Contraction of the proximal mapping and applications to the equilibrium problem

Abstract

In this paper, we introduce a new definition of Lipschitz-type continuity of a bifunction. Using this definition, we prove the contraction of the proximal mapping and apply it to the equilibrium problem over the fixed-point set of a nonexpansive mapping. We present a new algorithm for this problem. Under classical conditions, the convergence of the algorithm is proved. Finally, we present some numerical results for the proposed algorithm.

Keywords:

• Proximal mapping
• equilibrium problem
• variational inequality problem
• fixed-point problem
• contractive mapping
• nonexpansive mapping

AMS Subject Classifications:

• 90C33
• 65K10
• 90C25

Acknowledgements

The author would like to thank the anonymous referees and the editor for constructive suggestions that led to improvements in the paper.

Notes

No potential conflict of interest was reported by the authors.