R-linear and strong convergence of Tseng methods for solving certain split and non-split problems

Abstract

In the recent past, authors have fondly assumed the co-coercive condition for solving certain class of split inverse problems. Recently, some attempts have been made to relax this very strict condition, however, only few results exists in literature in such case. In this paper, we present some new Mann-type Tseng methods for solving split monotone variational inclusion and common fixed point problems for a finite collection of quasi-pseudocontractive operators. We prove strong convergence and $R-$ linear convergence rate results of our methods, while the co-coerciveness property is dispensed with. Our methods incorporate the relaxation, inertial extrapolation and self-adaptive step size techniques to achieve better performance and faster convergence. With the aid of relevant numerical experiments, we showcase the performance profile of our methods in comparison with recent methods in literature.

Keywords:

• Mann-Tseng method
• split inverse
• monotone inclusions
• Lipschitz continuity
• inertial
• relaxation technique
• linear & strong convergence
• self-adaptive step size
• Quasi-pseudocontractive operators

AMS Subject Classifications:

• 65Y05
• 65K15
• 65J15
• 47H09
• 49J40
• 68W10

Acknowledgments

The authors sincerely thank the Editor and the anonymous reviewers for their careful reading, constructive comments and useful suggestions. Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the IMU, NRF and (CoE-MaSS).

Disclosure statement

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

Additional information

Funding

The first author acknowledges with thanks the International Mathematical Union (IMU) Breakout Graduate Fellowship (BGF) Award for his doctoral study. The second author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers [grant number 119903] and Department of Science and Innovation DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa [grant number 2022-087-OPA].