An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.

KEYWORDS:

• Variational inequalities
• Fixed points
• Strong convergence
• Subgradient extragradient method
• Projection and contraction method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 47H05
• 47J20
• 47J25
• 65K15
• 90C25

Acknowledgments

We thank the anonymous referees for their useful comments which helped improve an earlier version of this paper. The research was carried out when the second author was an Alexander von Humboldt Postdoctoral Fellow at the Institute of Mathematics, University of Würzburg, Germany. He is grateful to the Alexander von Humboldt Foundation, Bonn, for the fellowship and the Institute of Mathematics, University of Würzburg, Germany, for the hospitality and facilities.

Disclosure statement

No potential conflict of interest was reported by the authors.