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We study variational inequalities and fixed point problems in real Hilbert spaces. A new algorithm is proposed for finding a common element of the solution set of a pseudo-monotone variational inequality and the fixed point set of a demicontractive mapping. The advantage of our algorithm is that it does not require prior information regarding the Lipschitz constant of the variational inequality operator and that it only computes one projection onto the feasible set per iteration. In addition, we do not need the sequential weak continuity of the variational inequality operator in order to establish our strong convergence theorem. Next, we also obtain an R-linear convergence rate for a related relaxed inertial gradient method under strong pseudo-monotonicity and Lipschitz continuity assumptions on the variational inequality operator. Finally, we present several numerical examples which illustrate the performance and the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]