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

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.