Iterative algorithms for solving fixed point problems and variational inequalities with uniformly continuous monotone operators

Using the double projection and Halpern methods, we prove two strong convergence results for finding a solution of a variational inequality problem involving uniformly continuous monotone operator which is also a fixed point of a quasi-nonexpansive mapping in a real Hilbert space. In our proposed methods, only two projections onto the feasible set in each iteration are performed, rather than one projection for each tentative step during the Armijo-type search, which represents a considerable saving especially when the projection is computationally expensive. We also give some numerical results which show that our proposed algorithms are efficient and implementable from the numerical point of view.