approximating the solution of the problem in Hilbert spaces. The proposed method uses variable step sizes, which do not depend on the norm of the bounded linear operator. We prove strong convergence results under some mild conditions. Finally, we present some numerical experiments to demonstrate the efficiency and applicability of the proposed method. Our result improves and extends several existing results in the current literature in this direction.

Original language	English
Pages (from-to)	95-119
Number of pages	25
Journal	Applied Set-Valued Analysis and Optimization
Volume	5
Issue number	1
DOIs	https://doi.org/10.23952/asvao.5.2023.1.07
Publication status	Published - 01 Apr 2023
Externally published	Yes

Bibliographical note

Funding Information:
The authors sincerely thank the reviewer for his careful reading, constructive comments, and useful suggestions. The research of the second author was wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The third author was 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 NRF.

Publisher Copyright:
©2023 Applied Set-Valued Analysis and Optimization.

Keywords

Adaptive step size
Forward-backward splitting method
Inclusion problem
Inertial technique
Multivalued demicontractive mappings
Split equilibrium problem

ASJC Scopus subject areas

Analysis
Numerical Analysis
Mathematics (miscellaneous)
Modelling and Simulation
Control and Optimization
Applied Mathematics

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10.23952/asvao.5.2023.1.07Licence: Unspecified

Cite this

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abstract = "In the recent time, the problem of finding common solutions of fixed point problems (FPPs) of nonlinear mappings and optimization problems (OPs) has received great research attention due to its potential applications to mathematical models whose constraints can be expressed as the FPPs and OPs. In this paper, we study the problem of finding a common solution of a split equilibrium problem (SEP), a variational inclusion problem (VIP) and the FPP with a finite family of multivalued demicontractive mappings. We propose a new inertial iterative method, which employs the forward-backward splitting technique together with the viscosity method for approximating the solution of the problem in Hilbert spaces. The proposed method uses variable step sizes, which do not depend on the norm of the bounded linear operator. We prove strong convergence results under some mild conditions. Finally, we present some numerical experiments to demonstrate the efficiency and applicability of the proposed method. Our result improves and extends several existing results in the current literature in this direction.",

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N1 - Funding Information: The authors sincerely thank the reviewer for his careful reading, constructive comments, and useful suggestions. The research of the second author was wholly supported by the University of KwaZulu-Natal, Durban, South Africa Postdoctoral Fellowship. He is grateful for the funding and financial support. The third author was 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 NRF. Publisher Copyright: ©2023 Applied Set-Valued Analysis and Optimization.

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N2 - In the recent time, the problem of finding common solutions of fixed point problems (FPPs) of nonlinear mappings and optimization problems (OPs) has received great research attention due to its potential applications to mathematical models whose constraints can be expressed as the FPPs and OPs. In this paper, we study the problem of finding a common solution of a split equilibrium problem (SEP), a variational inclusion problem (VIP) and the FPP with a finite family of multivalued demicontractive mappings. We propose a new inertial iterative method, which employs the forward-backward splitting technique together with the viscosity method for approximating the solution of the problem in Hilbert spaces. The proposed method uses variable step sizes, which do not depend on the norm of the bounded linear operator. We prove strong convergence results under some mild conditions. Finally, we present some numerical experiments to demonstrate the efficiency and applicability of the proposed method. Our result improves and extends several existing results in the current literature in this direction.

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KW - Inclusion problem

KW - Inertial technique

KW - Multivalued demicontractive mappings

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Modified forward-backward splitting method for split equilibrium, variational inclusion, and fixed point problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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