Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

Abstract

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.

Keywords:

• Iterative method
• variational inequality
• zero point
• fixed point
• nonexpansive mapping
• weak convergence
• hybrid steepest descent method

Acknowledgements

The authors thank the referees for their helping comments, which notably improved the presentation of this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Fundamental Research Funds for the Central Universities [grant number 3122016L006]; Ming Tian was supported by the Foundation of Tianjin Key Laboratory for Advanced Signal Processing.