In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the setting of Hilbert space. We propose a modified inertial viscosity subgradient extragradient algorithm with self-adaptive stepsize in which the two projections are made onto some half-spaces. Moreover, we obtain a strong convergence result for approximating a common solution of the variational inequality and fixed point of quasi-nonexpansive mappings under some mild conditions. The main advantages of our method are: the self adaptive step-size which avoids the need to know apriori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which speeds up the rate of convergence of the algorithm. Numerical experiments are presented to demonstrate the efficiency of our algorithm in comparison with other existing algorithms in literature.

In this paper, we propose a new viscosity type inertial extragradient method with Armijo line-search technique for approximating a common solution of equilibrium problem with pseudo-monotone bifunction and fixed points of relatively nonexpansive mapping in a real Hilbert space. Two advantages of our algorithm are that its convergence does not require the bifunction to satisfy any Lipschitz-type condition and only one strongly convex program and one projection onto the feasible set are perform at each iteration. Under some mild conditions on the control sequences, we state and prove a strong convergence theorem and also present two numerical examples to illustrate the performance of our algorithm. The results in this paper improve and generalize many recent results in this direction in the literature.