An inertial projection and contraction method with a line search technique for variational inequality and fixed point problems

Abstract

In this paper, we study the approximation of common elements in the set of solutions of a variational inequality problem with monotone and Lipschitz continuous operator and the set of fixed points of a relatively nonexpansive mapping in real Hilbert space. We introduce an inertial projection and contraction method with a line search technique for solving the considered problem. A strong convergence result is proved without any prior estimate of the Lipschitz constant of the variational inequality operator. Furthermore, we provide some numerical experiments using performance profile metrics and application to image deblurring problem to illustrate the efficiency and accuracy of our algorithm.

Keywords:

• Variational inequalities
• extragradient method
• line search
• monotone operator
• projection and contraction
• Banach spaces

2010 Mathematics Subject Classifications:

• 65K15
• 47J25
• 65J15
• 90C33

Acknowledgments

The author sincerely thank the editor and anonymous referees for the careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The author acknowledge with thanks, the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University for making their facilities available for the research. This research is supported by the postdoctoral research fund at the Sefako Makgatho Health Science University, Pretoria, South Africa.

Disclosure statement

No potential conflict of interest was reported by the author(s).