Inertial projection methods for finding a minimum-norm solution of pseudomonotone variational inequality and fixed-point problems

In this work, we introduce two new iterative methods for finding the common element of the set of fixed points of a demicontractive mapping and the set of solutions of a pseudomonotone variational inequality problem in real Hilbert spaces. It is shown that the proposed algorithms converge strongly under mild conditions. The advantage of the proposed algorithms is that it does not require prior knowledge of the Lipschitz-type constants of the variational inequality mapping and only requires computing one projection onto a feasible set per iteration as well as without the sequentially weakly continuity of the variational inequality mapping. The novelty of the proposed algorithm may improve the efficiency of algorithms. Our results improve and extend some known results existing in the literature.