Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

ABSTRACT

In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

KEYWORDS:

• Split inverse problems
• mann-type algorithm
• inertial technique
• self-adaptive step size

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65K15
• 47J25
• 65J15
• 90C33

Acknowledgments

The authors sincerely thank the anonymous referees for their careful reading, constructive comments and useful suggestions. This work was completed during the research visit of the second author to the Department of Mathematics, Clarkson University, Potsdam New York, United States. He is thankful to the Department of Mathematics at Clarkson University for hospitality. In particular, he is grateful to the fourth author for the invitation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Authors' contributions

Conceptualization of the article was given by TOA, OTM and OSI, methodology by TOA and OTM, formal analysis, investigation and writingoriginal draft preparation by MUW,TOA, OTM and OSI, software and validation by MUW,OTM and OSI, writingreview and editing by MUW, TOA, OTM and OSI, project administration and supervision by OTM and OSI. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Availability of data and material

Not applicable.

Code availability

The Matlab codes employed to run the numerical experiments are available upon request to the authors.

Additional information

Funding

The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers [grant number 119903] and DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa [grant number 2022-087-OPA]. The third author is funded by University of KwaZulu-Natal [Inyuvesi Yakwazulu-Natali], Durban, South Africa Postdoctoral Fellowship.