Self-adaptive ergodic algorithm for equilibrium problems over the fixed point set

ABSTRACT

We introduce a new ergodic algorithm for solving equilibrium problems over the fixed point set of a nonexpansive mapping. In contrast to the existing one in Kim [The Bruck's ergodic iteration method for the Ky Fan inequality over the fixed point set. Int. J. Comput. Math. 94 (2017), pp. 2466–2480], our algorithm uses self-adaptive step sizes. Thanks to that, the proposed algorithm converges under milder conditions. Moreover, at each step of our algorithm, instead of solving strongly convex problems, we only have to compute a subgradient of a convex function. Hence, our algorithm has lower computational cost.

Keywords:

• Equilibrium problem
• monotonicity
• nonexpansive mapping
• fixed point problem
• ergodic algorithm

2010 Mathematics Subject Classifications:

• 47H05
• 47H09
• 47H10
• 47J25
• 65K10

Acknowledgements

The authors thank two anonymous referees and the editor for their constructive comments which helped to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the author.