Inertial algorithm with self-adaptive step size for split common null point and common fixed point problems for multivalued mappings in Banach spaces

ABSTRACT

In this article we introduce an inertial iterative scheme with self-adaptive step size for finding a common solution of split common null point problem for a finite family of maximal monotone operators and fixed point problem for a finite family of multivalued demicontractive mappings between a Banach space and Hilbert space. Strong convergence result is obtained for the proposed algorithm. The self-adaptive step size ensures no requirement for a prior knowledge or estimate of the norm of the operator. The inertial term introduced in the algorithm is efficient, it helps to avoid imposing some strong conditions usually used for inertial-type algorithms by many authors. We give some applications of our results to game theory, split equilibrium and minimum-norm problems. Numerical experiment is also presented to demonstrate the efficiency of our proposed method as well as comparing with other existing method in the literature. Our results improve and generalize many well known results in this direction in the literature.

Keywords:

• Split common null point
• inertial algorithm
• self-adaptive
• multivalued mappings
• demicontractive mappings
• strong convergence

2010 Mathematics Subject Classifications:

• 65K15
• 47J25
• 65J15
• 90C33

Acknowledgements

The authors sincerely thank the editor and anonymous reviewers for their careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The second author acknowledges with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. The third author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. 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.

Disclosure statement

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

Additional information

Funding

The second author acknowledges with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. The third author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The fourth author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers [grant number 119903].