A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings

Abstract

We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.

Keywords:

• Bilevel Equilibria
• splitting
• parallel algorithm
• nonexpansive mapping
• equilibrium constraint
• maximal monotone inclusion

AMS Subject Classifications:

• 65 K10
• 90 C25

Acknowledgements

We would like to thank the Associate Editor and the referees for their useful remarks, comments and suggestions that helped us very much in revising the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Foundation for Science and Technology Development (NAFOSTED), Vietnam.