Fast and Simple Bregman Projection Methods for Solving Variational Inequalities and Related Problems in Banach Spaces

In this paper, we study the problem of finding a common solution to variational inequality and fixed point problems for a countable family of Bregman weak relatively nonexpansive mappings in real reflexive Banach spaces. Two inertial-type algorithms with adaptive step size rules for solving the problem are presented and their strong convergence theorems are established. The usage of the Bregman distances and the Armijo line search technique (which avoids the need to know a priori the Lipschitz constant of the involved operators), enable great flexibility of the proposed scheme, and besides their theoretical extensions, it might also have a practical potential.