A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces

ABSTRACT

The purpose of this article is to present a new iterative technique for approximating solutions of split generalized equilibrium problem and common fixed points of multivalued demicontractive mappings satisfying the gate conditions in real Hilbert spaces. Unlike the earlier results in this direction, we obtain a strong convergence result using an Armijo line search rule for determining the best appropriate step size for the next iteration. Also, our algorithm is designed in such a way that it does not require a projection onto the feasible set. We further give an analysis of the convergence rate of our method which is shown to be $O\left(1/t\right)$. Finally, the performances and comparisons with some existing methods are presented through numerical experiments. This result extends, generalizes and improves many of the existing related results in a unified way.

Keywords:

• Split generalized equilibrium problem
• split feasibility problem
• multivalued demicontractive mapping
• gate condition
• fixed point problem

2010 Mathematics Subject Classifications:

• 47H06
• 47H09
• 47J05
• 47J25

Acknowledgments

The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first and second authors 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 NRF and CoE-MaSS.

Disclosure statement

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

Additional information

Funding

The first and second authors 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 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 and CoE-MaSS.